1.1 What Is Data Mining?

Data mining is the process of automatically discovering useful information in

large data repositories. Data mining techniques are deployed to scour large

data sets in order to find novel and useful patterns that might otherwise

remain unknown. They also provide the capability to predict the outcome of a

future observation, such as the amount a customer will spend at an online or a

brick-and-mortar store.

Not all information discovery tasks are considered to be data mining.

Examples include queries, e.g., looking up individual records in a database or

finding web pages that contain a particular set of keywords. This is because

such tasks can be accomplished through simple interactions with a database

management system or an information retrieval system. These systems rely

on traditional computer science techniques, which include sophisticated

indexing structures and query processing algorithms, for efficiently

organizing and retrieving information from large data repositories.

Nonetheless, data mining techniques have been used to enhance the

performance of such systems by improving the quality of the search results

based on their relevance to the input queries.

Data Mining and Knowledge

Discovery in Databases

Data mining is an integral part of knowledge discovery in databases

(KDD), which is the overall process of converting raw data into useful

information, as shown in Figure 1.1. This process consists of a series of steps,

from data preprocessing to postprocessing of data mining results.

Figure 1.1.

The process of knowledge discovery in databases (KDD).

Figure 1.1. Full Alternative Text

The input data can be stored in a variety of formats (flat files, spreadsheets, or

relational tables) and may reside in a centralized data repository or be

distributed across multiple sites. The purpose of preprocessing is to

transform the raw input data into an appropriate format for subsequent

analysis. The steps involved in data preprocessing include fusing data from

multiple sources, cleaning data to remove noise and duplicate observations,

and selecting records and features that are relevant to the data mining task at

hand. Because of the many ways data can be collected and stored, data

preprocessing is perhaps the most laborious and time-consuming step in the

overall knowledge discovery process.

“Closing the loop” is a phrase often used to refer to the process of integrating

data mining results into decision support systems. For example, in business

applications, the insights offered by data mining results can be integrated

with campaign management tools so that effective marketing promotions can

be conducted and tested. Such integration requires a postprocessing step to

ensure that only valid and useful results are incorporated into the decision

support system. An example of postprocessing is visualization, which allows

analysts to explore the data and the data mining results from a variety of

viewpoints. Hypothesis testing methods can also be applied during

postprocessing to eliminate spurious data mining results. (See Chapter 10.)

1.2 Motivating Challenges

As mentioned earlier, traditional data analysis techniques have often

encountered practical difficulties in meeting the challenges posed by big data

applications. The following are some of the specific challenges that

motivated the development of data mining.

Scalability

Because of advances in data generation and collection, data sets with sizes of

terabytes, petabytes, or even exabytes are becoming common. If data mining

algorithms are to handle these massive data sets, they must be scalable. Many

data mining algorithms employ special search strategies to handle

exponential search problems. Scalability may also require the implementation

of novel data structures to access individual records in an efficient manner.

For instance, out-of-core algorithms may be necessary when processing data

sets that cannot fit into main memory. Scalability can also be improved by

using sampling or developing parallel and distributed algorithms. A general

overview of techniques for scaling up data mining algorithms is given in

Appendix F.

High Dimensionality

It is now common to encounter data sets with hundreds or thousands of

attributes instead of the handful common a few decades ago. In

bioinformatics, progress in microarray technology has produced gene

expression data involving thousands of features. Data sets with temporal or

spatial components also tend to have high dimensionality. For example,

consider a data set that contains measurements of temperature at various

locations. If the temperature measurements are taken repeatedly for an

extended period, the number of dimensions (features) increases in proportion

to the number of measurements taken. Traditional data analysis techniques

that were developed for low-dimensional data often do not work well for

such high-dimensional data due to issues such as curse of dimensionality (to

be discussed in Chapter 2). Also, for some data analysis algorithms, the

computational complexity increases rapidly as the dimensionality (the

number of features) increases.

Heterogeneous and Complex Data

Traditional data analysis methods often deal with data sets containing

attributes of the same type, either continuous or categorical. As the role of

data mining in business, science, medicine, and other fields has grown, so has

the need for techniques that can handle heterogeneous attributes. Recent

years have also seen the emergence of more complex data objects. Examples

of such non-traditional types of data include web and social media data

containing text, hyperlinks, images, audio, and videos; DNA data with

sequential and three-dimensional structure; and climate data that consists of

measurements (temperature, pressure, etc.) at various times and locations on

the Earth’s surface. Techniques developed for mining such complex objects

should take into consideration relationships in the data, such as temporal and

spatial autocorrelation, graph connectivity, and parent-child relationships

between the elements in semi-structured text and XML documents.

Data Ownership and Distribution

Sometimes, the data needed for an analysis is not stored in one location or

owned by one organization. Instead, the data is geographically distributed

among resources belonging to multiple entities. This requires the

development of distributed data mining techniques. The key challenges faced

by distributed data mining algorithms include the following: (1) how to

reduce the amount of communication needed to perform the distributed

computation, (2) how to effectively consolidate the data mining results

obtained from multiple sources, and (3) how to address data security and

privacy issues.

Non-traditional Analysis

The traditional statistical approach is based on a hypothesize-and-test

paradigm. In other words, a hypothesis is proposed, an experiment is

designed to gather the data, and then the data is analyzed with respect to the

hypothesis. Unfortunately, this process is extremely labor-intensive. Current

data analysis tasks often require the generation and evaluation of thousands

of hypotheses, and consequently, the development of some data mining

techniques has been motivated by the desire to automate the process of

hypothesis generation and evaluation. Furthermore, the data sets analyzed in

data mining are typically not the result of a carefully designed experiment

and often represent opportunistic samples of the data, rather than random

samples.

1.3 The Origins of Data Mining

While data mining has traditionally been viewed as an intermediate process

within the KDD framework, as shown in Figure 1.1, it has emerged over the

years as an academic field within computer science, focusing on all aspects of

KDD, including data preprocessing, mining, and postprocessing. Its origin

can be traced back to the late 1980s, following a series of workshops

organized on the topic of knowledge discovery in databases. The workshops

brought together researchers from different disciplines to discuss the

challenges and opportunities in applying computational techniques to extract

actionable knowledge from large databases. The workshops quickly grew

into hugely popular conferences that were attended by researchers and

practitioners from both the academia and industry. The success of these

conferences, along with the interest shown by businesses and industry in

recruiting new hires with data mining background, have fueled the

tremendous growth of this field.

The field was initially built upon the methodology and algorithms that

researchers had previously used. In particular, data mining researchers draw

upon ideas, such as (1) sampling, estimation, and hypothesis testing from

statistics and (2) search algorithms, modeling techniques, and learning

theories from artificial intelligence, pattern recognition, and machine

learning. Data mining has also been quick to adopt ideas from other areas,

including optimization, evolutionary computing, information theory, signal

processing, visualization, and information retrieval, and extending them to

solve the challenges of mining big data.

A number of other areas also play key supporting roles. In particular,

database systems are needed to provide support for efficient storage,

indexing, and query processing. Techniques from high performance (parallel)

computing are often important in addressing the massive size of some data

sets. Distributed techniques can also help address the issue of size and are

essential when the data cannot be gathered in one location. Figure 1.2 shows

the relationship of data mining to other areas.

Figure 1.2.

Data mining as a confluence of many disciplines.

Data Science and Data-Driven

Discovery

Data science is an interdisciplinary field that studies and applies tools and

techniques for deriving useful insights from data. Although data science is

regarded as an emerging field with a distinct identity of its own, the tools and

techniques often come from many different areas of data analysis, such as

data mining, statistics, AI, machine learning, pattern recognition, database

technology, and distributed and parallel computing. (See Figure 1.2.)

The emergence of data science as a new field is a recognition that, often,

none of the existing areas of data analysis provides a complete set of tools for

the data analysis tasks that are often encountered in emerging applications.

Instead, a broad range of computational, mathematical, and statistical skills is

often required. To illustrate the challenges that arise in analyzing such data,

consider the following example. Social media and the Web present new

opportunities for social scientists to observe and quantitatively measure

human behavior on a large scale. To conduct such a study, social scientists

work with analysts who possess skills in areas such as web mining, natural

language processing (NLP), network analysis, data mining, and statistics.

Compared to more traditional research in social science, which is often based

on surveys, this analysis requires a broader range of skills and tools, and

involves far larger amounts of data. Thus, data science is, by necessity, a

highly interdisciplinary field that builds on the continuing work of many

fields.

The data-driven approach of data science emphasizes the direct discovery of

patterns and relationships from data, especially in large quantities of data,

often without the need for extensive domain knowledge. A notable example

of the success of this approach is represented by advances in neural networks,

i.e., deep learning, which have been particularly successful in areas which

have long proved challenging, e.g., recognizing objects in photos or videos

and words in speech, as well as in other application areas. However, note that

this is just one example of the success of data-driven approaches, and

dramatic improvements have also occurred in many other areas of data

analysis. Many of these developments are topics described later in this book.

Some cautions on potential limitations of a purely data-driven approach are

given in the Bibliographic Notes.

1.4 Data Mining Tasks

Data mining tasks are generally divided into two major categories:

Predictive tasks The objective of these tasks is to predict the value of a

particular attribute based on the values of other attributes. The attribute to be

predicted is commonly known as the target or dependent variable, while

the attributes used for making the prediction are known as the explanatory or

independent variables.

Descriptive tasks Here, the objective is to derive patterns (correlations,

trends, clusters, trajectories, and anomalies) that summarize the underlying

relationships in data. Descriptive data mining tasks are often exploratory in

nature and frequently require postprocessing techniques to validate and

explain the results.

Figure 1.3 illustrates four of the core data mining tasks that are described in

the remainder of this book.

Figure 1.3.

Four of the core data mining tasks.

Figure 1.3. Full Alternative Text

Predictive modeling refers to the task of building a model for the target

variable as a function of the explanatory variables. There are two types of

predictive modeling tasks: classification, which is used for discrete target

variables, and regression, which is used for continuous target variables. For

example, predicting whether a web user will make a purchase at an online

bookstore is a classification task because the target variable is binary-valued.

On the other hand, forecasting the future price of a stock is a regression task

because price is a continuous-valued attribute. The goal of both tasks is to

learn a model that minimizes the error between the predicted and true values

of the target variable. Predictive modeling can be used to identify customers

who will respond to a marketing campaign, predict disturbances in the

Earth’s ecosystem, or judge whether a patient has a particular disease based

on the results of medical tests.

Example 1.1 (Predicting the Type of

a Flower).

Consider the task of predicting a species of flower based on the

characteristics of the flower. In particular, consider classifying an Iris flower

as one of the following three Iris species: Setosa, Versicolour, or Virginica.

To perform this task, we need a data set containing the characteristics of

various flowers of these three species. A data set with this type of

information is the well-known Iris data set from the UCI Machine Learning

Repository at http://www.ics.uci.edu/~mlearn. In addition to the species of a

flower, this data set contains four other attributes: sepal width, sepal length,

petal length, and petal width. Figure 1.4 shows a plot of petal width versus

petal length for the 150 flowers in the Iris data set. Petal width is broken into

the categories *low*, *medium*, and *high*, which correspond to the intervals [0,

0.75), [0.75, 1.75), [1.75, ∞), respectively. Also, petal length is broken into

categories *low*, *medium*,and *high*, which correspond to the intervals [0, 2.5),

[2.5, 5), [5, ∞), respectively. Based on these categories of petal width and

length, the following rules can be derived:

Petal width low and petal length low implies Setosa.

Petal width medium and petal length medium implies Versicolour.

Petal width high and petal length high implies Virginica.

While these rules do not classify all the flowers, they do a good (but not

perfect) job of classifying most of the flowers. Note that flowers from the

Setosa species are well separated from the Versicolour and Virginica species

with respect to petal width and length, but the latter two species overlap

somewhat with respect to these attributes.

Figure 1.4.

Petal width versus petal length for 150 Iris flowers.

Figure 1.4. Full Alternative Text

Association analysis is used to discover patterns that describe strongly

associated features in the data. The discovered patterns are typically

represented in the form of implication rules or feature subsets. Because of the

exponential size of its search space, the goal of association analysis is to

extract the most interesting patterns in an efficient manner. Useful

applications of association analysis include finding groups of genes that have

related functionality, identifying web pages that are accessed together, or

understanding the relationships between different elements of Earth’s climate

system.

Example 1.2 (Market Basket

Analysis).

The transactions shown in Table 1.1 illustrate point-of-sale data collected at

the checkout counters of a grocery store. Association analysis can be applied

to find items that are frequently bought together by customers. For example,

we may discover the rule {Diapers}→{Milk}, which suggests that customers

who buy diapers also tend to buy milk. This type of rule can be used to

identify potential cross-selling opportunities among related items.

Table 1.1. Market basket data.

Transaction ID Items

1 {Bread, Butter, Diapers, Milk}

2 {Coffee, Sugar, Cookies, Salmon}

3 {Bread, Butter, Coffee, Diapers, Milk, Eggs}

4 {Bread, Butter, Salmon, Chicken}

5 {Eggs, Bread, Butter}

6 {Salmon, Diapers, Milk}

7 {Bread, Tea, Sugar, Eggs}

8 {Coffee, Sugar, Chicken, Eggs}

9 {Bread, Diapers, Milk, Salt}

10 {Tea, Eggs, Cookies, Diapers, Milk}

Cluster analysis seeks to find groups of closely related observations so that

observations that belong to the same cluster are more similar to each other

than observations that belong to other clusters. Clustering has been used to

group sets of related customers, find areas of the ocean that have a significant

impact on the Earth’s climate, and compress data.

Example 1.3 (Document Clustering).

The collection of news articles shown in Table 1.2 can be grouped based on

their respective topics. Each article is represented as a set of word-frequency

pairs (*w *: *c*), where *w *is a word and *c *is the number of times the word appears

in the article. There are two natural clusters in the data set. The first cluster

consists of the first four articles, which correspond to news about the

economy, while the second cluster contains the last four articles, which

correspond to news about health care. A good clustering algorithm should be

able to identify these two clusters based on the similarity between words that

appear in the articles.

Table 1.2. Collection of news

articles.

Article Word-frequency pairs

1 dollar: 1, industry: 4, country: 2, loan: 3, deal: 2,

government: 2

2 machinery: 2, labor: 3, market: 4, industry: 2, work: 3,

country: 1

3 job: 5, inflation: 3, rise: 2, jobless: 2, market: 3, country: 2,

index: 3

4 domestic: 3, forecast: 2, gain: 1, market: 2, sale: 3, price: 2

5 patient: 4, symptom: 2, drug: 3, health: 2, clinic: 2, doctor:

2

6 pharmaceutical: 2, company: 3, drug: 2, vaccine: 1, flu: 3

7 death: 2, cancer: 4, drug: 3, public: 4, health: 3, director: 2

8

medical: 2, cost: 3, increase: 2, patient: 2, health: 3, care: 1

Anomaly detection is the task of identifying observations whose

characteristics are significantly different from the rest of the data. Such

observations are known as anomalies or outliers. The goal of an anomaly

detection algorithm is to discover the real anomalies and avoid falsely

labeling normal objects as anomalous. In other words, a good anomaly

detector must have a high detection rate and a low false alarm rate.

Applications of anomaly detection include the detection of fraud, network

intrusions, unusual patterns of disease, and ecosystem disturbances, such as

droughts, floods, fires, hurricanes, etc.

Example 1.4 (Credit Card Fraud

Detection).

A credit card company records the transactions made by every credit card

holder, along with personal information such as credit limit, age, annual

income, and address. Since the number of fraudulent cases is relatively small

compared to the number of legitimate transactions, anomaly detection

techniques can be applied to build a profile of legitimate transactions for the

users. When a new transaction arrives, it is compared against the profile of

the user. If the characteristics of the transaction are very different from the

previously created profile, then the transaction is flagged as potentially

fraudulent.

1.5 Scope and Organization of the

Book

This book introduces the major principles and techniques used in data mining

from an algorithmic perspective. A study of these principles and techniques is

essential for developing a better understanding of how data mining

technology can be applied to various kinds of data. This book also serves as a

starting point for readers who are interested in doing research in this field.

We begin the technical discussion of this book with a chapter on data

(Chapter 2), which discusses the basic types of data, data quality,

preprocessing techniques, and measures of similarity and dissimilarity.

Although this material can be covered quickly, it provides an essential

foundation for data analysis. Chapters 3 and 4 cover classification. Chapter 3

provides a foundation by discussing decision tree classifiers and several

issues that are important to all classification: overfitting, underfitting, model

selection, and performance evaluation. Using this foundation, Chapter 4

describes a number of other important classification techniques: rule-based

systems, nearest neighbor classifiers, Bayesian classifiers, artificial neural

networks, including deep learning, support vector machines, and ensemble

classifiers, which are collections of classifiers. The multiclass and

imbalanced class problems are also discussed. These topics can be covered

independently.

Association analysis is explored in Chapters 5 and 6. Chapter 5 describes the

basics of association analysis: frequent itemsets, association rules, and some

of the algorithms used to generate them. Specific types of frequent itemsets—

maximal, closed, and hyperclique—that are important for data mining are

also discussed, and the chapter concludes with a discussion of evaluation

measures for association analysis. Chapter 6 considers a variety of more

advanced topics, including how association analysis can be applied to

categorical and continuous data or to data that has a concept hierarchy. (A

concept hierarchy is a hierarchical categorization of objects, e.g., store

itemsstore items→clothing→shoes→sneakers.) This chapter also describes

how association analysis can be extended to find sequential patterns (patterns

involving order), patterns in graphs, and negative relationships (if one item is

present, then the other is not).

Cluster analysis is discussed in Chapters 7 and 8. Chapter 7 first describes the

different types of clusters, and then presents three specific clustering

techniques: K-means, agglomerative hierarchical clustering, and DBSCAN.

This is followed by a discussion of techniques for validating the results of a

clustering algorithm. Additional clustering concepts and techniques are

explored in Chapter 8, including fuzzy and probabilistic clustering, SelfOrganizing Maps (SOM), graph-based clustering, spectral clustering, and

density-based clustering. There is also a discussion of scalability issues and

factors to consider when selecting a clustering algorithm.

Chapter 9, is on anomaly detection. After some basic definitions, several

different types of anomaly detection are considered: statistical, distancebased, density-based, clustering-based, reconstruction-based, one-class

classification, and information theoretic. The last chapter, Chapter 10,

supplements the discussions in the other Chapters with a discussion of the

statistical concepts important for avoiding spurious results, and then

discusses those concepts in the context of data mining techniques studied in

the previous chapters. These techniques include statistical hypothesis testing,

p-values, the false discovery rate, and permutation testing. Appendices A

through F give a brief review of important topics that are used in portions of

the book: linear algebra, dimensionality reduction, statistics, regression,

optimization, and scaling up data mining techniques for big data.

The subject of data mining, while relatively young compared to statistics or

machine learning, is already too large to cover in a single book. Selected

references to topics that are only briefly covered, such as data quality, are

provided in the Bibliographic Notes section of the appropriate chapter.

References to topics not covered in this book, such as mining streaming data

and privacy-preserving data mining are provided in the Bibliographic Notes

of this chapter.

1.6 Bibliographic Notes

The topic of data mining has inspired many textbooks. Introductory textbooks

include those by Dunham [16], Han et al. [29], Hand et al. [31], Roiger and

Geatz [50], Zaki and Meira [61], and Aggarwal [2]. Data mining books with

a stronger emphasis on business applications include the works by Berry and

Linoff [5], Pyle [47], and Parr Rud [45]. Books with an emphasis on

statistical learning include those by Cherkassky and Mulier [11], and Hastie

et al. [32]. Similar books with an emphasis on machine learning or pattern

recognition are those by Duda et al. [15], Kantardzic [34], Mitchell [43],

Webb [57], and Witten and Frank [58]. There are also some more specialized

books: Chakrabarti [9] (web mining), Fayyad et al. [20] (collection of early

articles on data mining), Fayyad et al. [18] (visualization), Grossman et al.

[25] (science and engineering), Kargupta and Chan [35] (distributed data

mining), Wang et al. [56] (bioinformatics), and Zaki and Ho [60] (parallel

data mining).

There are several conferences related to data mining. Some of the main

conferences dedicated to this field include the ACM SIGKDD International

Conference on Knowledge Discovery and Data Mining (KDD), the IEEE

International Conference on Data Mining (ICDM), the SIAM International

Conference on Data Mining (SDM), the European Conference on Principles

and Practice of Knowledge Discovery in Databases (PKDD), and the PacificAsia Conference on Knowledge Discovery and Data Mining (PAKDD). Data

mining papers can also be found in other major conferences such as the

Conference and Workshop on Neural Information Processing Systems

(NIPS),the International Conference on Machine Learning (ICML), the ACM

SIGMOD/PODS conference, the International Conference on Very Large

Data Bases (VLDB), the Conference on Information and Knowledge

Management (CIKM), the International Conference on Data Engineering

(ICDE), the National Conference on Artificial Intelligence (AAAI), the IEEE

International Conference on Big Data, and the IEEE International Conference

on Data Science and Advanced Analytics (DSAA).

Journal publications on data mining include IEEE Transactions on

Knowledge and Data Engineering, Data Mining and Knowledge Discovery,

Knowledge and Information Systems, ACM Transactions on Knowledge

Discovery from Data, Statistical Analysis and Data Mining, and Information

Systems. There are various open-source data mining software available,

including Weka [27] and Scikit-learn [46]. More recently, data mining

software such as Apache Mahout and Apache Spark have been developed for

large-scale problems on the distributed computing platform.

There have been a number of general articles on data mining that define the

field or its relationship to other fields, particularly statistics. Fayyad et al.

[19] describe data mining and how it fits into the total knowledge discovery

process. Chen et al. [10] give a database perspective on data mining.

Ramakrishnan and Grama [48] provide a general discussion of data mining

and present several viewpoints. Hand [30] describes how data mining differs

from statistics, as does Friedman [21]. Lambert [40] explores the use of

statistics for large data sets and provides some comments on the respective

roles of data mining and statistics. Glymour et al. [23] consider the lessons

that statistics may have for data mining. Smyth et al. [53] describe how the

evolution of data mining is being driven by new types of data and

applications, such as those involving streams, graphs, and text. Han et al. [28]

consider emerging applications in data mining and Smyth [52] describes

some research challenges in data mining. Wu et al. [59] discuss how

developments in data mining research can be turned into practical tools. Data

mining standards are the subject of a paper by Grossman et al. [24]. Bradley

[7] discusses how data mining algorithms can be scaled to large data sets.

The emergence of new data mining applications has produced new challenges

that need to be addressed. For instance, concerns about privacy breaches as a

result of data mining have escalated in recent years, particularly in

application domains such as web commerce and health care. As a result, there

is growing interest in developing data mining algorithms that maintain user

privacy. Developing techniques for mining encrypted or randomized data is

known as privacy-preserving data mining. Some general references in this

area include papers by Agrawal and Srikant [3], Clifton et al. [12] and

Kargupta et al. [36]. Vassilios et al. [55] provide a survey. Another area of

concern is the bias in predictive models that may be used for some

applications, e.g., screening job applicants or deciding prison parole [39].

Assessing whether such applications are producing biased results is made

more difficult by the fact that the predictive models used for such

applications are often black box models, i.e., models that are not interpretable

in any straightforward way.

Data science, its constituent fields, and more generally, the new paradigm of

knowledge discovery they represent [33], have great potential, some of which

has been realized. However, it is important to emphasize that data science

works mostly with observational data, i.e., data that was collected by various

organizations as part of their normal operation. The consequence of this is

that sampling biases are common and the determination of causal factors

becomes more problematic. For this and a number of other reasons, it is often

hard to interpret the predictive models built from this data [42, 49]. Thus,

theory, experimentation and computational simulations will continue to be

the methods of choice in many areas, especially those related to science.

More importantly, a purely data-driven approach often ignores the existing

knowledge in a particular field. Such models may perform poorly, for

example, predicting impossible outcomes or failing to generalize to new

situations. However, if the model does work well, e.g., has high predictive

accuracy, then this approach may be sufficient for practical purposes in some

fields. But in many areas, such as medicine and science, gaining insight into

the underlying domain is often the goal. Some recent work attempts to

address these issues in order to create theory-guided data science, which

takes pre-existing domain knowledge into account [17, 37].

Recent years have witnessed a growing number of applications that rapidly

generate continuous streams of data. Examples of stream data include

network traffic, multimedia streams, and stock prices. Several issues must be

considered when mining data streams, such as the limited amount of memory

available, the need for online analysis, and the change of the data over time.

Data mining for stream data has become an important area in data mining.

Some selected publications are Domingos and Hulten [14] (classification),

Giannella et al. [22] (association analysis), Guha et al. [26] (clustering), Kifer

et al. [38] (change detection), Papadimitriou et al. [44] (time series), and Law

et al. [41] (dimensionality reduction).

Another area of interest is recommender and collaborative filtering systems

[1, 6, 8, 13, 54], which suggest movies, television shows, books, products,

etc. that a person might like. In many cases, this problem, or at least a

component of it, is treated as a prediction problem and thus, data mining

techniques can be applied [4, 51].

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1.7 Exercises

1. 1. Discuss whether or not each of the following activities is a data

mining task.

1. Dividing the customers of a company according to their gender.

2. Dividing the customers of a company according to their

profitability.

3. Computing the total sales of a company.

4. Sorting a student database based on student identification numbers.

5. Predicting the outcomes of tossing a (fair) pair of dice.

6. Predicting the future stock price of a company using historical

records.

7. Monitoring the heart rate of a patient for abnormalities.

8. Monitoring seismic waves for earthquake activities.

9. Extracting the frequencies of a sound wave.

2. 2. Suppose that you are employed as a data mining consultant for an

Internet search engine company. Describe how data mining can help the

company by giving specific examples of how techniques, such as

clustering, classification, association rule mining, and anomaly detection

can be applied.

3. 3. For each of the following data sets, explain whether or not data

privacy is an important issue.

1. Census data collected from 1900–1950.

2. IP addresses and visit times of web users who visit your website.

3. Images from Earth-orbiting satellites.

4. Names and addresses of people from the telephone book.

5. Names and email addresses collected from the Web.

2 Data

This chapter discusses several data-related issues that are important for

successful data mining:

The Type of Data Data sets differ in a number of ways. For example, the

attributes used to describe data objects can be of different types—quantitative

or qualitative—and data sets often have special characteristics; e.g., some

data sets contain time series or objects with explicit relationships to one

another. Not surprisingly, the type of data determines which tools and

techniques can be used to analyze the data. Indeed, new research in data

mining is often driven by the need to accommodate new application areas and

their new types of data.

The Quality of the Data Data is often far from perfect. While most data

mining techniques can tolerate some level of imperfection in the data, a focus

on understanding and improving data quality typically improves the quality

of the resulting analysis. Data quality issues that often need to be addressed

include the presence of noise and outliers; missing, inconsistent, or duplicate

data; and data that is biased or, in some other way, unrepresentative of the

phenomenon or population that the data is supposed to describe.

Preprocessing Steps to Make the Data More Suitable for Data Mining Often,

the raw data must be processed in order to make it suitable for analysis.

While one objective may be to improve data quality, other goals focus on

modifying the data so that it better fits a specified data mining technique or

tool. For example, a continuous attribute, e.g., length, sometimes needs to be

transformed into an attribute with discrete categories, e.g., *short*, *medium*, or

*long*, in order to apply a particular technique. As another example, the

number of attributes in a data set is often reduced because many techniques

are more effective when the data has a relatively small number of attributes.

Analyzing Data in Terms of Its Relationships One approach to data analysis

is to find relationships among the data objects and then perform the

remaining analysis using these relationships rather than the data objects

themselves. For instance, we can compute the similarity or distance between

pairs of objects and then perform the analysis—clustering, classification, or

anomaly detection—based on these similarities or distances. There are many

such similarity or distance measures, and the proper choice depends on the

type of data and the particular application.

Example 2.1 (An Illustration of

Data-Related Issues).

To further illustrate the importance of these issues, consider the following

hypothetical situation. You receive an email from a medical researcher

concerning a project that you are eager to work on.

Hi,

I’ve attached the data file that I mentioned in my previous email. Each

line contains the information for a single patient and consists of five

fields. We want to predict the last field using the other fields. I don’t

have time to provide any more information about the data since I’m

going out of town for a couple of days, but hopefully that won’t slow

you down too much. And if you don’t mind, could we meet when I get

back to discuss your preliminary results? I might invite a few other

members of my team.

Thanks and see you in a couple of days.

Despite some misgivings, you proceed to analyze the data. The first few rows

of the file are as follows:

012 232 33.5 0 10.7

020 121 16.9 2 210.1

027 165 24.0 0 427.6

⋮

A brief look at the data reveals nothing strange. You put your doubts aside

and start the analysis. There are only 1000 lines, a smaller data file than you

had hoped for, but two days later, you feel that you have made some

progress. You arrive for the meeting, and while waiting for others to arrive,

you strike up a conversation with a statistician who is working on the project.

When she learns that you have also been analyzing the data from the project,

she asks if you would mind giving her a brief overview of your results.

Statistician: So, you got the data for all the patients?

Data Miner: Yes. I haven’t had much time for analysis, but I do have a few

interesting results.

Statistician: Amazing. There were so many data issues with this set of

patients that I couldn’t do much.

Data Miner: Oh? I didn’t hear about any possible problems.

Statistician: Well, first there is field 5, the variable we want to predict. It’s

common knowledge among people who analyze this type of data that results

are better if you work with the log of the values, but I didn’t discover this

until later. Was it mentioned to you?

Data Miner: No.

Statistician: But surely you heard about what happened to field 4? It’s

supposed to be measured on a scale from 1 to 10, with 0 indicating a missing

value, but because of a data entry error, all 10’s were changed into 0’s.

Unfortunately, since some of the patients have missing values for this field,

it’s impossible to say whether a 0 in this field is a real 0 or a 10. Quite a few

of the records have that problem.

Data Miner: Interesting. Were there any other problems?

Statistician: Yes, fields 2 and 3 are basically the same, but I assume that you

probably noticed that.

Data Miner: Yes, but these fields were only weak predictors of field 5.

Statistician: Anyway, given all those problems, I’m surprised you were able

to accomplish anything.

Data Miner: True, but my results are really quite good. Field 1 is a very

strong predictor of field 5. I’m surprised that this wasn’t noticed before.

Statistician: What? Field 1 is just an identification number.

Data Miner: Nonetheless, my results speak for themselves.

Statistician: Oh, no! I just remembered. We assigned ID numbers after we

sorted the records based on field 5. There is a strong connection, but it’s

meaningless. Sorry.

Although this scenario represents an extreme situation, it emphasizes the

importance of “knowing your data.” To that end, this chapter will address

each of the four issues mentioned above, outlining some of the basic

challenges and standard approaches.

2.1 Types of Data

A data set can often be viewed as a collection of data objects. Other names

for a data object are *record*, *point*, *vector*, *pattern*, *event*, *case*, *sample*,

*instance*, *observation*, or *entity*. In turn, data objects are described by a

number of attributes that capture the characteristics of an object, such as the

mass of a physical object or the time at which an event occurred. Other

names for an attribute are *variable*, *characteristic*, *field*, *feature*, or

*dimension*.

Example 2.2 (Student Information).

Often, a data set is a file, in which the objects are records (or rows) in the file

and each field (or column) corresponds to an attribute. For example, Table

2.1 shows a data set that consists of student information. Each row

corresponds to a student and each column is an attribute that describes some

aspect of a student, such as grade point average (GPA) or identification

number (ID).

Table 2.1. A sample data set

containing student information.

Student ID Year Grade Point Average (GPA) …

⋮

1034262 Senior 3.24 …

1052663 Freshman 3.51 …

1082246 Sophomore 3.62 …

Although record-based data sets are common, either in flat files or relational

database systems, there are other important types of data sets and systems for

storing data. In Section 2.1.2, we will discuss some of the types of data sets

that are commonly encountered in data mining. However, we first consider

attributes.

2.1.1 Attributes and Measurement

In this section, we consider the types of attributes used to describe data

objects. We first define an attribute, then consider what we mean by the type

of an attribute, and finally describe the types of attributes that are commonly

encountered.

What Is an Attribute?

We start with a more detailed definition of an attribute.

Definition 2.1.

An attribute is a property or characteristic of an object that can vary, either

from one object to another or from one time to another.

For example, eye color varies from person to person, while the temperature

of an object varies over time. Note that eye color is a symbolic attribute with

a small number of possible values *{brown, black, blue, green, hazel, etc.} *,

while temperature is a numerical attribute with a potentially unlimited

number of values.

At the most basic level, attributes are not about numbers or symbols.

However, to discuss and more precisely analyze the characteristics of objects,

we assign numbers or symbols to them. To do this in a well-defined way, we

need a measurement scale.

Definition 2.2.

A measurement scale is a rule (function) that associates a numerical or

symbolic value with an attribute of an object.

Formally, the process of measurement is the application of a measurement

scale to associate a value with a particular attribute of a specific object. While

this may seem a bit abstract, we engage in the process of measurement all the

time. For instance, we step on a bathroom scale to determine our weight, we

classify someone as male or female, or we count the number of chairs in a

room to see if there will be enough to seat all the people coming to a meeting.

In all these cases, the “physical value” of an attribute of an object is mapped

to a numerical or symbolic value.

With this background, we can discuss the type of an attribute, a concept that

is important in determining if a particular data analysis technique is

consistent with a specific type of attribute.

The Type of an Attribute

It is common to refer to the type of an attribute as the type of a

measurement scale. It should be apparent from the previous discussion that

an attribute can be described using different measurement scales and that the

properties of an attribute need not be the same as the properties of the values

used to measure it. In other words, the values used to represent an attribute

can have properties that are not properties of the attribute itself, and vice

versa. This is illustrated with two examples.

Example 2.3 (Employee Age and ID

Number).

Two attributes that might be associated with an employee are *ID *and *age *(in

years). Both of these attributes can be represented as integers. However,

while it is reasonable to talk about the average age of an employee, it makes

no sense to talk about the average employee ID. Indeed, the only aspect of

employees that we want to capture with the ID attribute is that they are

distinct. Consequently, the only valid operation for employee IDs is to test

whether they are equal. There is no hint of this limitation, however, when

integers are used to represent the employee ID attribute. For the age attribute,

the properties of the integers used to represent age are very much the

properties of the attribute. Even so, the correspondence is not complete

because, for example, ages have a maximum, while integers do not.

Example 2.4 (Length of Line

Segments).

Consider Figure 2.1, which shows some objects—line segments—and how

the length attribute of these objects can be mapped to numbers in two

different ways. Each successive line segment, going from the top to the

bottom, is formed by appending the topmost line segment to itself. Thus, the

second line segment from the top is formed by appending the topmost line

segment to itself twice, the third line segment from the top is formed by

appending the topmost line segment to itself three times, and so forth. In a

very real (physical) sense, all the line segments are multiples of the first. This

fact is captured by the measurements on the right side of the figure, but not

by those on the left side. More specifically, the measurement scale on the left

side captures only the ordering of the length attribute, while the scale on the

right side captures both the ordering and additivity properties. Thus, an

attribute can be measured in a way that does not capture all the properties of

the attribute.

Figure 2.1.

The measurement of the length of line segments on two different

scales of measurement.

Figure 2.1. Full Alternative Text

Knowing the type of an attribute is important because it tells us which

properties of the measured values are consistent with the underlying

properties of the attribute, and therefore, it allows us to avoid foolish actions,

such as computing the average employee ID.

The Different Types of Attributes

A useful (and simple) way to specify the type of an attribute is to identify the

properties of numbers that correspond to underlying properties of the

attribute. For example, an attribute such as length has many of the properties

of numbers. It makes sense to compare and order objects by length, as well as

to talk about the differences and ratios of length. The following properties

(operations) of numbers are typically used to describe attributes.

1. Distinctness = and ≠

2. Order <, ≤, >, and ≥

3. Addition + and −

4. Multiplication × and */*

Given these properties, we can define four types of attributes: nominal ,

ordinal, interval , and ratio. Table 2.2 gives the definitions of these types,

along with information about the statistical operations that are valid for each

type. Each attribute type possesses all of the properties and operations of the

attribute types above it. Consequently, any property or operation that is valid

for nominal, ordinal, and interval attributes is also valid for ratio attributes. In

other words, the definition of the attribute types is cumulative. However, this

does not mean that the statistical operations appropriate for one attribute type

are appropriate for the attribute types above it.

Table 2.2. Different attribute

types.

Attribute Type Description Examples Operations

Nominal

The values of a

nominal attribute

are just different

names; i.e.,

nominal values

provide only

enough

information to

zip codes,

employee

ID

numbers,

eye color,

gender

mode,

entropy,

contingency

correlation,

χ2 test

Categorical

(Qualitative)

distinguish one

object from

another. (=, ≠)

Ordinal

The values of an

ordinal attribute

provide enough

information to

order objects.

(<, >)

hardness of

minerals,

*{good,*

*better,*

*best}*,

grades,

street

numbers

median,

percentiles,

rank

correlation,

run tests,

sign tests

Numeric

(Quantitative)

Interval

For interval

attributes, the

differences

between values

are meaningful,

i.e., a unit of

measurement

exists. (+, −)

calendar

dates,

temperature

in Celsius

or

Fahrenheit

mean,

standard

deviation,

Pearson’s

correlation,

*t *and *F *tests

Ratio

For ratio

variables, both

differences and

ratios are

meaningful. (×, /)

temperature

in Kelvin,

monetary

quantities,

counts, age,

mass,

length,

electrical

current

geometric

mean,

harmonic

mean,

percent

variation

Nominal and ordinal attributes are collectively referred to as categorical or

qualitative attributes. As the name suggests, qualitative attributes, such as

employee ID, lack most of the properties of numbers. Even if they are

represented by numbers, i.e., integers, they should be treated more like

symbols. The remaining two types of attributes, interval and ratio, are

collectively referred to as quantitative or numeric attributes. Quantitative

attributes are represented by numbers and have most of the properties of

numbers. Note that quantitative attributes can be integer-valued or

continuous.

The types of attributes can also be described in terms of transformations that

do not change the meaning of an attribute. Indeed, S. Smith Stevens, the

psychologist who originally defined the types of attributes shown in Table

2.2, defined them in terms of these permissible transformations. For

example, the meaning of a length attribute is unchanged if it is measured in

meters instead of feet.

The statistical operations that make sense for a particular type of attribute are

those that will yield the same results when the attribute is transformed by

using a transformation that preserves the attribute’s meaning. To illustrate,

the average length of a set of objects is different when measured in meters

rather than in feet, but both averages represent the same length. Table 2.3

shows the meaning-preserving transformations for the four attribute types of

Table 2.2.

Table 2.3. Transformations

that define attribute levels.

Attribute Type Transformation Comment

Categorical

(Qualitative)

Nominal

Any one-to-one mapping,

e.g., a permutation of

values

If all

employee ID

numbers are

reassigned, it

will not make

any

difference.

Ordinal

An order-preserving

change of values, i.e.,

new_value=f(old_value),

where *f *is a monotonic

An attribute

encompassing

the notion of

good, better,

best can be

represented

function. equally well

by the values

{1, 2, 3} or by

{0.5, 1, 10}.

Numeric

(Quantitative)

Interval new_value=a×old_value+b,

*a *and *b *constants.

The

Fahrenheit

and Celsius

temperature

scales differ in

the location of

their zero

value and the

size of a

degree (unit).

Ratio new_value=a×old_value

Length can be

measured in

meters or feet.

Example 2.5 (Temperature Scales).

Temperature provides a good illustration of some of the concepts that have

been described. First, temperature can be either an interval or a ratio attribute,

depending on its measurement scale. When measured on the Kelvin scale, a

temperature of 2*◦ *is, in a physically meaningful way, twice that of a

temperature of 1*◦*. This is not true when temperature is measured on either the

Celsius or Fahrenheit scales, because, physically, a temperature of 1*◦*

Fahrenheit (Celsius) is not much different than a temperature of 2*◦ *Fahrenheit

(Celsius). The problem is that the zero points of the Fahrenheit and Celsius

scales are, in a physical sense, arbitrary, and therefore, the ratio of two

Celsius or Fahrenheit temperatures is not physically meaningful.

Describing Attributes by the

Number of Values

An independent way of distinguishing between attributes is by the number of

values they can take.

Discrete A discrete attribute has a finite or countably infinite set of

values. Such attributes can be categorical, such as zip codes or ID

numbers, or numeric, such as counts. Discrete attributes are often

represented using integer variables. Binary attributes are a special case

of discrete attributes and assume only two values, e.g., true/false, yes/no,

male/female, or 0/1. Binary attributes are often represented as Boolean

variables, or as integer variables that only take the values 0 or 1.

Continuous A continuous attribute is one whose values are real numbers.

Examples include attributes such as temperature, height, or weight.

Continuous attributes are typically represented as floating-point

variables. Practically, real values can be measured and represented only

with limited precision.

In theory, any of the measurement scale types—nominal, ordinal, interval,

and ratio—could be combined with any of the types based on the number of

attribute values—binary, discrete, and continuous. However, some

combinations occur only infrequently or do not make much sense. For

instance, it is difficult to think of a realistic data set that contains a continuous

binary attribute. Typically, nominal and ordinal attributes are binary or

discrete, while interval and ratio attributes are continuous. However, count

attributes , which are discrete, are also ratio attributes.

Asymmetric Attributes

For asymmetric attributes, only presence—a non-zero attribute value—is

regarded as important. Consider a data set in which each object is a student

and each attribute records whether a student took a particular course at a

university. For a specific student, an attribute has a value of 1 if the student

took the course associated with that attribute and a value of 0 otherwise.

Because students take only a small fraction of all available courses, most of

the values in such a data set would be 0. Therefore, it is more meaningful and

more efficient to focus on the non-zero values. To illustrate, if students are

compared on the basis of the courses they don’t take, then most students

would seem very similar, at least if the number of courses is large. Binary

attributes where only non-zero values are important are called asymmetric

binary attributes. This type of attribute is particularly important for

association analysis, which is discussed in Chapter 5. It is also possible to

have discrete or continuous asymmetric features. For instance, if the number

of credits associated with each course is recorded, then the resulting data set

will consist of asymmetric discrete or continuous attributes.

General Comments on Levels of

Measurement

As described in the rest of this chapter, there are many diverse types of data.

The previous discussion of measurement scales, while useful, is not complete

and has some limitations. We provide the following comments and guidance.

Distinctness, order, and meaningful intervals and ratios are only four

properties of data—many others are possible. For instance, some data is

inherently cyclical, e.g., position on the surface of the Earth or time. As

another example, consider set valued attributes, where each attribute

value is a set of elements, e.g., the set of movies seen in the last year.

Define one set of elements (movies) to be greater (larger) than a second

set if the second set is a subset of the first. However, such a relationship

defines only a partial order that does not match any of the attribute types

just defined.

The numbers or symbols used to capture attribute values may not

capture all the properties of the attributes or may suggest properties that

are not there. An illustration of this for integers was presented in

Example 2.3 , i.e., averages of IDs and out of range ages.

Data is often transformed for the purpose of analysis—see Section 2.3.7.

This often changes the distribution of the observed variable to a

distribution that is easier to analyze, e.g., a Gaussian (normal)

distribution. Often, such transformations only preserve the order of the

original values, and other properties are lost. Nonetheless, if the desired

outcome is a statistical test of differences or a predictive model, such a

transformation is justified.

The final evaluation of any data analysis, including operations on

attributes, is whether the results make sense from a domain point of

view.

In summary, it can be challenging to determine which operations can be

performed on a particular attribute or a collection of attributes without

compromising the integrity of the analysis. Fortunately, established practice

often serves as a reliable guide. Occasionally, however, standard practices are

erroneous or have limitations.

2.1.2 Types of Data Sets

There are many types of data sets, and as the field of data mining develops

and matures, a greater variety of data sets become available for analysis. In

this section, we describe some of the most common types. For convenience,

we have grouped the types of data sets into three groups: record data, graphbased data, and ordered data. These categories do not cover all possibilities

and other groupings are certainly possible.

General Characteristics of Data Sets

Before providing details of specific kinds of data sets, we discuss three

characteristics that apply to many data sets and have a significant impact on

the data mining techniques that are used: dimensionality, distribution, and

resolution.

Dimensionality

The dimensionality of a data set is the number of attributes that the objects in

the data set possess. Analyzing data with a small number of dimensions tends

to be qualitatively different from analyzing moderate or high-dimensional

data. Indeed, the difficulties associated with the analysis of high-dimensional

data are sometimes referred to as the curse of dimensionality. Because of

this, an important motivation in preprocessing the data is dimensionality

reduction. These issues are discussed in more depth later in this chapter and

in Appendix B.

Distribution

The distribution of a data set is the frequency of occurrence of various values

or sets of values for the attributes comprising data objects. Equivalently, the

distribution of a data set can be considered as a description of the

concentration of objects in various regions of the data space. Statisticians

have enumerated many types of distributions, e.g., Gaussian (normal), and

described their properties. (See Appendix C.) Although statistical approaches

for describing distributions can yield powerful analysis techniques, many

data sets have distributions that are not well captured by standard statistical

distributions.

As a result, many data mining algorithms do not assume a particular

statistical distribution for the data they analyze. However, some general

aspects of distributions often have a strong impact. For example, suppose a

categorical attribute is used as a class variable, where one of the categories

occurs 95% of the time, while the other categories together occur only 5% of

the time. This skewness in the distribution can make classification difficult as

discussed in Section 4.11. (Skewness has other impacts on data analysis that

are not discussed here.)

A special case of skewed data is sparsity. For sparse binary, count or

continuous data, most attributes of an object have values of 0. In many cases,

fewer than 1% of the values are non-zero. In practical terms, sparsity is an

advantage because usually only the non-zero values need to be stored and

manipulated. This results in significant savings with respect to computation

time and storage. Indeed, some data mining algorithms, such as the

association rule mining algorithms described in Chapter 5, work well only for

sparse data. Finally, note that often the attributes in sparse data sets are

asymmetric attributes.

Resolution

It is frequently possible to obtain data at different levels of resolution, and

often the properties of the data are different at different resolutions. For

instance, the surface of the Earth seems very uneven at a resolution of a few

meters, but is relatively smooth at a resolution of tens of kilometers. The

patterns in the data also depend on the level of resolution. If the resolution is

too fine, a pattern may not be visible or may be buried in noise; if the

resolution is too coarse, the pattern can disappear. For example, variations in

atmospheric pressure on a scale of hours reflect the movement of storms and

other weather systems. On a scale of months, such phenomena are not

detectable.

Record Data

Much data mining work assumes that the data set is a collection of records

(data objects), each of which consists of a fixed set of data fields (attributes).

See Figure 2.2(a). For the most basic form of record data, there is no explicit

relationship among records or data fields, and every record (object) has the

same set of attributes. Record data is usually stored either in flat files or in

relational databases. Relational databases are certainly more than a collection

of records, but data mining often does not use any of the additional

information available in a relational database. Rather, the database serves as a

convenient place to find records. Different types of record data are described

below and are illustrated in Figure 2.2.

Figure 2.2.

Different variations of record data.

Figure 2.2. Full Alternative Text

Transaction or Market Basket Data

Transaction data is a special type of record data, where each record

(transaction) involves a set of items. Consider a grocery store. The set of

products purchased by a customer during one shopping trip constitutes a

transaction, while the individual products that were purchased are the items.

This type of data is called market basket data because the items in each

record are the products in a person’s “market basket.” Transaction data is a

collection of sets of items, but it can be viewed as a set of records whose

fields are asymmetric attributes. Most often, the attributes are binary,

indicating whether an item was purchased, but more generally, the attributes

can be discrete or continuous, such as the number of items purchased or the

amount spent on those items. Figure 2.2(b) shows a sample transaction data

set. Each row represents the purchases of a particular customer at a particular

time.

The Data Matrix

If all the data objects in a collection of data have the same fixed set of

numeric attributes, then the data objects can be thought of as points (vectors)

in a multidimensional space, where each dimension represents a distinct

attribute describing the object. A set of such data objects can be interpreted as

an *m *by *n *matrix, where there are *m *rows, one for each object, and *n*

columns, one for each attribute. (A representation that has data objects as

columns and attributes as rows is also fine.) This matrix is called a data

matrix or a pattern matrix. A data matrix is a variation of record data, but

because it consists of numeric attributes, standard matrix operation can be

applied to transform and manipulate the data. Therefore, the data matrix is the

standard data format for most statistical data. Figure 2.2(c) shows a sample

data matrix.

The Sparse Data Matrix

A sparse data matrix is a special case of a data matrix where the attributes are

of the same type and are asymmetric; i.e., only non-zero values are important.

Transaction data is an example of a sparse data matrix that has only 0–1

entries. Another common example is document data. In particular, if the order

of the terms (words) in a document is ignored—the “bag of words” approach

—then a document can be represented as a term vector, where each term is a

component (attribute) of the vector and the value of each component is the

number of times the corresponding term occurs in the document. This

representation of a collection of documents is often called a document-term

matrix. Figure 2.2(d) shows a sample document-term matrix. The documents

are the rows of this matrix, while the terms are the columns. In practice, only

the non-zero entries of sparse data matrices are stored.

Graph-Based Data

A graph can sometimes be a convenient and powerful representation for data.

We consider two specific cases: (1) the graph captures relationships among

data objects and (2) the data objects themselves are represented as graphs.

Data with Relationships among

Objects

The relationships among objects frequently convey important information. In

such cases, the data is often represented as a graph. In particular, the data

objects are mapped to nodes of the graph, while the relationships among

objects are captured by the links between objects and link properties, such as

direction and weight. Consider web pages on the World Wide Web, which

contain both text and links to other pages. In order to process search queries,

web search engines collect and process web pages to extract their contents. It

is well-known, however, that the links to and from each page provide a great

deal of information about the relevance of a web page to a query, and thus,

must also be taken into consideration. Figure 2.3(a) shows a set of linked web

pages. Another important example of such graph data are the social networks,

where data objects are people and the relationships among them are their

interactions via social media.

Data with Objects That Are Graphs

If objects have structure, that is, the objects contain subobjects that have

relationships, then such objects are frequently represented as graphs. For

example, the structure of chemical compounds can be represented by a graph,

where the nodes are atoms and the links between nodes are chemical bonds.

Figure 2.3(b) shows a ball-and-stick diagram of the chemical compound

benzene, which contains atoms of carbon (black) and hydrogen (gray). A

graph representation makes it possible to determine which substructures

occur frequently in a set of compounds and to ascertain whether the presence

of any of these substructures is associated with the presence or absence of

certain chemical properties, such as melting point or heat of formation.

Frequent graph mining, which is a branch of data mining that analyzes such

data, is considered in Section 6.5.

Figure 2.3.

Different variations of graph data.

Figure 2.3. Full Alternative Text

Ordered Data

For some types of data, the attributes have relationships that involve order in

time or space. Different types of ordered data are described next and are

shown in Figure 2.4.

Sequential Transaction Data

Sequential transaction data can be thought of as an extension of transaction

data, where each transaction has a time associated with it. Consider a retail

transaction data set that also stores the time at which the transaction took

place. This time information makes it possible to find patterns such as “candy

sales peak before Halloween.” A time can also be associated with each

attribute. For example, each record could be the purchase history of a

customer, with a listing of items purchased at different times. Using this

information, it is possible to find patterns such as “people who buy DVD

players tend to buy DVDs in the period immediately following the purchase.”

Figure 2.4(a) shows an example of sequential transaction data. There are five

different times—*t1*, *t2*, *t3*, *t4*, and *t5*; three different customers—C1, C2, and

C3; and five different items—A, B, C, D, and E. In the top table, each row

corresponds to the items purchased at a particular time by each customer. For

instance, at time *t3*, customer C2 purchased items A and D. In the bottom

table, the same information is displayed, but each row corresponds to a

particular customer. Each row contains information about each transaction

involving the customer, where a transaction is considered to be a set of items

and the time at which those items were purchased. For example, customer C3

bought items A and C at time *t2*.

Time Series Data

Time series data is a special type of ordered data where each record is a time

series , i.e., a series of measurements taken over time. For example, a

financial data set might contain objects that are time series of the daily prices

of various stocks. As another example, consider Figure 2.4(c), which shows a

time series of the average monthly temperature for Minneapolis during the

years 1982 to 1994. When working with temporal data, such as time series, it

is important to consider temporal autocorrelation; i.e., if two measurements

are close in time, then the values of those measurements are often very

similar.

Figure 2.4.

Different variations of ordered data.

Figure 2.4. Full Alternative Text

Sequence Data

Sequence data consists of a data set that is a sequence of individual entities,

such as a sequence of words or letters. It is quite similar to sequential data,

except that there are no time stamps; instead, there are positions in an ordered

sequence. For example, the genetic information of plants and animals can be

represented in the form of sequences of nucleotides that are known as genes.

Many of the problems associated with genetic sequence data involve

predicting similarities in the structure and function of genes from similarities

in nucleotide sequences. Figure 2.4(b) shows a section of the human genetic

code expressed using the four nucleotides from which all DNA is

constructed: A, T, G, and C.

Spatial and Spatio-Temporal Data

Some objects have spatial attributes, such as positions or areas, in addition to

other types of attributes. An example of spatial data is weather data

(precipitation, temperature, pressure) that is collected for a variety of

geographical locations. Often such measurements are collected over time, and

thus, the data consists of time series at various locations. In that case, we

refer to the data as spatio-temporal data. Although analysis can be conducted

separately for each specific time or location, a more complete analysis of

spatio-temporal data requires consideration of both the spatial and temporal

aspects of the data.

An important aspect of spatial data is spatial autocorrelation; i.e., objects

that are physically close tend to be similar in other ways as well. Thus, two

points on the Earth that are close to each other usually have similar values for

temperature and rainfall. Note that spatial autocorrelation is analogous to

temporal autocorrelation.

Important examples of spatial and spatio-temporal data are the science and

engineering data sets that are the result of measurements or model output

taken at regularly or irregularly distributed points on a two- or threedimensional grid or mesh. For instance, Earth science data sets record the

temperature or pressure measured at points (grid cells) on latitude–longitude

spherical grids of various resolutions, e.g., 1° by 1°. See Figure 2.4(d). As

another example, in the simulation of the flow of a gas, the speed and

direction of flow at various instants in time can be recorded for each grid

point in the simulation. A different type of spatio-temporal data arises from

tracking the trajectories of objects, e.g., vehicles, in time and space.

Handling Non-Record Data

Most data mining algorithms are designed for record data or its variations,

such as transaction data and data matrices. Record-oriented techniques can be

applied to non-record data by extracting features from data objects and using

these features to create a record corresponding to each object. Consider the

chemical structure data that was described earlier. Given a set of common

substructures, each compound can be represented as a record with binary

attributes that indicate whether a compound contains a specific substructure.

Such a representation is actually a transaction data set, where the transactions

are the compounds and the items are the substructures.

In some cases, it is easy to represent the data in a record format, but this type

of representation does not capture all the information in the data. Consider

spatio-temporal data consisting of a time series from each point on a spatial

grid. This data is often stored in a data matrix, where each row represents a

location and each column represents a particular point in time. However, such

a representation does not explicitly capture the time relationships that are

present among attributes and the spatial relationships that exist among

objects. This does not mean that such a representation is inappropriate, but

rather that these relationships must be taken into consideration during the

analysis. For example, it would not be a good idea to use a data mining

technique that ignores the temporal autocorrelation of the attributes or the

spatial autocorrelation of the data objects, i.e., the locations on the spatial

grid.

2.2 Data Quality

Data mining algorithms are often applied to data that was collected for

another purpose, or for future, but unspecified applications. For that reason,

data mining cannot usually take advantage of the significant benefits of “addressing quality issues at the source.” In contrast, much of statistics deals

with the design of experiments or surveys that achieve a prespecified level of

data quality. Because preventing data quality problems is typically not an

option, data mining focuses on (1) the detection and correction of data quality

problems and (2) the use of algorithms that can tolerate poor data quality.

The first step, detection and correction, is often called data cleaning.

The following sections discuss specific aspects of data quality. The focus is

on measurement and data collection issues, although some application-related

issues are also discussed.

2.2.1 Measurement and Data

Collection Issues

It is unrealistic to expect that data will be perfect. There may be problems due

to human error, limitations of measuring devices, or flaws in the data

collection process. Values or even entire data objects can be missing. In other

cases, there can be spurious or duplicate objects; i.e., multiple data objects

that all correspond to a single “real” object. For example, there might be two

different records for a person who has recently lived at two different

addresses. Even if all the data is present and “looks fine,” there may be

inconsistencies—a person has a height of 2 meters, but weighs only 2

kilograms.

In the next few sections, we focus on aspects of data quality that are related to

data measurement and collection. We begin with a definition of measurement

and data collection errors and then consider a variety of problems that

involve measurement error: noise, artifacts, bias, precision, and accuracy. We

conclude by discussing data quality issues that involve both measurement and

data collection problems: outliers, missing and inconsistent values, and

duplicate data.

Measurement and Data Collection

Errors

The term measurement error refers to any problem resulting from the

measurement process. A common problem is that the value recorded differs

from the true value to some extent. For continuous attributes, the numerical

difference of the measured and true value is called the error. The term data

collection error refers to errors such as omitting data objects or attribute

values, or inappropriately including a data object. For example, a study of

animals of a certain species might include animals of a related species that

are similar in appearance to the species of interest. Both measurement errors

and data collection errors can be either systematic or random.

We will only consider general types of errors. Within particular domains,

certain types of data errors are commonplace, and well-developed techniques

often exist for detecting and/or correcting these errors. For example,

keyboard errors are common when data is entered manually, and as a result,

many data entry programs have techniques for detecting and, with human

intervention, correcting such errors.

Noise and Artifacts

Noise is the random component of a measurement error. It typically involves

the distortion of a value or the addition of spurious objects. Figure 2.5 shows

a time series before and after it has been disrupted by random noise. If a bit

more noise were added to the time series, its shape would be lost. Figure 2.6

shows a set of data points before and after some noise points (indicated by

‘+’s) have been added. Notice that some of the noise points are intermixed

with the non-noise points.

Figure 2.5.

Noise in a time series context.

Figure 2.5. Full Alternative Text

Figure 2.6.

Noise in a spatial context.

Figure 2.6. Full Alternative Text

The term noise is often used in connection with data that has a spatial or

temporal component. In such cases, techniques from signal or image

processing can frequently be used to reduce noise and thus, help to discover

patterns (signals) that might be “lost in the noise.” Nonetheless, the

elimination of noise is frequently difficult, and much work in data mining

focuses on devising robust algorithms that produce acceptable results even

when noise is present.

Data errors can be the result of a more deterministic phenomenon, such as a

streak in the same place on a set of photographs. Such deterministic

distortions of the data are often referred to as artifacts.

Precision, Bias, and Accuracy

In statistics and experimental science, the quality of the measurement process

and the resulting data are measured by precision and bias. We provide the

standard definitions, followed by a brief discussion. For the following

definitions, we assume that we make repeated measurements of the same

underlying quantity.

Definition 2.3 (Precision).

The closeness of repeated measurements (of the same quantity) to one

another.

Definition 2.4 (Bias).

A systematic variation of measurements from the quantity being measured.

Precision is often measured by the standard deviation of a set of values, while

bias is measured by taking the difference between the mean of the set of

values and the known value of the quantity being measured. Bias can be

determined only for objects whose measured quantity is known by means

external to the current situation. Suppose that we have a standard laboratory

weight with a mass of 1g and want to assess the precision and bias of our new

laboratory scale. We weigh the mass five times, and obtain the following five

values:*{ *1*.*015*, *0*.*990*, *1*.*013*, *1*.*001*, *0*.*986*}*. The mean of these values is

1.001, and hence, the bias is 0.001. The precision, as measured by the

standard deviation, is 0.013.

It is common to use the more general term, accuracy , to refer to the degree

of measurement error in data.

Definition 2.5 (Accuracy)

The closeness of measurements to the true value of the quantity being

measured.

Accuracy depends on precision and bias, but there is no specific formula for

accuracy in terms of these two quantities.

One important aspect of accuracy is the use of significant digits. The goal is

to use only as many digits to represent the result of a measurement or

calculation as are justified by the precision of the data. For example, if the

length of an object is measured with a meter stick whose smallest markings

are millimeters, then we should record the length of data only to the nearest

millimeter. The precision of such a measurement would be ± 0.5mm. We do

not review the details of working with significant digits because most readers

will have encountered them in previous courses and they are covered in

considerable depth in science, engineering, and statistics textbooks.

Issues such as significant digits, precision, bias, and accuracy are sometimes

overlooked, but they are important for data mining as well as statistics and

science. Many times, data sets do not come with information about the

precision of the data, and furthermore, the programs used for analysis return

results without any such information. Nonetheless, without some

understanding of the accuracy of the data and the results, an analyst runs the

risk of committing serious data analysis blunders.

Outliers

Outliers are either (1) data objects that, in some sense, have characteristics

that are different from most of the other data objects in the data set, or (2)

values of an attribute that are unusual with respect to the typical values for

that attribute. Alternatively, they can be referred to as anomalous objects or

values. There is considerable leeway in the definition of an outlier, and many

different definitions have been proposed by the statistics and data mining

communities. Furthermore, it is important to distinguish between the notions

of noise and outliers. Unlike noise, outliers can be legitimate data objects or

values that we are interested in detecting. For instance, in fraud and network

intrusion detection, the goal is to find unusual objects or events from among a

large number of normal ones. Chapter 9 discusses anomaly detection in more

detail.

Missing Values

It is not unusual for an object to be missing one or more attribute values. In

some cases, the information was not collected; e.g., some people decline to

give their age or weight. In other cases, some attributes are not applicable to

all objects; e.g., often, forms have conditional parts that are filled out only

when a person answers a previous question in a certain way, but for

simplicity, all fields are stored. Regardless, missing values should be taken

into account during the data analysis.

There are several strategies (and variations on these strategies) for dealing

with missing data, each of which is appropriate in certain circumstances.

These strategies are listed next, along with an indication of their advantages

and disadvantages.

Eliminate Data Objects or

Attributes

A simple and effective strategy is to eliminate objects with missing values.

However, even a partially specified data object contains some information,

and if many objects have missing values, then a reliable analysis can be

difficult or impossible. Nonetheless, if a data set has only a few objects that

have missing values, then it may be expedient to omit them. A related

strategy is to eliminate attributes that have missing values. This should be

done with caution, however, because the eliminated attributes may be the

ones that are critical to the analysis.

Estimate Missing Values

Sometimes missing data can be reliably estimated. For example, consider a

time series that changes in a reasonably smooth fashion, but has a few,

widely scattered missing values. In such cases, the missing values can be

estimated (interpolated) by using the remaining values. As another example,

consider a data set that has many similar data points. In this situation, the

attribute values of the points closest to the point with the missing value are

often used to estimate the missing value. If the attribute is continuous, then

the average attribute value of the nearest neighbors is used; if the attribute is

categorical, then the most commonly occurring attribute value can be taken.

For a concrete illustration, consider precipitation measurements that are

recorded by ground stations. For areas not containing a ground station, the

precipitation can be estimated using values observed at nearby ground

stations.

Ignore the Missing Value during

Analysis

Many data mining approaches can be modified to ignore missing values. For

example, suppose that objects are being clustered and the similarity between

pairs of data objects needs to be calculated. If one or both objects of a pair

have missing values for some attributes, then the similarity can be calculated

by using only the attributes that do not have missing values. It is true that the

similarity will only be approximate, but unless the total number of attributes

is small or the number of missing values is high, this degree of inaccuracy

may not matter much. Likewise, many classification schemes can be

modified to work with missing values.

Inconsistent Values

Data can contain inconsistent values. Consider an address field, where both a

zip code and city are listed, but the specified zip code area is not contained in

that city. It is possible that the individual entering this information transposed

two digits, or perhaps a digit was misread when the information was scanned

from a handwritten form. Regardless of the cause of the inconsistent values,

it is important to detect and, if possible, correct such problems.

Some types of inconsistences are easy to detect. For instance, a person’s

height should not be negative. In other cases, it can be necessary to consult an

external source of information. For example, when an insurance company

processes claims for reimbursement, it checks the names and addresses on the

reimbursement forms against a database of its customers.

Once an inconsistency has been detected, it is sometimes possible to correct

the data. A product code may have “check” digits, or it may be possible to

double-check a product code against a list of known product codes, and then

correct the code if it is incorrect, but close to a known code. The correction of

an inconsistency requires additional or redundant information.

Example 2.6 (Inconsistent Sea

Surface Temperature).

This example illustrates an inconsistency in actual time series data that

measures the sea surface temperature (SST) at various points on the ocean.

SST data was originally collected using ocean-based measurements from

ships or buoys, but more recently, satellites have been used to gather the data.

To create a long-term data set, both sources of data must be used. However,

because the data comes from different sources, the two parts of the data are

subtly different. This discrepancy is visually displayed in Figure 2.7, which

shows the correlation of SST values between pairs of years. If a pair of years

has a positive correlation, then the location corresponding to the pair of years

is colored white; otherwise it is colored black. (Seasonal variations were

removed from the data since, otherwise, all the years would be highly

correlated.) There is a distinct change in behavior where the data has been put

together in 1983. Years within each of the two groups, 1958–1982 and 1983–

1999, tend to have a positive correlation with one another, but a negative

correlation with years in the other group. This does not mean that this data

should not be used, only that the analyst should consider the potential impact

of such discrepancies on the data mining analysis.

Figure 2.7.

Correlation of SST data between pairs of years. White areas

indicate positive correlation. Black areas indicate negative

correlation.

Figure 2.7. Full Alternative Text

Duplicate Data

A data set can include data objects that are duplicates, or almost duplicates,

of one another. Many people receive duplicate mailings because they appear

in a database multiple times under slightly different names. To detect and

eliminate such duplicates, two main issues must be addressed. First, if there

are two objects that actually represent a single object, then one or more

values of corresponding attributes are usually different, and these inconsistent

values must be resolved. Second, care needs to be taken to avoid accidentally

combining data objects that are similar, but not duplicates, such as two

distinct people with identical names. The term deduplication is often used to

refer to the process of dealing with these issues.

In some cases, two or more objects are identical with respect to the attributes

measured by the database, but they still represent different objects. Here, the

duplicates are legitimate, but can still cause problems for some algorithms if

the possibility of identical objects is not specifically accounted for in their

design. An example of this is given in Exercise 13 on page 108.

2.2.2 Issues Related to Applications

Data quality issues can also be considered from an application viewpoint as

expressed by the statement “data is of high quality if it is suitable for its

intended use.” This approach to data quality has proven quite useful,

particularly in business and industry. A similar viewpoint is also present in

statistics and the experimental sciences, with their emphasis on the careful

design of experiments to collect the data relevant to a specific hypothesis. As

with quality issues at the measurement and data collection level, many issues

are specific to particular applications and fields. Again, we consider only a

few of the general issues.

Timeliness

Some data starts to age as soon as it has been collected. In particular, if the

data provides a snapshot of some ongoing phenomenon or process, such as

the purchasing behavior of customers or web browsing patterns, then this

snapshot represents reality for only a limited time. If the data is out of date,

then so are the models and patterns that are based on it.

Relevance

The available data must contain the information necessary for the application.

Consider the task of building a model that predicts the accident rate for

drivers. If information about the age and gender of the driver is omitted, then

it is likely that the model will have limited accuracy unless this information is

indirectly available through other attributes.

Making sure that the objects in a data set are relevant is also challenging. A

common problem is sampling bias, which occurs when a sample does not

contain different types of objects in proportion to their actual occurrence in

the population. For example, survey data describes only those who respond to

the survey. (Other aspects of sampling are discussed further in Section 2.3.2.)

Because the results of a data analysis can reflect only the data that is present,

sampling bias will typically lead to erroneous results when applied to the

broader population.

Knowledge about the Data

Ideally, data sets are accompanied by documentation that describes different

aspects of the data; the quality of this documentation can either aid or hinder

the subsequent analysis. For example, if the documentation identifies several

attributes as being strongly related, these attributes are likely to provide

highly redundant information, and we usually decide to keep just one.

(Consider sales tax and purchase price.) If the documentation is poor,

however, and fails to tell us, for example, that the missing values for a

particular field are indicated with a -9999, then our analysis of the data may

be faulty. Other important characteristics are the precision of the data, the

type of features (nominal, ordinal, interval, ratio), the scale of measurement

(e.g., meters or feet for length), and the origin of the data.

2.3 Data Preprocessing

In this section, we consider which preprocessing steps should be applied to

make the data more suitable for data mining. Data preprocessing is a broad

area and consists of a number of different strategies and techniques that are

interrelated in complex ways. We will present some of the most important

ideas and approaches, and try to point out the interrelationships among them.

Specifically, we will discuss the following topics:

Aggregation

Sampling

Dimensionality reduction

Feature subset selection

Feature creation

Discretization and binarization

Variable transformation

Roughly speaking, these topics fall into two categories: selecting data objects

and attributes for the analysis or for creating/changing the attributes. In both

cases, the goal is to improve the data mining analysis with respect to time,

cost, and quality. Details are provided in the following sections.

A quick note about terminology: In the following, we sometimes use

synonyms for attribute, such as feature or variable, in order to follow

common usage.

2.3.1 Aggregation

Sometimes “less is more,” and this is the case with aggregation , the

combining of two or more objects into a single object. Consider a data set

consisting of transactions (data objects) recording the daily sales of products

in various store locations (Minneapolis, Chicago, Paris, *…*) for different days

over the course of a year. See Table 2.4. One way to aggregate transactions

for this data set is to replace all the transactions of a single store with a single

storewide transaction. This reduces the hundreds or thousands of transactions

that occur daily at a specific store to a single daily transaction, and the

number of data objects per day is reduced to the number of stores.

Table 2.4. Data set containing

information about customer

purchases.

Transaction ID Item Store Location Date Price …

⋮ ⋮ ⋮ ⋮ ⋮

101123 Watch Chicago 09/06/04 $25.99 …

101123 Battery Chicago 09/06/04 $5.99 …

101124 Shoes Minneapolis 09/06/04 $75.00 …

An obvious issue is how an aggregate transaction is created; i.e., how the

values of each attribute are combined across all the records corresponding to

a particular location to create the aggregate transaction that represents the

sales of a single store or date. Quantitative attributes, such as *price*, are

typically aggregated by taking a sum or an average. A qualitative attribute,

such as *item*, can either be omitted or summarized in terms of a higher level

category, e.g., televisions versus electronics.

The data in Table 2.4 can also be viewed as a multidimensional array, where

each attribute is a dimension. From this viewpoint, aggregation is the process

of eliminating attributes, such as the type of item, or reducing the number of

values for a particular attribute; e.g., reducing the possible values for *date*

from 365 days to 12 months. This type of aggregation is commonly used in

Online Analytical Processing (OLAP). References to OLAP are given in the

bibliographic Notes.

There are several motivations for aggregation. First, the smaller data sets

resulting from data reduction require less memory and processing time, and

hence, aggregation often enables the use of more expensive data mining

algorithms. Second, aggregation can act as a change of scope or scale by

providing a high-level view of the data instead of a low-level view. In the

previous example, aggregating over store locations and months gives us a

monthly, per store view of the data instead of a daily, per item view. Finally,

the behavior of groups of objects or attributes is often more stable than that of

individual objects or attributes. This statement reflects the statistical fact that

aggregate quantities, such as averages or totals, have less variability than the

individual values being aggregated. For totals, the actual amount of variation

is larger than that of individual objects (on average), but the percentage of the

variation is smaller, while for means, the actual amount of variation is less

than that of individual objects (on average). A disadvantage of aggregation is

the potential loss of interesting details. In the store example, aggregating over

months loses information about which day of the week has the highest sales.

Example 2.7 (Australian

Precipitation).

This example is based on precipitation in Australia from the period 1982–

1993. Figure 2.8(a) shows a histogram for the standard deviation of average

monthly precipitation for 3,030 0.5° by 0.5° grid cells in Australia, while

Figure 2.8(b) shows a histogram for the standard deviation of the average

yearly precipitation for the same locations. The average yearly precipitation

has less variability than the average monthly precipitation. All precipitation

measurements (and their standard deviations) are in centimeters.

Figure 2.8.

Histograms of standard deviation for monthly and yearly

precipitation in Australia for the period 1982–1993.

Figure 2.8. Full Alternative Text

2.3.2 Sampling

Sampling is a commonly used approach for selecting a subset of the data

objects to be analyzed. In statistics, it has long been used for both the

preliminary investigation of the data and the final data analysis. Sampling can

also be very useful in data mining. However, the motivations for sampling in

statistics and data mining are often different. Statisticians use sampling

because obtaining the entire set of data of interest is too expensive or time

consuming, while data miners usually sample because it is too

computationally expensive in terms of the memory or time required to

process all the data. In some cases, using a sampling algorithm can reduce the

data size to the point where a better, but more computationally expensive

algorithm can be used.

The key principle for effective sampling is the following: Using a sample will

work almost as well as using the entire data set if the sample is

representative. In turn, a sample is representative if it has approximately the

same property (of interest) as the original set of data. If the mean (average) of

the data objects is the property of interest, then a sample is representative if it

has a mean that is close to that of the original data. Because sampling is a

statistical process, the representativeness of any particular sample will vary,

and the best that we can do is choose a sampling scheme that guarantees a

high probability of getting a representative sample. As discussed next, this

involves choosing the appropriate sample size and sampling technique.

Sampling Approaches

There are many sampling techniques, but only a few of the most basic ones

and their variations will be covered here. The simplest type of sampling is

simple random sampling. For this type of sampling, there is an equal

probability of selecting any particular object. There are two variations on

random sampling (and other sampling techniques as well): (1) sampling

without replacement —as each object is selected, it is removed from the set

of all objects that together constitute the population , and (2) sampling with

replacement —objects are not removed from the population as they are

selected for the sample. In sampling with replacement, the same object can be

picked more than once. The samples produced by the two methods are not

much different when samples are relatively small compared to the data set

size, but sampling with replacement is simpler to analyze because the

probability of selecting any object remains constant during the sampling

process.

When the population consists of different types of objects, with widely

different numbers of objects, simple random sampling can fail to adequately

represent those types of objects that are less frequent. This can cause

problems when the analysis requires proper representation of all object types.

For example, when building classification models for rare classes, it is critical

that the rare classes be adequately represented in the sample. Hence, a

sampling scheme that can accommodate differing frequencies for the object

types of interest is needed. Stratified sampling , which starts with

prespecified groups of objects, is such an approach. In the simplest version,

equal numbers of objects are drawn from each group even though the groups

are of different sizes. In another variation, the number of objects drawn from

each group is proportional to the size of that group.

Example 2.8 (Sampling and Loss of

Information).

Once a sampling technique has been selected, it is still necessary to choose

the sample size. Larger sample sizes increase the probability that a sample

will be representative, but they also eliminate much of the advantage of

sampling. Conversely, with smaller sample sizes, patterns can be missed or

erroneous patterns can be detected. Figure 2.9(a) shows a data set that

contains 8000 two-dimensional points, while Figures 2.9(b) and 2.9(c) show

samples from this data set of size 2000 and 500, respectively. Although most

of the structure of this data set is present in the sample of 2000 points, much

of the structure is missing in the sample of 500 points.

Figure 2.9.

Example of the loss of structure with sampling.

Example 2.9 (Determining the

Proper Sample Size).

To illustrate that determining the proper sample size requires a methodical

approach, consider the following task.

Given a set of data consisting of a small number of almost equalsized

groups, find at least one representative point for each of the groups.

Assume that the objects in each group are highly similar to each other,

but not very similar to objects in different groups. Figure 2.10(a) shows

an idealized set of clusters (groups) from which these points might be

drawn.

Figure 2.10.

Finding representative points from 10 groups.

Figure 2.10. Full Alternative Text

This problem can be efficiently solved using sampling. One approach is to

take a small sample of data points, compute the pairwise similarities between

points, and then form groups of points that are highly similar. The desired set

of representative points is then obtained by taking one point from each of

these groups. To follow this approach, however, we need to determine a

sample size that would guarantee, with a high probability, the desired

outcome; that is, that at least one point will be obtained from each cluster.

Figure 2.10(b) shows the probability of getting one object from each of the

10 groups as the sample size runs from 10 to 60. Interestingly, with a sample

size of 20, there is little chance (20%) of getting a sample that includes all 10

clusters. Even with a sample size of 30, there is still a moderate chance

(almost 40%) of getting a sample that doesn’t contain objects from all 10

clusters. This issue is further explored in the context of clustering by Exercise

4 on page 603.

Progressive Sampling

The proper sample size can be difficult to determine, so adaptive or

progressive sampling schemes are sometimes used. These approaches start

with a small sample, and then increase the sample size until a sample of

sufficient size has been obtained. While this technique eliminates the need to

determine the correct sample size initially, it requires that there be a way to

evaluate the sample to judge if it is large enough.

Suppose, for instance, that progressive sampling is used to learn a predictive

model. Although the accuracy of predictive models increases as the sample

size increases, at some point the increase in accuracy levels off. We want to

stop increasing the sample size at this leveling-off point. By keeping track of

the change in accuracy of the model as we take progressively larger samples,

and by taking other samples close to the size of the current one, we can get an

estimate of how close we are to this leveling-off point, and thus, stop

sampling.

2.3.3 Dimensionality Reduction

Data sets can have a large number of features. Consider a set of documents,

where each document is represented by a vector whose components are the

frequencies with which each word occurs in the document. In such cases,

there are typically thousands or tens of thousands of attributes (components),

one for each word in the vocabulary. As another example, consider a set of

time series consisting of the daily closing price of various stocks over a

period of 30 years. In this case, the attributes, which are the prices on specific

days, again number in the thousands.

There are a variety of benefits to dimensionality reduction. A key benefit is

that many data mining algorithms work better if the dimensionality—the

number of attributes in the data—is lower. This is partly because

dimensionality reduction can eliminate irrelevant features and reduce noise

and partly because of the curse of dimensionality, which is explained below.

Another benefit is that a reduction of dimensionality can lead to a more

understandable model because the model usually involves fewer attributes.

Also, dimensionality reduction may allow the data to be more easily

visualized. Even if dimensionality reduction doesn’t reduce the data to two or

three dimensions, data is often visualized by looking at pairs or triplets of

attributes, and the number of such combinations is greatly reduced. Finally,

the amount of time and memory required by the data mining algorithm is

reduced with a reduction in dimensionality.

The term dimensionality reduction is often reserved for those techniques that

reduce the dimensionality of a data set by creating new attributes that are a

combination of the old attributes. The reduction of dimensionality by

selecting attributes that are a subset of the old is known as feature subset

selection or feature selection. It will be discussed in Section 2.3.4.

In the remainder of this section, we briefly introduce two important topics:

the curse of dimensionality and dimensionality reduction techniques based on

linear algebra approaches such as principal components analysis (PCA).

More details on dimensionality reduction can be found in Appendix B.

The Curse of Dimensionality

The curse of dimensionality refers to the phenomenon that many types of

data analysis become significantly harder as the dimensionality of the data

increases. Specifically, as dimensionality increases, the data becomes

increasingly sparse in the space that it occupies. Thus, the data objects we

observe are quite possibly not a representative sample of all possible objects.

For classification, this can mean that there are not enough data objects to

allow the creation of a model that reliably assigns a class to all possible

objects. For clustering, the differences in density and in the distances between

points, which are critical for clustering, become less meaningful. (This is

discussed further in Sections 8.1.2, 8.4.6, and 8.4.8.) As a result, many

clustering and classification algorithms (and other data analysis algorithms)

have trouble with high-dimensional data leading to reduced classification

accuracy and poor quality clusters.

Linear Algebra Techniques for

Dimensionality Reduction

Some of the most common approaches for dimensionality reduction,

particularly for continuous data, use techniques from linear algebra to project

the data from a high-dimensional space into a lower-dimensional space.

Principal Components Analysis (PCA) is a linear algebra technique for

continuous attributes that finds new attributes (principal components) that (1)

are linear combinations of the original attributes, (2) are orthogonal

(perpendicular) to each other, and (3) capture the maximum amount of

variation in the data. For example, the first two principal components capture

as much of the variation in the data as is possible with two orthogonal

attributes that are linear combinations of the original attributes. Singular

Value Decomposition (SVD) is a linear algebra technique that is related to

PCA and is also commonly used for dimensionality reduction. For additional

details, see Appendices A and B.

2.3.4 Feature Subset Selection

Another way to reduce the dimensionality is to use only a subset of the

features. While it might seem that such an approach would lose information,

this is not the case if redundant and irrelevant features are present.

Redundant features duplicate much or all of the information contained in

one or more other attributes. For example, the purchase price of a product and

the amount of sales tax paid contain much of the same information.

Irrelevant features contain almost no useful information for the data mining

task at hand. For instance, students’ ID numbers are irrelevant to the task of

predicting students’ grade point averages. Redundant and irrelevant features

can reduce classification accuracy and the quality of the clusters that are

found.

While some irrelevant and redundant attributes can be eliminated

immediately by using common sense or domain knowledge, selecting the best

subset of features frequently requires a systematic approach. The ideal

approach to feature selection is to try all possible subsets of features as input

to the data mining algorithm of interest, and then take the subset that

produces the best results. This method has the advantage of reflecting the

objective and bias of the data mining algorithm that will eventually be used.

Unfortunately, since the number of subsets involving *n *attributes is 2*n*, such

an approach is impractical in most situations and alternative strategies are

needed. There are three standard approaches to feature selection: embedded,

filter, and wrapper.

Embedded approaches

Feature selection occurs naturally as part of the data mining algorithm.

Specifically, during the operation of the data mining algorithm, the algorithm

itself decides which attributes to use and which to ignore. Algorithms for

building decision tree classifiers, which are discussed in Chapter 3, often

operate in this manner.

Filter approaches

Features are selected before the data mining algorithm is run, using some

approach that is independent of the data mining task. For example, we might

select sets of attributes whose pairwise correlation is as low as possible so

that the attributes are non-redundant.

Wrapper approaches

These methods use the target data mining algorithm as a black box to find the

best subset of attributes, in a way similar to that of the ideal algorithm

described above, but typically without enumerating all possible subsets.

Because the embedded approaches are algorithm-specific, only the filter and

wrapper approaches will be discussed further here.

An Architecture for Feature Subset

Selection

It is possible to encompass both the filter and wrapper approaches within a

common architecture. The feature selection process is viewed as consisting of

four parts: a measure for evaluating a subset, a search strategy that controls

the generation of a new subset of features, a stopping criterion, and a

validation procedure. Filter methods and wrapper methods differ only in the

way in which they evaluate a subset of features. For a wrapper method,

subset evaluation uses the target data mining algorithm, while for a filter

approach, the evaluation technique is distinct from the target data mining

algorithm. The following discussion provides some details of this approach,

which is summarized in Figure 2.11.

Figure 2.11.

Flowchart of a feature subset selection process.

Figure 2.11. Full Alternative Text

Conceptually, feature subset selection is a search over all possible subsets of

features. Many different types of search strategies can be used, but the search

strategy should be computationally inexpensive and should find optimal or

near optimal sets of features. It is usually not possible to satisfy both

requirements, and thus, trade-offs are necessary.

An integral part of the search is an evaluation step to judge how the current

subset of features compares to others that have been considered. This requires

an evaluation measure that attempts to determine the goodness of a subset of

attributes with respect to a particular data mining task, such as classification

or clustering. For the filter approach, such measures attempt to predict how

well the actual data mining algorithm will perform on a given set of

attributes. For the wrapper approach, where evaluation consists of actually

running the target data mining algorithm, the subset evaluation function is

simply the criterion normally used to measure the result of the data mining.

Because the number of subsets can be enormous and it is impractical to

examine them all, some sort of stopping criterion is necessary. This strategy

is usually based on one or more conditions involving the following: the

number of iterations, whether the value of the subset evaluation measure is

optimal or exceeds a certain threshold, whether a subset of a certain size has

been obtained, and whether any improvement can be achieved by the options

available to the search strategy.

Finally, once a subset of features has been selected, the results of the target

data mining algorithm on the selected subset should be validated. A

straightforward validation approach is to run the algorithm with the full set of

features and compare the full results to results obtained using the subset of

features. Hopefully, the subset of features will produce results that are better

than or almost as good as those produced when using all features. Another

validation approach is to use a number of different feature selection

algorithms to obtain subsets of features and then compare the results of

running the data mining algorithm on each subset.

Feature Weighting

Feature weighting is an alternative to keeping or eliminating features. More

important features are assigned a higher weight, while less important features

are given a lower weight. These weights are sometimes assigned based on

domain knowledge about the relative importance of features. Alternatively,

they can sometimes be determined automatically. For example, some

classification schemes, such as support vector machines (Chapter 4), produce

classification models in which each feature is given a weight. Features with

larger weights play a more important role in the model. The normalization of

objects that takes place when computing the cosine similarity (Section 2.4.5)

can also be regarded as a type of feature weighting.

2.3.5 Feature Creation

It is frequently possible to create, from the original attributes, a new set of

attributes that captures the important information in a data set much more

effectively. Furthermore, the number of new attributes can be smaller than

the number of original attributes, allowing us to reap all the previously

described benefits of dimensionality reduction. Two related methodologies

for creating new attributes are described next: feature extraction and mapping

the data to a new space.

Feature Extraction

The creation of a new set of features from the original raw data is known as

feature extraction. Consider a set of photographs, where each photograph is

to be classified according to whether it contains a human face. The raw data

is a set of pixels, and as such, is not suitable for many types of classification

algorithms. However, if the data is processed to provide higher-level features,

such as the presence or absence of certain types of edges and areas that are

highly correlated with the presence of human faces, then a much broader set

of classification techniques can be applied to this problem.

Unfortunately, in the sense in which it is most commonly used, feature

extraction is highly domain-specific. For a particular field, such as image

processing, various features and the techniques to extract them have been

developed over a period of time, and often these techniques have limited

applicability to other fields. Consequently, whenever data mining is applied

to a relatively new area, a key task is the development of new features and

feature extraction methods.

Although feature extraction is often complicated, Example 2.10 illustrates

that it can be relatively straightforward.

Example 2.10 (Density).

Consider a data set consisting of information about historical artifacts, which,

along with other information, contains the volume and mass of each artifact.

For simplicity, assume that these artifacts are made of a small number of

materials (wood, clay, bronze, gold) and that we want to classify the artifacts

with respect to the material of which they are made. In this case, a density

feature constructed from the mass and volume features, i.e., *density*

=*mass/volume *, would most directly yield an accurate classification.

Although there have been some attempts to automatically perform such

simple feature extraction by exploring basic mathematical combinations of

existing attributes, the most common approach is to construct features using

domain expertise.

Mapping the Data to a New Space

A totally different view of the data can reveal important and interesting

features. Consider, for example, time series data, which often contains

periodic patterns. If there is only a single periodic pattern and not much

noise, then the pattern is easily detected. If, on the other hand, there are a

number of periodic patterns and a significant amount of noise, then these

patterns are hard to detect. Such patterns can, nonetheless, often be detected

by applying a Fourier transform to the time series in order to change to a

representation in which frequency information is explicit. In Example 2.11, it

will not be necessary to know the details of the Fourier transform. It is

enough to know that, for each time series, the Fourier transform produces a

new data object whose attributes are related to frequencies.

Example 2.11 (Fourier Analysis).

The time series presented in Figure 2.12(b) is the sum of three other time

series, two of which are shown in Figure 2.12(a) and have frequencies of 7

and 17 cycles per second, respectively. The third time series is random noise.

Figure 2.12(c) shows the power spectrum that can be computed after

applying a Fourier transform to the original time series. (Informally, the

power spectrum is proportional to the square of each frequency attribute.) In

spite of the noise, there are two peaks that correspond to the periods of the

two original, non-noisy time series. Again, the main point is that better

features can reveal important aspects of the data.

Figure 2.12.

Application of the Fourier transform to identify the underlying

frequencies in time series data.

Figure 2.12. Full Alternative Text

Many other sorts of transformations are also possible. Besides the Fourier

transform, the wavelet transform has also proven very useful for time series

and other types of data.

2.3.6 Discretization and

Binarization

Some data mining algorithms, especially certain classification algorithms,

require that the data be in the form of categorical attributes. Algorithms that

find association patterns require that the data be in the form of binary

attributes. Thus, it is often necessary to transform a continuous attribute into

a categorical attribute (discretization), and both continuous and discrete

attributes may need to be transformed into one or more binary attributes

(binarization). Additionally, if a categorical attribute has a large number of

values (categories), or some values occur infrequently, then it can be

beneficial for certain data mining tasks to reduce the number of categories by

combining some of the values.

As with feature selection, the best discretization or binarization approach is

the one that “produces the best result for the data mining algorithm that will

be used to analyze the data.” It is typically not practical to apply such a

criterion directly. Consequently, discretization or binarization is performed in

a way that satisfies a criterion that is thought to have a relationship to good

performance for the data mining task being considered. In general, the best

discretization depends on the algorithm being used, as well as the other

attributes being considered. Typically, however, the discretization of each

attribute is considered in isolation.

Binarization

A simple technique to binarize a categorical attribute is the following: If there

are *m *categorical values, then uniquely assign each original value to an

integer in the interval [0, m−1]. If the attribute is ordinal, then order must be

maintained by the assignment. (Note that even if the attribute is originally

represented using integers, this process is necessary if the integers are not in

the interval [0, m−1].) Next, convert each of these *m *integers to a binary

number. Since n=[log2(m)] binary digits are required to represent these

integers, represent these binary numbers using *n *binary attributes. To

illustrate, a categorical variable with 5 values *{awful*, *poor*, *OK*, *good*, *great}*

would require three binary variables x1, x2, and x3. The conversion is shown

in Table 2.5.

Table 2.5. Conversion of a

categorical attribute to three

binary attributes.

Categorical Value Integer Value x1 x2 x3

*awful *0 0 0 0

*poor *1 0 0 1

*OK *2 0 1 0

*good *3 0 1 1

*great *4 1 0 0

Such a transformation can cause complications, such as creating unintended

relationships among the transformed attributes. For example, in Table 2.5,

attributes x2 and x3 are correlated because information about the *good *value

is encoded using both attributes. Furthermore, association analysis requires

asymmetric binary attributes, where only the presence of the attribute

(value =1). is important. For association problems, it is therefore necessary to

introduce one asymmetric binary attribute for each categorical value, as

shown in Table 2.6. If the number of resulting attributes is too large, then the

techniques described in the following sections can be used to reduce the

number of categorical values before binarization.

Table 2.6. Conversion of a

categorical attribute to five

asymmetric binary attributes.

Categorical Value Integer Value x1 x2 x3 x4 x5

*awful *0 1 0 0 0 0

*poor *1 0 1 0 0 0

*OK *2 0 0 1 0 0

*good *3 0 0 0 1 0

*great *4 0 0 0 0 1

Likewise, for association problems, it can be necessary to replace a single

binary attribute with two asymmetric binary attributes. Consider a binary

attribute that records a person’s gender, male or female. For traditional

association rule algorithms, this information needs to be transformed into two

asymmetric binary attributes, one that is a 1 only when the person is male and

one that is a 1 only when the person is female. (For asymmetric binary

attributes, the information representation is somewhat inefficient in that two

bits of storage are required to represent each bit of information.)

Discretization of Continuous

Attributes

Discretization is typically applied to attributes that are used in classification

or association analysis. Transformation of a continuous attribute to a

categorical attribute involves two subtasks: deciding how many categories,*n *,

to have and determining how to map the values of the continuous attribute to

these categories. In the first step, after the values of the continuous attribute

are sorted, they are then divided into *n *intervals by specifying n−1 split

points. In the second, rather trivial step, all the values in one interval are

mapped to the same categorical value. Therefore, the problem of

discretization is one of deciding how many split points to choose and where

to place them. The result can be represented either as a set of intervals

{(x0, x1], (x1, x2],…, (xn−1, xn)}, where x0 and xn can be +∞ or −∞,

respectively, or equivalently, as a series of inequalities x0<x≤x1, …, xn

−1<x<xn.

Unsupervised Discretization

A basic distinction between discretization methods for classification is

whether class information is used (supervised) or not (unsupervised). If class

information is not used, then relatively simple approaches are common. For

instance, the equal width approach divides the range of the attribute into a

user-specified number of intervals each having the same width. Such an

approach can be badly affected by outliers, and for that reason, an equal

frequency (equal depth) approach, which tries to put the same number of

objects into each interval, is often preferred. As another example of

unsupervised discretization, a clustering method, such as K-means (see

Chapter 7), can also be used. Finally, visually inspecting the data can

sometimes be an effective approach.

Example 2.12 (Discretization

Techniques).

This example demonstrates how these approaches work on an actual data set.

Figure 2.13(a) shows data points belonging to four different groups, along

with two outliers—the large dots on either end. The techniques of the

previous paragraph were applied to discretize the *x *values of these data points

into four categorical values. (Points in the data set have a random *y*

component to make it easy to see how many points are in each group.)

Visually inspecting the data works quite well, but is not automatic, and thus,

we focus on the other three approaches. The split points produced by the

techniques equal width, equal frequency, and K-means are shown in Figures

2.13(b), 2.13(c), and 2.13(d), respectively. The split points are represented as

dashed lines.

Figure 2.13.

Different discretization techniques.

Figure 2.13. Full Alternative Text

In this particular example, if we measure the performance of a discretization

technique by the extent to which different objects that clump together have

the same categorical value, then K-means performs best, followed by equal

frequency, and finally, equal width. More generally, the best discretization

will depend on the application and often involves domain-specific

discretization. For example, the discretization of people into low income,

middle income, and high income is based on economic factors.

Supervised Discretization

If classification is our application and class labels are known for some data

objects, then discretization approaches that use class labels often produce

better classification. This should not be surprising, since an interval

constructed with no knowledge of class labels often contains a mixture of

class labels. A conceptually simple approach is to place the splits in a way

that maximizes the purity of the intervals, i.e., the extent to which an interval

contains a single class label. In practice, however, such an approach requires

potentially arbitrary decisions about the purity of an interval and the

minimum size of an interval.

To overcome such concerns, some statistically based approaches start with

each attribute value in a separate interval and create larger intervals by

merging adjacent intervals that are similar according to a statistical test. An

alternative to this bottom-up approach is a top-down approach that starts by

bisecting the initial values so that the resulting two intervals give minimum

entropy. This technique only needs to consider each value as a possible split

point, because it is assumed that intervals contain ordered sets of values. The

splitting process is then repeated with another interval, typically choosing the

interval with the worst (highest) entropy, until a user-specified number of

intervals is reached, or a stopping criterion is satisfied.

Entropy-based approaches are one of the most promising approaches to

discretization, whether bottom-up or top-down. First, it is necessary to define

entropy. Let *k *be the number of different class labels, *m*

i

be the number of

values in the *i**th *interval of a partition, and *m**ij *be the number of values of

class *j *in interval *i*. Then the entropy *e**i *of the *i**th *interval is given by the

equation

ei=−∑j=1kpijlog2 pij,

where pij=mij/mi is the probability (fraction of values) of class *j *in the ith

interval. The total entropy, *e*, of the partition is the weighted average of the

individual interval entropies, i.e.,

e=∑i=1nwiei,

where *m *is the number of values, wi=mi/m is the fraction of values in the ith

interval, and *n *is the number of intervals. Intuitively, the entropy of an

interval is a measure of the purity of an interval. If an interval contains only

values of one class (is perfectly pure), then the entropy is 0 and it contributes

nothing to the overall entropy. If the classes of values in an interval occur

equally often (the interval is as impure as possible), then the entropy is a

maximum.

Example 2.13 (Discretization of Two

Attributes).

The top-down method based on entropy was used to independently discretize

both the *x *and *y *attributes of the two-dimensional data shown in Figure 2.14.

In the first discretization, shown in Figure 2.14(a), the *x *and *y *attributes were

both split into three intervals. (The dashed lines indicate the split points.) In

the second discretization, shown in Figure 2.14(b), the *x *and *y *attributes were

both split into five intervals.

Figure 2.14.

Discretizing *x *and *y *attributes for four groups (classes) of points.

Figure 2.14. Full Alternative Text

This simple example illustrates two aspects of discretization. First, in two

dimensions, the classes of points are well separated, but in one dimension,

this is not so. In general, discretizing each attribute separately often

guarantees suboptimal results. Second, five intervals work better than three,

but six intervals do not improve the discretization much, at least in terms of

entropy. (Entropy values and results for six intervals are not shown.)

Consequently, it is desirable to have a stopping criterion that automatically

finds the right number of partitions.

Categorical Attributes with Too

Many Values

Categorical attributes can sometimes have too many values. If the categorical

attribute is an ordinal attribute, then techniques similar to those for

continuous attributes can be used to reduce the number of categories. If the

categorical attribute is nominal, however, then other approaches are needed.

Consider a university that has a large number of departments. Consequently,

a *department name *attribute might have dozens of different values. In this

situation, we could use our knowledge of the relationships among different

departments to combine departments into larger groups, such as *engineering*,

*social sciences*, or *biological sciences*. If domain knowledge does not serve

as a useful guide or such an approach results in poor classification

performance, then it is necessary to use a more empirical approach, such as

grouping values together only if such a grouping results in improved

classification accuracy or achieves some other data mining objective.

2.3.7 Variable Transformation

A variable transformation refers to a transformation that is applied to all the

values of a variable. (We use the term variable instead of attribute to adhere

to common usage, although we will also refer to attribute transformation on

occasion.) In other words, for each object, the transformation is applied to the

value of the variable for that object. For example, if only the magnitude of a

variable is important, then the values of the variable can be transformed by

taking the absolute value. In the following section, we discuss two important

types of variable transformations: simple functional transformations and

normalization.

Simple Functions

For this type of variable transformation, a simple mathematical function is

applied to each value individually. If *x *is a variable, then examples of such

transformations include xk, log x, ex, x, 1/x, sin x, or |x|. In statistics, variable

transformations, especially sqrt, log, and 1*/x*, are often used to transform data

that does not have a Gaussian (normal) distribution into data that does. While

this can be important, other reasons often take precedence in data mining.

Suppose the variable of interest is the number of data bytes in a session, and

the number of bytes ranges from 1 to 1 billion. This is a huge range, and it

can be advantageous to compress it by using a log 10transformation. In this

case, sessions that transferred 108 and 109 bytes would be more similar to

each other than sessions that transferred 10 and 1000 bytes

(9−8=1 versus 3−1=3). For some applications, such as network intrusion

detection, this may be what is desired, since the first two sessions most likely

represent transfers of large files, while the latter two sessions could be two

quite distinct types of sessions.

Variable transformations should be applied with caution because they change

the nature of the data. While this is what is desired, there can be problems if

the nature of the transformation is not fully appreciated. For instance, the

transformation 1*/x *reduces the magnitude of values that are 1 or larger, but

increases the magnitude of values between 0 and 1. To illustrate, the values

*{*1*, *2*, *3*} *go to { 1, 12, 13 }, but the values { 1, 12, 13 } go to {1, 2, 3}. Thus,

for all sets of values, the transformation 1*/x *reverses the order. To help clarify

the effect of a transformation, it is important to ask questions such as the

following: What is the desired property of the transformed attribute? Does the

order need to be maintained? Does the transformation apply to all values,

especially negative values and 0? What is the effect of the transformation on

the values between 0 and 1? Exercise 17 on page 109 explores other aspects

of variable transformation.

Normalization or Standardization

The goal of standardization or normalization is to make an entire set of values

have a particular property. A traditional example is that of “standardizing a

variable” in statistics. If x¯ is the mean (average) of the attribute values and

sx is their standard deviation, then the transformation x′=(x−x¯)/sx creates a

new variable that has a mean of 0 and a standard deviation of 1. If different

variables are to be used together, e.g., for clustering, then such a

transformation is often necessary to avoid having a variable with large values

dominate the results of the analysis. To illustrate, consider comparing people

based on two variables: age and income. For any two people, the difference

in income will likely be much higher in absolute terms (hundreds or

thousands of dollars) than the difference in age (less than 150). If the

differences in the range of values of age and income are not taken into

account, then the comparison between people will be dominated by

differences in income. In particular, if the similarity or dissimilarity of two

people is calculated using the similarity or dissimilarity measures defined

later in this chapter, then in many cases, such as that of Euclidean distance,

the income values will dominate the calculation.

The mean and standard deviation are strongly affected by outliers, so the

above transformation is often modified. First, the mean is replaced by the

median, i.e., the middle value. Second, the standard deviation is replaced by

the absolute standard deviation. Specifically, if *x *is a variable, then the

absolute standard deviation of *x *is given by σA=∑i=1m|xi−μ|, where xi is the

ith value of the variable, *m *is the number of objects, and μ is either the mean

or median. Other approaches for computing estimates of the location (center)

and spread of a set of values in the presence of outliers are described in

statistics books. These more robust measures can also be used to define a

standardization transformation.

2.4 Measures of Similarity and

Dissimilarity

Similarity and dissimilarity are important because they are used by a number

of data mining techniques, such as clustering, nearest neighbor classification,

and anomaly detection. In many cases, the initial data set is not needed once

these similarities or dissimilarities have been computed. Such approaches can

be viewed as transforming the data to a similarity (dissimilarity) space and

then performing the analysis. Indeed, kernel methods are a powerful

realization of this idea. These methods are introduced in Section 2.4.7 and are

discussed more fully in the context of classification in Section 4.9.4.

We begin with a discussion of the basics: high-level definitions of similarity

and dissimilarity, and a discussion of how they are related. For convenience,

the term proximity is used to refer to either similarity or dissimilarity. Since

the proximity between two objects is a function of the proximity between the

corresponding attributes of the two objects, we first describe how to measure

the proximity between objects having only one attribute.

We then consider proximity measures for objects with multiple attributes.

This includes measures such as the Jaccard and cosine similarity measures,

which are useful for sparse data, such as documents, as well as correlation

and Euclidean distance, which are useful for non-sparse (dense) data, such as

time series or multi-dimensional points. We also consider mutual

information, which can be applied to many types of data and is good for

detecting nonlinear relationships. In this discussion, we restrict ourselves to

objects with relatively homogeneous attribute types, typically binary or

continuous.

Next, we consider several important issues concerning proximity measures.

This includes how to compute proximity between objects when they have

heterogeneous types of attributes, and approaches to account for differences

of scale and correlation among variables when computing distance between

numerical objects. The section concludes with a brief discussion of how to

select the right proximity measure.

Although this section focuses on the computation of proximity between data

objects, proximity can also be computed between attributes. For example, for

the document-term matrix of Figure 2.2(d), the cosine measure can be used to

compute similarity between a pair of documents or a pair of terms (words).

Knowing that two variables are strongly related can, for example, be helpful

for eliminating redundancy. In particular, the correlation and mutual

information measures discussed later are often used for that purpose.

2.4.1 Basics

Definitions

Informally, the similarity between two objects is a numerical measure of the

degree to which the two objects are alike. Consequently, similarities are

*higher *for pairs of objects that are more alike. Similarities are usually nonnegative and are often between 0 (no similarity) and 1 (complete similarity).

The dissimilarity between two objects is a numerical measure of the degree

to which the two objects are different. Dissimilarities are *lower *for more

similar pairs of objects. Frequently, the term distance is used as a synonym

for dissimilarity, although, as we shall see, distance often refers to a special

class of dissimilarities. Dissimilarities sometimes fall in the interval [0*, *1],

but it is also common for them to range from 0 to ∞.

Transformations

Transformations are often applied to convert a similarity to a dissimilarity, or

vice versa, or to transform a proximity measure to fall within a particular

range, such as [0,1]. For instance, we may have similarities that range from 1

to 10, but the particular algorithm or software package that we want to use

may be designed to work only with dissimilarities, or it may work only with

similarities in the interval [0,1]. We discuss these issues here because we will

employ such transformations later in our discussion of proximity. In addition,

these issues are relatively independent of the details of specific proximity

measures.

Frequently, proximity measures, especially similarities, are defined or

transformed to have values in the interval [0,1]. Informally, the motivation

for this is to use a scale in which a proximity value indicates the fraction of

similarity (or dissimilarity) between two objects. Such a transformation is

often relatively straightforward. For example, if the similarities between

objects range from 1 (not at all similar) to 10 (completely similar), we can

make them fall within the range [0*, *1] by using the transformation s′=(s−1)/9,

where *s *and *s*′ are the original and new similarity values, respectively. In the

more general case, the transformation of similarities to the interval [0, 1] is

given by the expression s′=(s−min_s)/(max_s−min_s), where *max_s *and

*min_s *are the maximum and minimum similarity values, respectively.

Likewise, dissimilarity measures with a finite range can be mapped to the

interval [0,1] by using the formula d′=(d−min_d)/(max_d−min_d). This is an

example of a linear transformation, which preserves the relative distances

between points. In other words, if points, x1 and x2, are twice as far apart as

points, x3 and x4, the same will be true after a linear transformation.

However, there can be complications in mapping proximity measures to the

interval [0, 1] using a linear transformation. If, for example, the proximity

measure originally takes values in the interval [0,∞], then *max_d *is not

defined and a nonlinear transformation is needed. Values will not have the

same relationship to one another on the new scale. Consider the

transformation d=d/(1+d) for a dissimilarity measure that ranges from 0 to ∞.

The dissimilarities 0, 0.5, 2, 10, 100, and 1000 will be transformed into the

new dissimilarities 0, 0.33, 0.67, 0.90, 0.99, and 0.999, respectively. Larger

values on the original dissimilarity scale are compressed into the range of

values near 1, but whether this is desirable depends on the application.

Note that mapping proximity measures to the interval [0, 1] can also change

the meaning of the proximity measure. For example, correlation, which is

discussed later, is a measure of similarity that takes values in the interval

[−1, 1]. Mapping these values to the interval [0,1] by taking the absolute

value loses information about the sign, which can be important in some

applications. See Exercise 22 on page 111.

Transforming similarities to dissimilarities and vice versa is also relatively

straightforward, although we again face the issues of preserving meaning and

changing a linear scale into a nonlinear scale. If the similarity (or

dissimilarity) falls in the interval [0,1], then the dissimilarity can be defined

as d=1−s(s=1−d). Another simple approach is to define similarity as the

negative of the dissimilarity (or vice versa). To illustrate, the dissimilarities 0,

1, 10, and 100 can be transformed into the similarities 0, −1, −10, and −100,

respectively.

The similarities resulting from the negation transformation are not restricted

to the range [0, 1], but if that is desired, then transformations such as

s=1d+1, s=e−d, or s=1−d−min_dmax_d−min_d can be used. For the

transformation s=1d+1, the dissimilarities 0, 1, 10, 100 are transformed into

1, 0.5, 0.09, 0.01, respectively. For s=e−d, they become 1.00, 0.37, 0.00,

0.00, respectively, while for s=1−d−min_dmax_d−min_d they become 1.00,

0.99, 0.90, 0.00, respectively. In this discussion, we have focused on

converting dissimilarities to similarities. Conversion in the opposite direction

is considered in Exercise 23 on page 111.

In general, any monotonic decreasing function can be used to convert

dissimilarities to similarities, or vice versa. Of course, other factors also must

be considered when transforming similarities to dissimilarities, or vice versa,

or when transforming the values of a proximity measure to a new scale. We

have mentioned issues related to preserving meaning, distortion of scale, and

requirements of data analysis tools, but this list is certainly not exhaustive.

2.4.2 Similarity and Dissimilarity

between Simple Attributes

The proximity of objects with a number of attributes is typically defined by

combining the proximities of individual attributes, and thus, we first discuss

proximity between objects having a single attribute. Consider objects

described by one nominal attribute. What would it mean for two such objects

to be similar? Because nominal attributes convey only information about the

distinctness of objects, all we can say is that two objects either have the same

value or they do not. Hence, in this case similarity is traditionally defined as

1 if attribute values match, and as 0 otherwise. A dissimilarity would be

defined in the opposite way: 0 if the attribute values match, and 1 if they do

not.

For objects with a single ordinal attribute, the situation is more complicated

because information about order should be taken into account. Consider an

attribute that measures the quality of a product, e.g., a candy bar, on the scale

*{poor*, *fair*, *OK*, *good*, *wonderful}*. It would seem reasonable that a product,

P1, which is rated *wonderful*, would be closer to a product P2, which is rated

*good*, than it would be to a product P3, which is rated *OK*. To make this

observation quantitative, the values of the ordinal attribute are often mapped

to successive integers, beginning at 0 or 1, e.g.,

{poor=0, fair=1, OK=2, good=3, wonderful=4}. Then, d(P1, P2)=3−2=1 or,

if we want the dissimilarity to fall between 0 and d(P1, P2)=3−24=0.25. A

similarity for ordinal attributes can then be defined as s=1−d.

This definition of similarity (dissimilarity) for an ordinal attribute should

make the reader a bit uneasy since this assumes equal intervals between

successive values of the attribute, and this is not necessarily so. Otherwise,

we would have an interval or ratio attribute. Is the difference between the

values *fair *and *good *really the same as that between the values *OK *and

*wonderful*? Probably not, but in practice, our options are limited, and in the

absence of more information, this is the standard approach for defining

proximity between ordinal attributes.

For interval or ratio attributes, the natural measure of dissimilarity between

two objects is the absolute difference of their values. For example, we might

compare our current weight and our weight a year ago by saying “I am ten

pounds heavier.” In cases such as these, the dissimilarities typically range

from 0 to ∞, rather than from 0 to 1. The similarity of interval or ratio

attributes is typically expressed by transforming a dissimilarity into a

similarity, as previously described.

Table 2.7 summarizes this discussion. In this table, *x *and *y *are two objects

that have one attribute of the indicated type. Also, *d*(*x, y*) and *s*(*x, y*) are the

dissimilarity and similarity between *x *and *y*, respectively. Other approaches

are possible; these are the most common ones.

Table 2.7. Similarity and

dissimilarity for simple

attributes

Attribute

Type Dissimilarity Similarity

Nominal d={

0if x=y1if x≠y s={ 1if x=y0if x≠y

Ordinal

d=|x−y|/(n−1)

(values

mapped to

integers 0 to n

−1, where *n *is

the number of

values)

s=1−d

Interval

or Ratio d=|x−y| s=−d, s=11+d, s=e−d,s=1−d−min_dmax_d−min_

The following two sections consider more complicated measures of

proximity between objects that involve multiple attributes: (1) dissimilarities

between data objects and (2) similarities between data objects. This division

allows us to more naturally display the underlying motivations for employing

various proximity measures. We emphasize, however, that similarities can be

transformed into dissimilarities and vice versa using the approaches described

earlier.

2.4.3 Dissimilarities between Data

Objects

In this section, we discuss various kinds of dissimilarities. We begin with a

discussion of distances, which are dissimilarities with certain properties, and

then provide examples of more general kinds of dissimilarities.

Distances

We first present some examples, and then offer a more formal description of

distances in terms of the properties common to all distances. The Euclidean

distance ,*d *, between two points, x and y , in one-, two-, three-, or higherdimensional space, is given by the following familiar formula:

d(x,y)=∑k=1n(xk−yk)2, (2.1)

where *n *is the number of dimensions and xk and yk are, respectively, the kth

attributes (components) of *x *and *y*. We illustrate this formula with Figure

2.15 and Tables 2.8 and 2.9, which show a set of points, the *x *and *y*

coordinates of these points, and the distance matrix containing the pairwise

distances of these points.

Figure 2.15.

Four two-dimensional points.

The Euclidean distance measure given in Equation 2.1 is generalized by the

Minkowski distance metric shown in Equation 2.2,

d(x,y)=(∑k=1n|xk−yk|r)1/r, (2.2)

where *r *is a parameter. The following are the three most common examples

of Minkowski distances.

r=1. City block (Manhattan, taxicab, L1 norm) distance. A common

example is the Hamming distance , which is the number of bits that is

different between two objects that have only binary attributes, i.e.,

between two binary vectors.

r=2. Euclidean distance (L2 norm).

r=∞. Supremum (Lmax or L∞ norm) distance. This is the maximum

difference between any attribute of the objects. More formally, the L∞

distance is defined by Equation 2.3

d(x, y)=limr→∞(∑k=1n|xk−yk|r)1/r. (2.3)

The *r *parameter should not be confused with the number of dimensions (attributes) *n*. The Euclidean, Manhattan, and supremum distances are defined

for all values of *n*: 1, 2, 3, …, and specify different ways of combining the

differences in each dimension (attribute) into an overall distance.

Tables 2.10 and 2.11, respectively, give the proximity matrices for the L1 and

L∞ distances using data from Table 2.8. Notice that all these distance

matrices are symmetric; i.e., the ijth entry is the same as the jith entry. In

Table 2.9, for instance, the fourth row of the first column and the fourth

column of the first row both contain the value 5.1.

Table 2.8. *x *and *y *coordinates

of four points.

point *x *coordinate *y *coordinate

p1 0 2

p2 2 0

p3 3 1

p4 5 1

Table 2.9. Euclidean distance

matrix for Table 2.8.

p1 p2 p3 p4

p1 0.0 2.8 3.2 5.1

p2 2.8 0.0 1.4 3.2

p3 3.2 1.4 0.0 2.0

p4 5.1 3.2 2.0 0.0

Table 2.10. L1 distance matrix

for Table 2.8.

L

1 p1 p2 p3 p4

p1 0.0 4.0 4.0 6.0

p2 4.0 0.0 2.0 4.0

p3 4.0 2.0 0.0 2.0

p4 6.0 4.0 2.0 0.0

Table 2.11. L∞ distance matrix

for Table 2.8.

L∞ p1 p2 p3 p4

p1 0.0 2.0 3.0 5.0

p2 2.0 0.0 1.0 3.0

p3 3.0 1.0 0.0 2.0

p4 5.0 3.0 2.0 0.0

Distances, such as the Euclidean distance, have some well-known properties.

If *d*(x, y) is the distance between two points, x and y, then the following

properties hold.

1. Positivity

1. d(x, y)≥0 for all x and y,

2. d(x, y)=0 only if x=y.

2. Symmetry d(x, y)=d(y, x) for all x and y.

3. Triangle Inequality d(x, z)≤d(x, y)+d(y, z) for all points x , y , and z.

Measures that satisfy all three properties are known as metrics. Some people

use the term distance only for dissimilarity measures that satisfy these

properties, but that practice is often violated. The three properties described

here are useful, as well as mathematically pleasing. Also, if the triangle

inequality holds, then this property can be used to increase the efficiency of

techniques (including clustering) that depend on distances possessing this

property. (See Exercise 25.) Nonetheless, many dissimilarities do not satisfy

one or more of the metric properties. Example 2.14 illustrates such a

measure.

Example 2.14 (Non-metric

Dissimilarities: Set Differences).

This example is based on the notion of the difference of two sets, as defined

in set theory. Given two sets *A *and *B*, A−B is the set of elements of A that are

not in

*B*. For example, if A={1, 2, 3, 4} and B={2, 3, 4}, then A−B={1} and B

−A=∅, the empty set. We can define the distance *d *between two sets *A *and *B*

as d(A, B)=size(A−B), where *size *is a function returning the number of

elements in a set. This distance measure, which is an integer value greater

than or equal to 0, does not satisfy the second part of the positivity property,

the symmetry property, or the triangle inequality. However, these properties

can be made to hold if the dissimilarity measure is modified as follows:

d(A, B)=size(A−B)+size(B−A). See Exercise 21 on page 110.

2.4.4 Similarities between Data

Objects

For similarities, the triangle inequality (or the analogous property) typically

does not hold, but symmetry and positivity typically do. To be explicit, if *s*(x,

y) is the similarity between points x and y, then the typical properties of

similarities are the following:

1. s(x, y)=1 only if x=y. (0≤s≤1)

2. s(x, y)=s(y, x) for all x and y. (Symmetry)

There is no general analog of the triangle inequality for similarity measures.

It is sometimes possible, however, to show that a similarity measure can

easily be converted to a metric distance. The cosine and Jaccard similarity

measures, which are discussed shortly, are two examples. Also, for specific

similarity measures, it is possible to derive mathematical bounds on the

similarity between two objects that are similar in spirit to the triangle

inequality.

Example 2.15 (A Non-symmetric

Similarity Measure).

Consider an experiment in which people are asked to classify a small set of

characters as they flash on a screen. The confusion matrix for this

experiment records how often each character is classified as itself, and how

often each is classified as another character. Using the confusion matrix, we

can define a similarity measure between a character *x *and a character *y *as the

number of times that *x *is misclassified as*y *, but note that this measure is not

symmetric. For example, suppose that “0” appeared 200 times and was

classified as a “0” 160 times, but as an “o” 40 times. Likewise, suppose that

“o” appeared 200 times and was classified as an “o” 170 times, but as “0”

only 30 times. Then, s(0,o)=40, but s(o, 0)=30. In such situations, the

similarity measure can be made symmetric by setting s′=(x,y)=s′(x,y)=

(s(x,y+s(y,x))/2, where *s *indicates the new similarity measure.

2.4.5 Examples of Proximity

Measures

This section provides specific examples of some similarity and dissimilarity

measures.

Similarity Measures for Binary

Data

Similarity measures between objects that contain only binary attributes are

called similarity coefficients , and typically have values between 0 and 1. A

value of 1 indicates that the two objects are completely similar, while a value

of 0 indicates that the objects are not at all similar. There are many rationales

for why one coefficient is better than another in specific instances.

Let x and y be two objects that consist of *n *binary attributes. The comparison

of two such objects, i.e., two binary vectors, leads to the following four

quantities (frequencies):

f00=the number of attributes where x is 0 and y is 0f01= the number of attributes where

Simple Matching Coefficient

One commonly used similarity coefficient is the simple matching coefficient

(*SMC*), which is defined as

SMC=number of matching attribute valuesnumber of attributes=f11+f00f01+f

(2.4)

This measure counts both presences and absences equally. Consequently, the

*SMC *could be used to find students who had answered questions similarly on

a test that consisted only of true/false questions.

Jaccard Coefficient

Suppose that x and y are data objects that represent two rows (two

transactions) of a transaction matrix (see Section 2.1.2). If each asymmetric

binary attribute corresponds to an item in a store, then a 1 indicates that the

item was purchased, while a 0 indicates that the product was not purchased.

Because the number of products not purchased by any customer far

outnumbers the number of products that were purchased, a similarity measure

such as *SMC *would say that all transactions are very similar. As a result, the

Jaccard coefficient is frequently used to handle objects consisting of

asymmetric binary attributes. The Jaccard coefficient , which is often

symbolized by *j*, is given by the following equation:

J=number of matching presencesnumber of attributes not involved in 00 matches

(2.5)

Example 2.16 (The SMC and

Jaccard Similarity Coefficients).

To illustrate the difference between these two similarity measures, we

calculate *SMC *and *j *for the following two binary vectors.

x = (1, 0, 0, 0, 0, 0, 0, 0, 0, 0)y = (0, 0, 0, 0, 0, 0, 1, 0, 0, 1)

f01=2the number of attributes where x was 0 and y was 1f10=1the number of attributes where

SMC=f11+f00f01+f10+f11+f00=0+72+1+0+7=0.7

J=f11f01+f10+f11=02+1+0=0

Cosine Similarity

Documents are often represented as vectors, where each component

(attribute) represents the frequency with which a particular term (word)

occurs in the document. Even though documents have thousands or tens of

thousands of attributes (terms), each document is sparse since it has relatively

few nonzero attributes. Thus, as with transaction data, similarity should not

depend on the number of shared 0 values because any two documents are

likely to “not contain” many of the same words, and therefore, if 0–0 matches

are counted, most documents will be highly similar to most other documents.

Therefore, a similarity measure for documents needs to ignores 0–0 matches

like the Jaccard measure, but also must be able to handle non-binary vectors.

The cosine similarity , defined next, is one of the most common measures of

document similarity. If x and y are two document vectors, then

cos(x, y)=〈 x, y 〉∥x∥∥y∥=x′y∥x∥∥y∥, (2.6)

where ′ indicates vector or matrix transpose and 〈 x, y 〉 indicates the inner

product of the two vectors,

〈 x, y 〉=∑k=1nxkyk=x′y, (2.7)

and ∥x∥ is the length of vector x, ∥x∥=∑k=1nxk2=〈 x, x 〉=x′x.

The inner product of two vectors works well for asymmetric attributes since it

depends only on components that are non-zero in both vectors. Hence, the

similarity between two documents depends only upon the words that appear

in both of them.

Example 2.17 (Cosine Similarity

between Two Document Vectors).

This example calculates the cosine similarity for the following two data

objects, which might represent document vectors:

x=(3, 2, 0, 5, 0, 0, 0, 2, 0, 0)y=(1, 0, 0, 0, 0, 0, 0, 1, 0, 2)

〈 x, y

〉=3×1+2×0+0×0+5×0+0×0+0×0+0×0+2×1+0×0×2=5∥x∥=3×3+2×2+0×0+5

As indicated by Figure 2.16, cosine similarity really is a measure of the

(cosine of the) angle between x and y. Thus, if the cosine similarity is 1, the

angle between x and y is 0°, and x and y are the same except for length. If the

cosine similarity is 0, then the angle between x and y is 90°, and they do not

share any terms (words).

Figure 2.16.

Geometric illustration of the cosine measure.

Equation 2.6 also can be written as Equation 2.8.

cos(x, y)=〈 x∥x∥, y∥y∥ 〉=〈 x′, y′ 〉, (2.8)

where x′=x/∥x∥ and y′=y/∥y∥. Dividing x and y by their lengths

normalizes them to have a length of 1. This means that cosine similarity does

not take the *length *of the two data objects into account when computing

similarity. (Euclidean distance might be a better choice when length is

important.) For vectors with a length of 1, the cosine measure can be

calculated by taking a simple inner product. Consequently, when many

cosine similarities between objects are being computed, normalizing the

objects to have unit length can reduce the time required.

Extended Jaccard Coefficient

(Tanimoto Coefficient)

The extended Jaccard coefficient can be used for document data and that

reduces to the Jaccard coefficient in the case of binary attributes. This

coefficient, which we shall represent as *EJ*, is defined by the following

equation:

EJ(x, y)=〈 x, y 〉ǁ x ǁ2+ǁ y ǁ2−〈 x, y 〉=x′yǁ x ǁ2+ǁ y ǁ2−x′y. (2.9)

Correlation

Correlation is frequently used to measure the linear relationship between two

sets of values that are observed together. Thus, correlation can measure the

relationship between two variables (height and weight) or between two

objects (a pair of temperature time series). Correlation is used much more

frequently to measure the similarity between attributes since the values in two

data objects come from different attributes, which can have very different

attribute types and scales. There are many types of correlation, and indeed

correlation is sometimes used in a general sense to mean the relationship

between two sets of values that are observed together. In this discussion, we

will focus on a measure appropriate for numerical values.

Specifically, Pearson’s correlation between two sets of numerical values,

i.e., two vectors, x and y, is defined by the following equation:

corr(x, y)=covariance(x, y)standard_deviation(x)×standard_deviation(y)=sxys

(2.10)

where we use the following standard statistical notation and definitions:

covariance(x, y)=sxy=1n−1∑k=1n(xk−x¯)(yk−y¯) (2.11)

standard_deviation(x)=sx=1n−1∑k=1n(xk−x¯)2

standard_deviation(y)=sy=1n−1∑k=1n(yk−y¯)2

x¯=1n∑k=1nxk is the mean of x

y¯=1n∑k=1nyk is the mean of y

Example 2.18 (Perfect Correlation).

Correlation is always in the range −1 to 1. A correlation of 1 (−1) means that

x and y have a perfect positive (negative) linear relationship; that is,

xk=ayk+b, where *a *and *b *are constants. The following two vectors x and y

illustrate cases where the correlation is −1 and +1, respectively. In the first

case, the means of x and y were chosen to be 0, for simplicity.

x=(−3, 6, 0, 3, −6)y=(1, −2, 0, −1, 2)corr(x, y)=−1xk=−3yk

x=(3, 6, 0, 3, 6)y=(1, 2, 0, 1, 2)corr(x, y)=1xk=3yk

Example 2.19 (Nonlinear

Relationships).

If the correlation is 0, then there is no linear relationship between the two sets

of values. However, nonlinear relationships can still exist. In the following

example, yk=xk2, but their correlation is 0.

x=(−3, −2, −1, 0, 1, 2, 3)y=(9, 4, 1, 0, 1, 4, 9)

Example 2.20 (Visualizing

Correlation).

It is also easy to judge the correlation between two vectors x and y by

plotting pairs of corresponding values of x and y in a scatter plot. Figure 2.17

shows a number of these scatter plots when x and y consist of a set of 30

pairs of values that are randomly generated (with a normal distribution) so

that the correlation of x and y ranges from −1 to 1. Each circle in a plot

represents one of the 30 pairs of x and y values; its *x *coordinate is the value

of that pair for x, while its *y *coordinate is the value of the same pair for y.

Figure 2.17.

Scatter plots illustrating correlations from −1 to 1.

Figure 2.17. Full Alternative Text

If we transform x and y by subtracting off their means and then normalizing

them so that their lengths are 1, then their correlation can be calculated by

taking the dot product. Let us refer to these transformed vectors of x and y as

x′ and y′, respectively. (Notice that this transformation is not the same as the

standardization used in other contexts, where we subtract the means and

divide by the standard deviations, as discussed in Section 2.3.7.) This

transformation highlights an interesting relationship between the correlation

measure and the cosine measure. Specifically, the correlation between x and

y is identical to the cosine between x′ and y′. However, the cosine between x

and y is not the same as the cosine between x′ and y′, even though they both

have the same correlation measure. In general, the correlation between two

vectors is equal to the cosine measure only in the special case when the

means of the two vectors are 0.

Differences Among Measures For

Continuous Attributes

In this section, we illustrate the difference among the three proximity

measures for continuous attributes that we have just defined: cosine,

correlation, and Minkowski distance. Specifically, we consider two types of

data transformations that are commonly used, namely, scaling

(multiplication) by a constant factor and translation (addition) by a constant

value. A proximity measure is considered to be invariant to a data

transformation if its value remains unchanged even after performing the

transformation. Table 2.12 compares the behavior of cosine, correlation, and

Minkowski distance measures regarding their invariance to scaling and

translation operations. It can be seen that while correlation is invariant to both

scaling and translation, cosine is only invariant to scaling but not to

translation. Minkowski distance measures, on the other hand, are sensitive to

both scaling and translation and are thus invariant to neither.

Table 2.12. Properties of cosine,

correlation, and Minkowski

distance measures.

Property Cosine Correlation Minkowski

Distance

Invariant to scaling

(multiplication) Yes Yes No

Invariant to translation

(addition) No Yes No

Let us consider an example to demonstrate the significance of these

differences among different proximity measures.

Example 2.21 (Comparing

proximity measures).

Consider the following two vectors x and y with seven numeric attributes.

x=(1, 2, 4, 3, 0, 0, 0)y=(1, 2, 3, 4, 0, 0, 0)

It can be seen that both x and y have 4 non-zero values, and the values in the

two vectors are mostly the same, except for the third and the fourth

components. The cosine, correlation, and Euclidean distance between the two

vectors can be computed as follows.

cos(x, y)=2930×30=0.9667correlation(x, y)=2.35711.5811×1.5811=0.9429Euclidean distance

x−y ǁ=1.4142

Not surprisingly, x and y have a cosine and correlation measure close to 1,

while the Euclidean distance between them is small, indicating that they are

quite similar. Now let us consider the vector ys, which is a scaled version of y

(multiplied by a constant factor of 2), and the vector yt, which is constructed

by translating y by 5 units as follows.

ys=2×y=(2, 4, 6, 8, 0, 0, 0)

yt=y+5=(6, 7, 8, 9, 5, 5, 5)

We are interested in finding whether ys and yt show the same proximity with

x as shown by the original vector y. Table 2.13 shows the different measures

of proximity computed for the pairs (x, y), (x, ys), and (x, yt). It can be seen

that the value of correlation between x and y remains unchanged even after

replacing y with ys or yt. However, the value of cosine remains equal to

0.9667 when computed for (x, y) and (x, ys), but significantly reduces to

0.7940 when computed for (x, yt). This highlights the fact that cosine is

invariant to the scaling operation but not to the translation operation, in

contrast with the correlation measure. The Euclidean distance, on the other

hand, shows different values for all three pairs of vectors, as it is sensitive to

both scaling and translation.

Table 2.13. Similarity between

(x, y), (x, ys), and (x, yt).

Measure (x, y) (x, ys) (x, yt)

Cosine 0.9667 0.9667 0.7940

Correlation 0.9429 0.9429 0.9429

Euclidean Distance 1.4142 5.8310 14.2127

We can observe from this example that different proximity measures behave

differently when scaling or translation operations are applied on the data. The

choice of the right proximity measure thus depends on the desired notion of

similarity between data objects that is meaningful for a given application. For

example, if x and y represented the frequencies of different words in a

document-term matrix, it would be meaningful to use a proximity measure

that remains unchanged when y is replaced by ys, because ys is just a scaled

version of y with the same distribution of words occurring in the document.

However, yt is different from y, since it contains a large number of words

with non-zero frequencies that do not occur in y. Because cosine is invariant

to scaling but not to translation, it will be an ideal choice of proximity

measure for this application.

Consider a different scenario in which x represents a location’s temperature

measured on the Celsius scale for seven days. Let y, ys, and yt be the

temperatures measured on those days at a different location, but using three

different measurement scales. Note that different units of temperature have

different offsets (e.g. Celsius and Kelvin) and different scaling factors (e.g.

Celsius and Fahrenheit). It is thus desirable to use a proximity measure that

captures the proximity between temperature values without being affected by

the measurement scale. Correlation would then be the ideal choice of

proximity measure for this application, as it is invariant to both scaling and

translation.

As another example, consider a scenario where x represents the amount of

precipitation (in cm) measured at seven locations. Let y, ys, and yt be

estimates of the precipitation at these locations, which are predicted using

three different models. Ideally, we would like to choose a model that

accurately reconstructs the measurements in x without making any error. It is

evident that y provides a good approximation of the values in x, whereas ys

and yt provide poor estimates of precipitation, even though they do capture

the trend in precipitation across locations. Hence, we need to choose a

proximity measure that penalizes any difference in the model estimates from

the actual observations, and is sensitive to both the scaling and translation

operations. The Euclidean distance satisfies this property and thus would be

the right choice of proximity measure for this application. Indeed, the

Euclidean distance is commonly used in computing the accuracy of models,

which will be discussed later in Chapter 3.

2.4.6 Mutual Information

Like correlation, mutual information is used as a measure of similarity

between two sets of paired values that is sometimes used as an alternative to

correlation, particularly when a nonlinear relationship is suspected between

the pairs of values. This measure comes from information theory, which is

the study of how to formally define and quantify information. Indeed, mutual

information is a measure of how much information one set of values provides

about another, given that the values come in pairs, e.g., height and weight. If

the two sets of values are independent, i.e., the value of one tells us nothing

about the other, then their mutual information is 0. On the other hand, if the

two sets of values are completely dependent, i.e., knowing the value of one

tells us the value of the other and vice-versa, then they have maximum

mutual information. Mutual information does not have a maximum value, but

we will define a normalized version of it that ranges between 0 and 1.

To define mutual information, we consider two sets of values, *X *and *Y *,

which occur in pairs (*X*, *Y*). We need to measure the average information in a

single set of values, i.e., either in *X *or in *Y *, and in the pairs of their values.

This is commonly measured by entropy. More specifically, assume *X *and *Y*

are discrete, that is, *X *can take *m *distinct values,u1, u2, …, um and *Y *can

take *n *distinct values,v1, v2, …, vn. Then their individual and joint entropy

can be defined in terms of the probabilities of each value and pair of values as

follows:

H(X)=−∑j=1mP(X=uj)log2 P(X=uj) (2.12)

H(Y)=−∑k=1nP(Y=vk)log2 P(Y=vk) (2.13)

H(X, Y)=−∑j=1m∑k=1nP(X=uj, Y=vk)log2 P(X=uj, Y=vk) (2.14)

where if the probability of a value or combination of values is 0, then

0 log2(0) is conventionally taken to be 0.

The mutual information of *X *and *Y *can now be defined straightforwardly:

I(X, Y)=H(X)+H(Y)−H(X, Y) (2.15)

Note that *H*(*X*, *Y*) is symmetric, i.e., H(X, Y)=H(Y, X), and thus mutual

information is also symmetric, i.e., I(X, Y)=I(Y).

Practically, *X *and *Y *are either the values in two attributes or two rows of the

same data set. In Example 2.22, we will represent those values as two vectors

x and y and calculate the probability of each value or pair of values from the

frequency with which values or pairs of values occur in x, y and (xi, yi),

where xi is the ith component of x and yi is the ith component of y. Let us

illustrate using a previous example.

Example 2.22 (Evaluating Nonlinear

Relationships with Mutual

Information).

Recall Example 2.19 where yk=xk2, but their correlation was 0.

x=(−3, −2, −1, 0, 1, 2, 3)y=(9, 4, 1, 0, 1, 4, 9)

From Figure 2.22, I(x, y)=H(x)+H(y)−H(x, y)=1.9502. Although a variety of

approaches to normalize mutual information are possible—see Bibliographic

Notes—for this example, we will apply one that divides the mutual

information by log2(min(m, n)) and produces a result between 0 and 1. This

yields a value of 1.9502/log2(4))=0.9751. Thus, we can see that x and y are

strongly related. They are not perfectly related because given a value of y

there is, except for y=0, some ambiguity about the value of x. Notice that for

y=−x, the normalized mutual information would be 1.

Figure 2.18.

Computation of mutual information.

Table 2.14. Entropy for x

xj P(x=xj) −P(x=xj)log2 P(x=xj)

−3 1/7 0.4011

−2 1/7 0.4011

−1 1/7 0.4011

0 1/7 0.4011

1 1/7 0.4011

2 1/7 0.4011

3 1/7 0.4011

*H*(x) 2.8074

Table 2.15. Entropy for y

yk P(y=yk) −P(y=yk)log2 P(y=yk)

9 2/7 0.5164

4 2/7 0.5164

1 2/7 0.5164

0 1/7 0.4011

*H*(y) 1.9502

Table 2.16. Joint entropy

for x and y

xj yk P(x=xj, y=xk) −P(x=xj, y=xk)log2 P(x=xj, y=xk)

−3 9 1/7 0.4011

−2 4 1/7 0.4011

−1 1 1/7 0.4011

0 0 1/7 0.4011

1 1 1/7 0.4011

2 4 1/7 0.4011

3 9 1/7 0.4011

*H*(x, y) 2.8074

2.4.7 Kernel Functions*

It is easy to understand how similarity and distance might be useful in an

application such as clustering, which tries to group similar objects together.

What is much less obvious is that many other data analysis tasks, including

predictive modeling and dimensionality reduction, can be expressed in terms

of pairwise “proximities” of data objects. More specifically, many data

analysis problems can be mathematically formulated to take as input, a

kernel matrix, K, which can be considered a type of proximity matrix. Thus,

an initial preprocessing step is used to convert the input data into a kernel

matrix, which is the input to the data analysis algorithm.

More formally, if a data set has *m *data objects, then K is an *m *by *m *matrix. If

xi and xj are the ith and jth data objects, respectively, then kij, the ijth entry

of K, is computed by a kernel function:

kij=κ(xi, xj) (2.16)

As we will see in the material that follows, the use of a kernel matrix allows

both wider applicability of an algorithm to various kinds of data and an

ability to model nonlinear relationships with algorithms that are designed

only for detecting linear relationships.

Kernels make an algorithm data

independent

If an algorithm uses a kernel matrix, then it can be used with any type of data

for which a kernel function can be designed. This is illustrated by Algorithm

2.1. Although only some data analysis algorithms can be modified to use a

kernel matrix as input, this approach is extremely powerful because it allows

such an algorithm to be used with almost any type of data for which an

appropriate kernel function can be defined. Thus, a classification algorithm

can be used, for example, with record data, string data, or graph data. If an

algorithm can be reformulated to use a kernel matrix, then its applicability to

different types of data increases dramatically. As we will see in later chapters,

many clustering, classification, and anomaly detection algorithms work only

with similarities or distances, and thus, can be easily modified to work with

kernels.

Algorithm 2.1 Basic kernel

algorithm.

1. Read in the *m *data objects in the data set.

2. Compute the kernel matrix, K by applying the kernel function, κ, to each

pair of data objects.

3. Run the data analysis algorithm with K as input.

4. Return the analysis result, e.g., predicted class or cluster labels.

Mapping data into a higher

dimensional data space can allow

modeling of nonlinear relationships

There is yet another, equally important, aspect of kernel based data

algorithms—their ability to model nonlinear relationships with algorithms

that model only linear relationships. Typically, this works by first

transforming (mapping) the data from a lower dimensional data space to a

higher dimensional space.

Example 2.23 (Mapping Data to a

Higher Dimensional Space).

Consider the relationship between two variables *x *and *y *given by the

following equation, which defines an ellipse in two dimensions (Figure

2.19(a)):

Figure 2.19.

Mapping data to a higher dimensional space: two to three

dimensions.

Figure 2.19. Full Alternative Text

4×2+9xy+7y2=10 (2.17)

We can map our two dimensional data to three dimensions by creating three

new variables, *u*, *v*, and *w*, which are defined as follows:

w=x2u=xyv=y2

As a result, we can now express Equation 2.17 as a linear one. This equation

describes a plane in three dimensions. Points on the ellipse will lie on that

plane, while points inside and outside the ellipse will lie on opposite sides of

the plane. See Figure 2.19(b). The viewpoint of this 3D plot is along the

surface of the separating plane so that the plane appears as a line.

4u+9v+7w=10 (2.18)

The Kernel Trick

The approach illustrated above shows the value in mapping data to higher

dimensional space, an operation that is integral to kernel-based methods.

Conceptually, we first define a function φ that maps data points x and y to

data points φ(x) and φ(y) in a higher dimensional space such that the inner

product 〈x, y〉 gives the desired measure of proximity of x and y. It may seem

that we have potentially sacrificed a great deal by using such an approach,

because we can greatly expand the size of our data, increase the

computational complexity of our analysis, and encounter problems with the

curse of dimensionality by computing similarity in a high-dimensional space.

However, this is not the case since these problems can be avoided by defining

a kernel function κ that can compute the same similarity value, but with the

data points in the original space, i.e., κ(x, y)=〈 φ(x), φ(y) 〉. This is known as

the *kernel trick*. Despite the name, the kernel trick has a very solid

mathematical foundation and is a remarkably powerful approach for data

analysis.

Not every function of a pair of data objects satisfies the properties needed for

a kernel function, but it has been possible to design many useful kernels for a

wide variety of data types. For example, three common kernel functions are

the polynomial, Gaussian (radial basis function (RBF)), and sigmoid kernels.

If x and y are two data objects, specifically, two data vectors, then these two

kernel functions can be expressed as follows, respectively:

κ(x, y)−(x′y+c)d (2.19)

κ(x, y)=exp(−ǁ x−y ǁ/2σ2) (2.20)

κ(x, y)=tanh(αx′y+c) (2.21)

where α and c≥0 are constants, *d *is an integer parameter that gives the

polynomial degree, ǁ x−y ǁ is the length of the vector x−y and σ>0 is a

parameter that governs the “spread” of a Gaussian.

Example 2.24 (The Polynomial

Kernel).

Note that the kernel functions presented in the previous section are

computing the same similarity value as would be computed if we actually

mapped the data to a higher dimensional space and then computed an inner

product there. For example, for the polynomial kernel of degree 2, let φ be

the function that maps a two-dimensional data vector x=(x1, x2) to the higher

dimensional space. Specifically, let

φ(x)=(x12, x22, 2x1x2, 2cx1, 2cx2, c). (2.22)

For the higher dimensional space, let the proximity be defined as the inner

product of φ(x) and φ(y), i.e., 〈 φ(x), φ(y) 〉. Then, as previously mentioned, it

can be shown that

κ(x, y)=〈 φ(x), φ(y) 〉 (2.23)

where κ is defined by Equation 2.19 above. Specifically, if x=(x1, x2) and y=

(y1, y2), then

κ(x, y)=〈 x, y 〉=x′y=(x12y12, x22y22, 2x1x2y1y2, 2cx1y1, 2cx2y2, c2).

(2.24)

More generally, the kernel trick depends on defining κ and φ so that Equation

2.23 holds. This has been done for a wide variety of kernels.

This discussion of kernel-based approaches was intended only to provide a

brief introduction to this topic and has omitted many details. A fuller

discussion of the kernel-based approach is provided in Section 4.9.4, which

discusses these issues in the context of nonlinear support vector machines for

classification. More general references for the kernel based analysis can be

found in the Bibliographic Notes of this chapter.

2.4.8 Bregman Divergence*

This section provides a brief description of Bregman divergences, which are

a family of proximity functions that share some common properties. As a

result, it is possible to construct general data mining algorithms, such as

clustering algorithms, that work with any Bregman divergence. A concrete

example is the K-means clustering algorithm (Section 7.2). Note that this

section requires knowledge of vector calculus.

Bregman divergences are loss or distortion functions. To understand the idea

of a loss function, consider the following. Let x and y be two points, where y

is regarded as the original point and x is some distortion or approximation of

it. For example, x may be a point that was generated by adding random noise

to y. The goal is to measure the resulting distortion or loss that results if y is

approximated by x. Of course, the more similar x and y are, the smaller the

loss or distortion. Thus, Bregman divergences can be used as dissimilarity

functions.

More formally, we have the following definition.

Definition 2.6 (Bregman divergence)

Given a strictly convex function ϕ (with a few modest restrictions that are

generally satisfied), the Bregman divergence (loss function) D(x, y)

generated by that function is given by the following equation:

D(x, y)=ϕ(x)−ϕ(y)−〈 ∇ϕ(y), (x−y) 〉 (2.25)

where ∇ϕ(y) is the gradient of ϕ evaluated at y, x−y, is the vector difference

between x and y, and 〈 ∇ϕ(y), (x−y) 〉 is the inner product between ∇ϕ(y)

and (x−y). For points in Euclidean space, the inner product is just the dot

product.

*D*(*x*, *y*) can be written as D(x, y)=ϕ(x)−L(x), where L(x)=ϕ(y)+〈 ∇ϕ(y), (x

−y) 〉 and represents the equation of a plane that is tangent to the function ϕ at

y. Using calculus terminology, *L*(x) is the linearization of ϕ around the point

y, and the Bregman divergence is just the difference between a function and a

linear approximation to that function. Different Bregman divergences are

obtained by using different choices for ϕ.

Example 2.25.

We provide a concrete example using squared Euclidean distance, but restrict

ourselves to one dimension to simplify the mathematics. Let *x *and *y *be real

numbers and ϕ(t) be the real-valued function, ϕ(t)=t2. In that case, the

gradient reduces to the derivative, and the dot product reduces to

multiplication. Specifically, Equation 2.25 becomes Equation 2.26.

D(x,y)=x2−y2−2y(x−y)=(x−y)2 (2.26)

The graph for this example, with y=1, is shown in Figure 2.20. The Bregman

divergence is shown for two values of *x*: x=2 and x=3.

Figure 2.20.

Illustration of Bregman divergence.

Figure 2.20. Full Alternative Text

2.4.9 Issues in Proximity

Calculation

This section discusses several important issues related to proximity measures:

(1) how to handle the case in which attributes have different scales and/or are

correlated, (2) how to calculate proximity between objects that are composed

of different types of attributes, e.g., quantitative and qualitative, (3) and how

to handle proximity calculations when attributes have different weights; i.e.,

when not all attributes contribute equally to the proximity of objects.

Standardization and Correlation for

Distance Measures

An important issue with distance measures is how to handle the situation

when attributes do not have the same range of values. (This situation is often

described by saying that “the variables have different scales.”) In a previous

example, Euclidean distance was used to measure the distance between

people based on two attributes: age and income. Unless these two attributes

are standardized, the distance between two people will be dominated by

income.

A related issue is how to compute distance when there is correlation between

some of the attributes, perhaps in addition to differences in the ranges of

values. A generalization of Euclidean distance, the Mahalanobis distance, is

useful when attributes are correlated, have different ranges of values

(different variances), and the distribution of the data is approximately

Gaussian (normal). Correlated variables have a large impact on standard

distance measures since a change in any of the correlated variables is

reflected in a change in all the correlated variables. Specifically, the

Mahalanobis distance between two objects (vectors) x and y is defined as

Mahalanobis(x, y)=(x−y)′∑−1(x−y), (2.27)

where ∑−1 is the inverse of the covariance matrix of the data. Note that the

covariance matrix ∑ is the matrix whose ijth entry is the covariance of the ith

and jth attributes as defined by Equation 2.11.

Example 2.26.

In Figure 2.21, there are 1000 points, whose *x *and *y *attributes have a

correlation of 0.6. The distance between the two large points at the opposite

ends of the long axis of the ellipse is 14.7 in terms of Euclidean distance, but

only 6 with respect to Mahalanobis distance. This is because the Mahalanobis

distance gives less emphasis to the direction of largest variance. In practice,

computing the Mahalanobis distance is expensive, but can be worthwhile for

data whose attributes are correlated. If the attributes are relatively

uncorrelated, but have different ranges, then standardizing the variables is

sufficient.

Figure 2.21.

Set of two-dimensional points. The Mahalanobis distance between

the two points represented by large dots is 6; their Euclidean

distance is 14.7.

Combining Similarities for

Heterogeneous Attributes

The previous definitions of similarity were based on approaches that assumed

all the attributes were of the same type. A general approach is needed when

the attributes are of different types. One straightforward approach is to

compute the similarity between each attribute separately using Table 2.7, and

then combine these similarities using a method that results in a similarity

between 0 and 1. One possible approach is to define the overall similarity as

the average of all the individual attribute similarities. Unfortunately, this

approach does not work well if some of the attributes are asymmetric

attributes. For example, if all the attributes are asymmetric binary attributes,

then the similarity measure suggested previously reduces to the simple

matching coefficient, a measure that is not appropriate for asymmetric binary

attributes. The easiest way to fix this problem is to omit asymmetric attributes

from the similarity calculation when their values are 0 for both of the objects

whose similarity is being computed. A similar approach also works well for

handling missing values.

In summary, Algorithm 2.2 is effective for computing an overall similarity

between two objects, x and y, with different types of attributes. This

procedure can be easily modified to work with dissimilarities.

Algorithm 2.2 Similarities of

heterogeneous objects.

1. 1: For the kth attribute, compute a similarity, sk(x, y), in the range [0, 1].

2. 2: Define an indicator variable, δk, for the kth attribute as follows:

δk={

0if the kth attribute is an asymmetric attribute andboth objects have a value of

3. 3: Compute the overall similarity between the two objects using the

following formula:

similarity (x, y)=∑k=1nδksk(x, y)∑k=1nδk (2.28)

Using Weights

In much of the previous discussion, all attributes were treated equally when

computing proximity. This is not desirable when some attributes are more

important to the definition of proximity than others. To address these

situations, the formulas for proximity can be modified by weighting the

contribution of each attribute.

With attribute weights, wk, (2.28) becomes

similarity (x, y)=∑k=1nwkδksk(x, y)∑k=1nwkδk. (2.29)

The definition of the Minkowski distance can also be modified as follows:

d (x, y)=(∑k=1nwk|xk−yk|r)1/r. (2.30)

2.4.10 Selecting the Right Proximity

Measure

A few general observations may be helpful. First, the type of proximity

measure should fit the type of data. For many types of dense, continuous

data, metric distance measures such as Euclidean distance are often used.

Proximity between continuous attributes is most often expressed in terms of

differences, and distance measures provide a well-defined way of combining

these differences into an overall proximity measure. Although attributes can

have different scales and be of differing importance, these issues can often be

dealt with as described earlier, such as normalization and weighting of

attributes.

For sparse data, which often consists of asymmetric attributes, we typically

employ similarity measures that ignore 0–0 matches. Conceptually, this

reflects the fact that, for a pair of complex objects, similarity depends on the

number of characteristics they both share, rather than the number of

characteristics they both lack. The cosine, Jaccard, and extended Jaccard

measures are appropriate for such data.

There are other characteristics of data vectors that often need to be

considered. Invariance to scaling (multiplication) and to translation (addition)

were previously discussed with respect to Euclidean distance and the cosine

and correlation measures. The practical implications of such considerations

are that, for example, cosine is more suitable for sparse document data where

only scaling is important, while correlation works better for time series,

where both scaling and translation are important. Euclidean distance or other

types of Minkowski distance are most appropriate when two data vectors are

to match as closely as possible across all components (features).

In some cases, transformation or normalization of the data is needed to obtain

a proper similarity measure. For instance, time series can have trends or

periodic patterns that significantly impact similarity. Also, a proper

computation of similarity often requires that time lags be taken into account.

Finally, two time series may be similar only over specific periods of time. For

example, there is a strong relationship between temperature and the use of

natural gas, but only during the heating season.

Practical consideration can also be important. Sometimes, one or more

proximity measures are already in use in a particular field, and thus, others

will have answered the question of which proximity measures should be

used. Other times, the software package or clustering algorithm being used

can drastically limit the choices. If efficiency is a concern, then we may want

to choose a proximity measure that has a property, such as the triangle

inequality, that can be used to reduce the number of proximity calculations.

(See Exercise 25.)

However, if common practice or practical restrictions do not dictate a choice,

then the proper choice of a proximity measure can be a time-consuming task

that requires careful consideration of both domain knowledge and the

purpose for which the measure is being used. A number of different similarity

measures may need to be evaluated to see which ones produce results that

make the most sense.

2.5 Bibliographic Notes

It is essential to understand the nature of the data that is being analyzed, and

at a fundamental level, this is the subject of measurement theory. In

particular, one of the initial motivations for defining types of attributes was to

be precise about which statistical operations were valid for what sorts of data.

We have presented the view of measurement theory that was initially

described in a classic paper by S. S. Stevens [112]. (Tables 2.2 and 2.3 are

derived from those presented by Stevens [113].) While this is the most

common view and is reasonably easy to understand and apply, there is, of

course, much more to measurement theory. An authoritative discussion can

be found in a three-volume series on the foundations of measurement theory

[88, 94, 114]. Also of interest is a wide-ranging article by Hand [77], which

discusses measurement theory and statistics, and is accompanied by

comments from other researchers in the field. Numerous critiques and

extensions of the approach of Stevens have been made [66, 97, 117]. Finally,

many books and articles describe measurement issues for particular areas of

science and engineering.

Data quality is a broad subject that spans every discipline that uses data.

Discussions of precision, bias, accuracy, and significant figures can be found

in many introductory science, engineering, and statistics textbooks. The view

of data quality as “fitness for use” is explained in more detail in the book by

Redman [103]. Those interested in data quality may also be interested in

MIT’s Information Quality (MITIQ) Program [95, 118]. However, the

knowledge needed to deal with specific data quality issues in a particular

domain is often best obtained by investigating the data quality practices of

researchers in that field.

Aggregation is a less well-defined subject than many other preprocessing

tasks. However, aggregation is one of the main techniques used by the

database area of Online Analytical Processing (OLAP) [68, 76, 102]. There

has also been relevant work in the area of symbolic data analysis (Bock and

Diday [64]). One of the goals in this area is to summarize traditional record

data in terms of symbolic data objects whose attributes are more complex

than traditional attributes. Specifically, these attributes can have values that

are sets of values (categories), intervals, or sets of values with weights

(histograms). Another goal of symbolic data analysis is to be able to perform

clustering, classification, and other kinds of data analysis on data that consists

of symbolic data objects.

Sampling is a subject that has been well studied in statistics and related

fields. Many introductory statistics books, such as the one by Lindgren [90],

have some discussion about sampling, and entire books are devoted to the

subject, such as the classic text by Cochran [67]. A survey of sampling for

data mining is provided by Gu and Liu [74], while a survey of sampling for

databases is provided by Olken and Rotem [98]. There are a number of other

data mining and database-related sampling references that may be of interest,

including papers by Palmer and Faloutsos [100], Provost et al. [101],

Toivonen [115], and Zaki et al. [119].

In statistics, the traditional techniques that have been used for dimensionality

reduction are multidimensional scaling (MDS) (Borg and Groenen [65],

Kruskal and Uslaner [89]) and principal component analysis (PCA) (Jolliffe

[80]), which is similar to singular value decomposition (SVD) (Demmel

[70]). Dimensionality reduction is discussed in more detail in Appendix B.

Discretization is a topic that has been extensively investigated in data mining.

Some classification algorithms work only with categorical data, and

association analysis requires binary data, and thus, there is a significant

motivation to investigate how to best binarize or discretize continuous

attributes. For association analysis, we refer the reader to work by Srikant

and Agrawal [111], while some useful references for discretization in the area

of classification include work by Dougherty et al. [71], Elomaa and Rousu

[72], Fayyad and Irani [73], and Hussain et al. [78].

Feature selection is another topic well investigated in data mining. A broad

coverage of this topic is provided in a survey by Molina et al. [96] and two

books by Liu and Motada [91, 92]. Other useful papers include those by

Blum and Langley [63], Kohavi and John [87], and Liu et al. [93].

It is difficult to provide references for the subject of feature transformations

because practices vary from one discipline to another. Many statistics books

have a discussion of transformations, but typically the discussion is restricted

to a particular purpose, such as ensuring the normality of a variable or

making sure that variables have equal variance. We offer two references:

Osborne [99] and Tukey [116].

While we have covered some of the most commonly used distance and

similarity measures, there are hundreds of such measures and more are being

created all the time. As with so many other topics in this chapter, many of

these measures are specific to particular fields, e.g., in the area of time series

see papers by Kalpakis et al. [81] and Keogh and Pazzani [83]. Clustering

books provide the best general discussions. In particular, see the books by

Anderberg [62], Jain and Dubes [79], Kaufman and Rousseeuw [82], and

Sneath and Sokal [109].

Information-based measures of similarity have become more popular lately

despite the computational difficulties and expense of calculating them. A

good introduction to information theory is provided by Cover and Thomas

[69]. Computing the mutual information for continuous variables can be

straightforward if they follow a well-know distribution, such as Gaussian.

However, this is often not the case, and many techniques have been

developed. As one example, the article by Khan, et al. [85] compares various

methods in the context of comparing short time series. See also the

information and mutual information packages for R and Matlab. Mutual

information has been the subject of considerable recent attention due to paper

by Reshef, et al. [104, 105] that introduced an alternative measure, albeit one

based on mutual information, which was claimed to have superior properties.

Although this approach had some early support, e.g., [110], others have

pointed out various limitations [75, 86, 108].

Two popular books on the topic of kernel methods are [106] and [107]. The

latter also has a website with links to kernel-related materials [84]. In

addition, many current data mining, machine learning, and statistical learning

textbooks have some material about kernel methods. Further references for

kernel methods in the context of support vector machine classifiers are

provided in the bibliographic Notes of Section 4.9.4.

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2.6 Exercises

1. 1. In the initial example of Chapter 2, the statistician says, “Yes, fields 2

and 3 are basically the same.” Can you tell from the three lines of

sample data that are shown why she says that?

2. 2. Classify the following attributes as binary, discrete, or continuous.

Also classify them as qualitative (nominal or ordinal) or quantitative

(interval or ratio). Some cases may have more than one interpretation, so

briefly indicate your reasoning if you think there may be some

ambiguity.

Example: Age in years. Answer: Discrete, quantitative, ratio

1. Time in terms of AM or PM.

2. Brightness as measured by a light meter.

3. Brightness as measured by people’s judgments.

4. Angles as measured in degrees between 0 and 360.

5. Bronze, Silver, and Gold medals as awarded at the Olympics.

6. Height above sea level.

7. Number of patients in a hospital.

8. ISBN numbers for books. (Look up the format on the Web.)

9. Ability to pass light in terms of the following values: opaque,

translucent, transparent.

10. Military rank.

11. Distance from the center of campus.

12. Density of a substance in grams per cubic centimeter.

13. Coat check number. (When you attend an event, you can often give

your coat to someone who, in turn, gives you a number that you can

use to claim your coat when you leave.)

3. 3. You are approached by the marketing director of a local company,

who believes that he has devised a foolproof way to measure customer

satisfaction. He explains his scheme as follows: “It’s so simple that I

can’t believe that no one has thought of it before. I just keep track of the

number of customer complaints for each product. I read in a data mining

book that counts are ratio attributes, and so, my measure of product

satisfaction must be a ratio attribute. But when I rated the products based

on my new customer satisfaction measure and showed them to my boss,

he told me that I had overlooked the obvious, and that my measure was

worthless. I think that he was just mad because our bestselling product

had the worst satisfaction since it had the most complaints. Could you

help me set him straight?”

1. Who is right, the marketing director or his boss? If you answered,

his boss, what would you do to fix the measure of satisfaction?

2. What can you say about the attribute type of the original product

satisfaction attribute?

4. 4. A few months later, you are again approached by the same marketing

director as in Exercise 3. This time, he has devised a better approach to

measure the extent to which a customer prefers one product over other

similar products. He explains, “When we develop new products, we

typically create several variations and evaluate which one customers

prefer. Our standard procedure is to give our test subjects all of the

product variations at one time and then ask them to rank the product

variations in order of preference. However, our test subjects are very

indecisive, especially when there are more than two products. As a

result, testing takes forever. I suggested that we perform the

comparisons in pairs and then use these comparisons to get the rankings.

Thus, if we have three product variations, we have the customers

compare variations 1 and 2, then 2 and 3, and finally 3 and 1. Our

testing time with my new procedure is a third of what it was for the old

procedure, but the employees conducting the tests complain that they

cannot come up with a consistent ranking from the results. And my boss

wants the latest product evaluations, yesterday. I should also mention

that he was the person who came up with the old product evaluation

approach. Can you help me?”

1. Is the marketing director in trouble? Will his approach work for

generating an ordinal ranking of the product variations in terms of

customer preference? Explain.

2. Is there a way to fix the marketing director’s approach? More

generally, what can you say about trying to create an ordinal

measurement scale based on pairwise comparisons?

3. For the original product evaluation scheme, the overall rankings of

each product variation are found by computing its average over all

test subjects. Comment on whether you think that this is a

reasonable approach. What other approaches might you take?

5. 5. Can you think of a situation in which identification numbers would be

useful for prediction?

6. 6. An educational psychologist wants to use association analysis to

analyze test results. The test consists of 100 questions with four possible

answers each.

1. How would you convert this data into a form suitable for

association analysis?

2. In particular, what type of attributes would you have and how many

of them are there?

7. 7. Which of the following quantities is likely to show more temporal

autocorrelation: daily rainfall or daily temperature? Why?

8. 8. Discuss why a document-term matrix is an example of a data set that

has asymmetric discrete or asymmetric continuous features.

9. 9. Many sciences rely on observation instead of (or in addition to)

designed experiments. Compare the data quality issues involved in

observational science with those of experimental science and data

mining.

10. 10. Discuss the difference between the precision of a measurement and

the terms single and double precision, as they are used in computer

science, typically to represent floating-point numbers that require 32 and

64 bits, respectively.

11. 11. Give at least two advantages to working with data stored in text files

instead of in a binary format.

12. 12. Distinguish between noise and outliers. Be sure to consider the

following questions.

1. Is noise ever interesting or desirable? Outliers?

2. Can noise objects be outliers?

3. Are noise objects always outliers?

4. Are outliers always noise objects?

5. Can noise make a typical value into an unusual one, or vice versa?

Algorithm 2.3 Algorithm for

finding *k*-nearest neighbors.

1. 1: for i=1 to *number of data objects *do

2. 2: Find the distances of the ith object to all other objects.

3. 3: Sort these distances in decreasing order.

(Keep track of which object is associated with each distance.)

4. 4: return the objects associated with the first *k *distances of the

sorted list

5. 5: end for

13. 13. Consider the problem of finding the *K*-nearest neighbors of a data

object. A programmer designs Algorithm 2.3 for this task.

1. Describe the potential problems with this algorithm if there are

duplicate objects in the data set. Assume the distance function will

return a distance of 0 only for objects that are the same.

2. How would you fix this problem?

14. 14. The following attributes are measured for members of a herd of

Asian elephants: *weight, height, tusk length, trunk length, *and *ear area*.

Based on these measurements, what sort of proximity measure from

Section 2.4 would you use to compare or group these elephants? Justify

your answer and explain any special circumstances.

15. 15. You are given a set of *m *objects that is divided into *k *groups, where

the *i**th *group is of size mi. If the goal is to obtain a sample of size n<m,

what is the difference between the following two sampling schemes?

(Assume sampling with replacement.)

1. We randomly select n×mi/m elements from each group.

2. We randomly select *n *elements from the data set, without regard

for the group to which an object belongs.

16. 16. Consider a document-term matrix, where tfij is the frequency of the

ith word (term) in the jth document and *m *is the number of documents.

Consider the variable transformation that is defined by

tfij′=tfij×logmdfi, (2.31)

where dfi is the number of documents in which the ith term appears,

which is known as the document frequency of the term. This

transformation is known as the inverse document frequency

transformation.

1. What is the effect of this transformation if a term occurs in one

document? In every document?

2. What might be the purpose of this transformation?

17. 17. Assume that we apply a square root transformation to a ratio

attribute *x *to obtain the new attribute x*. As part of your analysis, you

identify an interval (*a*, *b*) in which x* has a linear relationship to another

attribute*y*.

1. What is the corresponding interval (*A*, *B*) in terms of *x *?

2. Give an equation that relates *y *to *x*.

18. 18. This exercise compares and contrasts some similarity and distance

measures.

1. For binary data, the L1 distance corresponds to the Hamming

distance; that is, the number of bits that are different between two

binary vectors. The Jaccard similarity is a measure of the similarity

between two binary vectors. Compute the Hamming distance and

the Jaccard similarity between the following two binary vectors.

x=0101010001y=0100011000

2. Which approach, Jaccard or Hamming distance, is more similar to

the Simple Matching Coefficient, and which approach is more

similar to the cosine measure? Explain. (Note: The Hamming

measure is a distance, while the other three measures are

similarities, but don’t let this confuse you.)

3. Suppose that you are comparing how similar two organisms of

different species are in terms of the number of genes they share.

Describe which measure, Hamming or Jaccard, you think would be

more appropriate for comparing the genetic makeup of two

organisms. Explain. (Assume that each animal is represented as a

binary vector, where each attribute is 1 if a particular gene is

present in the organism and 0 otherwise.)

4. If you wanted to compare the genetic makeup of two organisms of

the same species, e.g., two human beings, would you use the

Hamming distance, the Jaccard coefficient, or a different measure

of similarity or distance? Explain. (Note that two human beings

share >99.9% of the same genes.)

19. 19. For the following vectors, x and y, calculate the indicated similarity

or distance measures.

1. x=(1, 1, 1, 1), y=(2, 2, 2, 2) cosine, correlation, Euclidean

2. x=(0, 1, 0, 1), y=(1, 0, 1, 0) cosine, correlation, Euclidean, Jaccard

3. x=(0, −1, 0, 1), y=(1, 0, −1, 0) cosine, correlation, Euclidean

4. x=(1, 1, 0, 1, 0, 1), y=(1, 1, 1, 0, 0, 1) cosine, correlation, Jaccard

5. x=(2, −1, 0, 2, 0, −3), y=( −1, 1, −1, 0, 0, −1) cosine, correlation

20. 20. Here, we further explore the cosine and correlation measures.

1. What is the range of values possible for the cosine measure?

2. If two objects have a cosine measure of 1, are they identical?

Explain.

3. What is the relationship of the cosine measure to correlation, if

any? (Hint: Look at statistical measures such as mean and standard

deviation in cases where cosine and correlation are the same and

different.)

4. Figure 2.22(a) shows the relationship of the cosine measure to

Euclidean distance for 100,000 randomly generated points that

have been normalized to have an L2 length of 1. What general

observation can you make about the relationship between

Euclidean distance and cosine similarity when vectors have an L2

norm of 1?

Figure 2.22.

Graphs for Exercise 20.

Figure 2.22. Full Alternative Text

5. Figure 2.22(b) shows the relationship of correlation to Euclidean

distance for 100,000 randomly generated points that have been

standardized to have a mean of 0 and a standard deviation of 1.

What general observation can you make about the relationship

between Euclidean distance and correlation when the vectors have

been standardized to have a mean of 0 and a standard deviation of

1?

6. Derive the mathematical relationship between cosine similarity and

Euclidean distance when each data object has an L2 length of 1.

7. Derive the mathematical relationship between correlation and

Euclidean distance when each data point has been been

standardized by subtracting its mean and dividing by its standard

deviation.

21. 21. Show that the set difference metric given by

d(A, B)=size(A−B)+size(B−A) (2.32)

satisfies the metric axioms given on page 77. *A *and *B *are sets and A−B

is the set difference.

22. 22. Discuss how you might map correlation values from the interval

[−1, 1] to the interval [0, 1]. Note that the type of transformation that

you use might depend on the application that you have in mind. Thus,

consider two applications: clustering time series and predicting the

behavior of one time series given another.

23. 23. Given a similarity measure with values in the interval [0, 1], describe

two ways to transform this similarity value into a dissimilarity value in

the interval [0, ∞].

24. 24. Proximity is typically defined between a pair of objects.

1. Define two ways in which you might define the proximity among a

group of objects.

2. How might you define the distance between two sets of points in

Euclidean space?

3. How might you define the proximity between two sets of data

objects? (Make no assumption about the data objects, except that a

proximity measure is defined between any pair of objects.)

25. 25. You are given a set of points *s *in Euclidean space, as well as the

distance of each point in *s *to a point x. (It does not matter if x∈S.)

1. If the goal is to find all points within a specified distance ε of point

y, y≠x, explain how you could use the triangle inequality and the

already calculated distances to x to potentially reduce the number

of distance calculations necessary? Hint: The triangle inequality,

d(x, z)≤d(x, y)+d(y, x), can be rewritten as d(x, y)≥d(x, z)−d(y, z).

2. In general, how would the distance between x and y affect the

number of distance calculations?

3. Suppose that you can find a small subset of points S′, from the

original data set, such that every point in the data set is within a

specified distance ε of at least one of the points in S′, and that you

also have the pairwise distance matrix for S′. Describe a technique

that uses this information to compute, with a minimum of distance

calculations, the set of all points within a distance of β of a

specified point from the data set.

26. 26. Show that 1 minus the Jaccard similarity is a distance measure

between two data objects, x and y, that satisfies the metric axioms given

on page 77. Specifically, d(x, y)=1−J(x, y).

27. 27. Show that the distance measure defined as the angle between two

data vectors, x and y, satisfies the metric axioms given on page 77.

Specifically, d(x, y)=arccos(cos(x, y)).

28. 28. Explain why computing the proximity between two attributes is

often simpler than computing the similarity between two objects.

3 Classification: Basic Concepts and

Techniques

Humans have an innate ability to classify things into categories, e.g.,

mundane tasks such as filtering spam email messages or more specialized

tasks such as recognizing celestial objects in telescope images (see Figure

3.1). While manual classification often suffices for small and simple data sets

with only a few attributes, larger and more complex data sets require an

automated solution.

Figure 3.1.

Classification of galaxies from telescope images taken from the

NASA website.

This chapter introduces the basic concepts of classification and describes

some of its key issues such as model overfitting, model selection, and model

evaluation. While these topics are illustrated using a classification technique

known as decision tree induction, most of the discussion in this chapter is

also applicable to other classification techniques, many of which are covered

in Chapter 4.

3.1 Basic Concepts

Figure 3.2 illustrates the general idea behind classification. The data for a

classification task consists of a collection of instances (records). Each such

instance is characterized by the tuple (x, *y*), where x is the set of attribute

values that describe the instance and *y *is the class label of the instance. The

attribute set x can contain attributes of any type, while the class label *y *must

be categorical.

Figure 3.2.

A schematic illustration of a classification task.

A classification model is an abstract representation of the relationship

between the attribute set and the class label. As will be seen in the next two

chapters, the model can be represented in many ways, e.g., as a tree, a

probability table, or simply, a vector of real-valued parameters. More

formally, we can express it mathematically as a target function *f *that takes as

input the attribute set x and produces an output corresponding to the predicted

class label. The model is said to classify an instance (x, *y*) correctly if f(x)=y.

Table 3.1 shows examples of attribute sets and class labels for various

classification tasks. Spam filtering and tumor identification are examples of

binary classification problems, in which each data instance can be

categorized into one of two classes. If the number of classes is larger than 2,

as in the galaxy classification example, then it is called a multiclass

classification problem.

Table 3.1. Examples of

classification tasks.

Task Attribute set Class label

Spam

filtering

Features extracted from email

message header and content spam or non-spam

Tumor

identification

Features extracted from magnetic

resonance imaging (MRI) scans

malignant or

benign

Galaxy

classification

Features extracted from telescope

images

elliptical, spiral,

or irregularshaped

We illustrate the basic concepts of classification in this chapter with the

following two examples.

3.1. Example Vertebrate

Classification

Table 3.2 shows a sample data set for classifying vertebrates into mammals,

reptiles, birds, fishes, and amphibians. The attribute set includes

characteristics of the vertebrate such as its body temperature, skin cover, and

ability to fly. The data set can also be used for a binary classification task

such as mammal classification, by grouping the reptiles, birds, fishes, and

amphibians into a single category called non-mammals.

Table 3.2. A sample data for

the vertebrate classification

problem.

Vertebrate

Name

Body

Temperature

Skin

Cover

Gives

Birth

Aquatic

Creature

Aerial

Creature

Has

Legs Hibernates

human warmblooded hair yes no no yes no

python cold-blooded scales no no no no yes

salmon cold-blooded scales no yes no no no

whale warmblooded hair yes yes no no no

frog cold-blooded none no semi no yes yes

komodo cold-blooded scales no no no yes no

dragon

bat warmblooded hair yes no yes yes yes

pigeon warmblooded feathers no no yes yes no

cat warmblooded fur yes no no yes no

leopard cold-blooded scales yes yes no no no

shark

turtle cold-blooded scales no semi no yes no

penguin warmblooded feathers no semi no yes no

porcupine warmblooded quills yes no no yes yes

eel cold-blooded scales no yes no no no

salamander cold-blooded none no semi no yes yes

3.2. Example Loan Borrower

Classification

Consider the problem of predicting whether a loan borrower will repay the

loan or default on the loan payments. The data set used to build the

classification model is shown in Table 3.3. The attribute set includes personal

information of the borrower such as marital status and annual income, while

the class label indicates whether the borrower had defaulted on the loan

payments.

Table 3.3. A sample data for

the loan borrower classification

problem.

ID Home Owner Marital Status Annual Income Defaulted?

1 Yes Single 125000 No

2 No Married 100000 No

3 No Single 70000 No

4 Yes Married 120000 No

5 No Divorced 95000 Yes

6 No Single 60000 No

7 Yes Divorced 220000 No

8 No Single 85000 Yes

9 No Married 75000 No

10 No Single 90000 Yes

A classification model serves two important roles in data mining. First, it is

used as a predictive model to classify previously unlabeled instances. A

good classification model must provide accurate predictions with a fast

response time. Second, it serves as a descriptive model to identify the

characteristics that distinguish instances from different classes. This is

particularly useful for critical applications, such as medical diagnosis, where

it is insufficient to have a model that makes a prediction without justifying

how it reaches such a decision.

For example, a classification model induced from the vertebrate data set

shown in Table 3.2 can be used to predict the class label of the following

vertebrate:

Vertebrate

Name

Body

Temperature

Skin

Cover

Gives

Birth

Aquatic

Creature

Aerial

Creature

Has

Legs Hibernates

gila

monster

cold-blooded scales no no no yes yes

In addition, it can be used as a descriptive model to help determine

characteristics that define a vertebrate as a mammal, a reptile, a bird, a fish,

or an amphibian. For example, the model may identify mammals as warmblooded vertebrates that give birth to their young.

There are several points worth noting regarding the previous example. First,

although all the attributes shown in Table 3.2 are qualitative, there are no

restrictions on the type of attributes that can be used as predictor variables.

The class label, on the other hand, must be of nominal type. This

distinguishes classification from other predictive modeling tasks such as

regression, where the predicted value is often quantitative. More information

about regression can be found in Appendix D.

Another point worth noting is that not all attributes may be relevant to the

classification task. For example, the average length or weight of a vertebrate

may not be useful for classifying mammals, as these attributes can show same

value for both mammals and non-mammals. Such an attribute is typically

discarded during preprocessing. The remaining attributes might not be able to

distinguish the classes by themselves, and thus, must be used in concert with

other attributes. For instance, the Body Temperature attribute is insufficient

to distinguish mammals from other vertebrates. When it is used together with

Gives Birth, the classification of mammals improves significantly. However,

when additional attributes, such as Skin Cover are included, the model

becomes overly specific and no longer covers all mammals. Finding the

optimal combination of attributes that best discriminates instances from

different classes is the key challenge in building classification models.

3.2 General Framework for

Classification

Classification is the task of assigning labels to unlabeled data instances and a

classifier is used to perform such a task. A classifier is typically described in

terms of a model as illustrated in the previous section. The model is created

using a given a set of instances, known as the training set, which contains

attribute values as well as class labels for each instance. The systematic

approach for learning a classification model given a training set is known as a

learning algorithm. The process of using a learning algorithm to build a

classification model from the training data is known as induction. This

process is also often described as “learning a model” or “building a model.”

This process of applying a classification model on unseen test instances to

predict their class labels is known as deduction. Thus, the process of

classification involves two steps: applying a learning algorithm to training

data to learn a model, and then applying the model to assign labels to

unlabeled instances. Figure 3.3 illustrates the general framework for

classification.

Figure 3.3.

General framework for building a classification model.

Figure 3.3. Full Alternative Text

A classification technique refers to a general approach to classification, e.g.,

the decision tree technique that we will study in this chapter. This

classification technique like most others, consists of a family of related

models and a number of algorithms for learning these models. In Chapter 4,

we will study additional classification techniques, including neural networks

and support vector machines.

A couple notes on terminology. First, the terms “classifier” and “model” are

often taken to be synonymous. If a classification technique builds a single,

global model, then this is fine. However, while every model defines a

classifier, not every classifier is defined by a single model. Some classifiers,

such as *k-*nearest neighbor classifiers, do not build an explicit model (Section

4.3), while other classifiers, such as ensemble classifiers, combine the output

of a collection of models (Section 4.10). Second, the term “classifier” is often

used in a more general sense to refer to a classification technique. Thus, for

example, “decision tree classifier” can refer to the decision tree classification

technique or a specific classifier built using that technique. Fortunately, the

meaning of “classifier” is usually clear from the context.

In the general framework shown in Figure 3.3, the induction and deduction

steps should be performed separately. In fact, as will be discussed later in

Section 3.6, the training and test sets should be independent of each other to

ensure that the induced model can accurately predict the class labels of

instances it has never encountered before. Models that deliver such predictive

insights are said to have good generalization performance. The

performance of a model (classifier) can be evaluated by comparing the

predicted labels against the true labels of instances. This information can be

summarized in a table called a confusion matrix. Table 3.4 depicts the

confusion matrix for a binary classification problem. Each entry fij denotes

the number of instances from class *i *predicted to be of class *j*. For example,

f01 is the number of instances from class 0 incorrectly predicted as class 1.

The number of correct predictions made by the model is (f11+f00) and the

number of incorrect predictions is (f10+f01).

Table 3.4. Confusion matrix for

a binary classification problem.

Predicted Class

Class=1 Class=0

Actual Class Class=1 f11 f10

Class=0 f01 f00

Although a confusion matrix provides the information needed to determine

how well a classification model performs, summarizing this information into

a single number makes it more convenient to compare the relative

performance of different models. This can be done using an evaluation

metric such as accuracy, which is computed in the following way:

Accuracy =

Accuracy=Number of correct predictionsTotal number of predictions. (3.1)

For binary classification problems, the accuracy of a model is given by

Accuracy=f11+f00f11+f10+f01+f00. (3.2)

Error rate is another related metric, which is defined as follows for binary

classification problems:

Error rate=Number of wrong predictionsTotal number of predictions=f10+f01

(3.3)

The learning algorithms of most classification techniques are designed to

learn models that attain the highest accuracy, or equivalently, the lowest error

rate when applied to the test set. We will revisit the topic of model evaluation

in Section 3.6.

3.3 Decision Tree Classifier

This section introduces a simple classification technique known as the

decision tree classifier. To illustrate how a decision tree works, consider the

classification problem of distinguishing mammals from non-mammals using

the vertebrate data set shown in Table 3.2. Suppose a new species is

discovered by scientists. How can we tell whether it is a mammal or a nonmammal? One approach is to pose a series of questions about the

characteristics of the species. The first question we may ask is whether the

species is cold- or warm-blooded. If it is cold-blooded, then it is definitely

not a mammal. Otherwise, it is either a bird or a mammal. In the latter case,

we need to ask a follow-up question: Do the females of the species give birth

to their young? Those that do give birth are definitely mammals, while those

that do not are likely to be non-mammals (with the exception of egg-laying

mammals such as the platypus and spiny anteater).

The previous example illustrates how we can solve a classification problem

by asking a series of carefully crafted questions about the attributes of the test

instance. Each time we receive an answer, we could ask a follow-up question

until we can conclusively decide on its class label. The series of questions

and their possible answers can be organized into a hierarchical structure

called a decision tree. Figure 3.4 shows an example of the decision tree for

the mammal classification problem. The tree has three types of nodes:

A root node, with no incoming links and zero or more outgoing links.

Internal nodes, each of which has exactly one incoming link and two or

more outgoing links.

Leaf or terminal nodes, each of which has exactly one incoming link

and no outgoing links.

Every leaf node in the decision tree is associated with a class label. The nonterminal nodes, which include the root and internal nodes, contain attribute

test conditions that are typically defined using a single attribute. Each

possible outcome of the attribute test condition is associated with exactly one

child of this node. For example, the root node of the tree shown in Figure 3.4

uses the attribute Body Temperature to define an attribute test condition that

has two outcomes, warm and cold, resulting in two child nodes.

Figure 3.4.

A decision tree for the mammal classification problem.

Figure 3.4. Full Alternative Text

Given a decision tree, classifying a test instance is straightforward. Starting

from the root node, we apply its attribute test condition and follow the

appropriate branch based on the outcome of the test. This will lead us either

to another internal node, for which a new attribute test condition is applied, or

to a leaf node. Once a leaf node is reached, we assign the class label

associated with the node to the test instance. As an illustration, Figure 3.5

traces the path used to predict the class label of a flamingo. The path

terminates at a leaf node labeled as Non-mammals.

Figure 3.5.

Classifying an unlabeled vertebrate. The dashed lines represent the

outcomes of applying various attribute test conditions on the

unlabeled vertebrate. The vertebrate is eventually assigned to the

Non-mammals class.

Figure 3.5. Full Alternative Text

3.3.1 A Basic Algorithm to Build a

Decision Tree

Many possible decision trees that can be constructed from a particular data

set. While some trees are better than others, finding an optimal one is

computationally expensive due to the exponential size of the search space.

Efficient algorithms have been developed to induce a reasonably accurate,

albeit suboptimal, decision tree in a reasonable amount of time. These

algorithms usually employ a greedy strategy to grow the decision tree in a

top-down fashion by making a series of locally optimal decisions about

which attribute to use when partitioning the training data. One of the earliest

method is Hunt’s algorithm, which is the basis for many current

implementations of decision tree classifiers, including ID3, C4.5, and CART.

This subsection presents Hunt’s algorithm and describes some of the design

issues that must be considered when building a decision tree.

Hunt’s Algorithm

In Hunt’s algorithm, a decision tree is grown in a recursive fashion. The tree

initially contains a single root node that is associated with all the training

instances. If a node is associated with instances from more than one class, it

is expanded using an attribute test condition that is determined using a

splitting criterion. A child leaf node is created for each outcome of the

attribute test condition and the instances associated with the parent node are

distributed to the children based on the test outcomes. This node expansion

step can then be recursively applied to each child node, as long as it has

labels of more than one class. If all the instances associated with a leaf node

have identical class labels, then the node is not expanded any further. Each

leaf node is assigned a class label that occurs most frequently in the training

instances associated with the node.

To illustrate how the algorithm works, consider the training set shown in

Table 3.3 for the loan borrower classification problem. Suppose we apply

Hunt’s algorithm to fit the training data. The tree initially contains only a

single leaf node as shown in Figure 3.6(a). This node is labeled as Defaulted

= No, since the majority of the borrowers did not default on their loan

payments. The training error of this tree is 30% as three out of the ten

training instances have the class label Defaulted = Yes. The leaf node can

therefore be further expanded because it contains training instances from

more than one class.

Figure 3.6.

Hunt’s algorithm for building decision trees.

Figure 3.6. Full Alternative Text

Let Home Owner be the attribute chosen to split the training instances. The

justification for choosing this attribute as the attribute test condition will be

discussed later. The resulting binary split on the Home Owner attribute is

shown in Figure 3.6(b). All the training instances for which Home Owner =

Yes are propagated to the left child of the root node and the rest are

propagated to the right child. Hunt’s algorithm is then recursively applied to

each child. The left child becomes a leaf node labeled Defaulted = No, since

all instances associated with this node have identical class label

Defaulted = No. The right child has instances from each class label. Hence,

we split it further. The resulting subtrees after recursively expanding the right

child are shown in Figures 3.6(c) and (d).

Hunt’s algorithm, as described above, makes some simplifying assumptions

that are often not true in practice. In the following, we describe these

assumptions and briefly discuss some of the possible ways for handling them.

1. Some of the child nodes created in Hunt’s algorithm can be empty if

none of the training instances have the particular attribute values. One

way to handle this is by declaring each of them as a leaf node with a

class label that occurs most frequently among the training instances

associated with their parent nodes.

2. If all training instances associated with a node have identical attribute

values but different class labels, it is not possible to expand this node

any further. One way to handle this case is to declare it a leaf node and

assign it the class label that occurs most frequently in the training

instances associated with this node.

Design Issues of Decision Tree

Induction

Hunt’s algorithm is a generic procedure for growing decision trees in a greedy

fashion. To implement the algorithm, there are two key design issues that

must be addressed.

1. What is the splitting criterion? At each recursive step, an attribute must

be selected to partition the training instances associated with a node into

smaller subsets associated with its child nodes. The splitting criterion

determines which attribute is chosen as the test condition and how the

training instances should be distributed to the child nodes. This will be

discussed in Sections 3.3.2 and 3.3.3.

2. What is the stopping criterion? The basic algorithm stops expanding a

node only when all the training instances associated with the node have

the same class labels or have identical attribute values. Although these

conditions are sufficient, there are reasons to stop expanding a node

much earlier even if the leaf node contains training instances from more

than one class. This process is called early termination and the condition

used to determine when a node should be stopped from expanding is

called a stopping criterion. The advantages of early termination are

discussed in Section 3.4.

3.3.2 Methods for Expressing

Attribute Test Conditions

Decision tree induction algorithms must provide a method for expressing an

attribute test condition and its corresponding outcomes for different attribute

types.

Binary Attributes

The test condition for a binary attribute generates two potential outcomes, as

shown in Figure 3.7.

Figure 3.7.

Attribute test condition for a binary attribute.

Nominal Attributes

Since a nominal attribute can have many values, its attribute test condition

can be expressed in two ways, as a multiway split or a binary split as shown

in Figure 3.8. For a multiway split (Figure 3.8(a)), the number of outcomes

depends on the number of distinct values for the corresponding attribute. For

example, if an attribute such as marital status has three distinct values—

single, married, or divorced—its test condition will produce a three-way split.

It is also possible to create a binary split by partitioning all values taken by

the nominal attribute into two groups. For example, some decision tree

algorithms, such as CART, produce only binary splits by considering all 2k

−1−1 ways of creating a binary partition of *k *attribute values. Figure 3.8(b)

illustrates three different ways of grouping the attribute values for marital

status into two subsets.

Figure 3.8.

Attribute test conditions for nominal attributes.

Figure 3.8. Full Alternative Text

Ordinal Attributes

Ordinal attributes can also produce binary or multi-way splits. Ordinal

attribute values can be grouped as long as the grouping does not violate the

order property of the attribute values. Figure 3.9 illustrates various ways of

splitting training records based on the Shirt Size attribute. The groupings

shown in Figures 3.9(a) and (b) preserve the order among the attribute values,

whereas the grouping shown in Figure 3.9(c) violates this property because it

combines the attribute values Small and Large into the same partition while

Medium and Extra Large are combined into another partition.

Figure 3.9.

Different ways of grouping ordinal attribute values.

Figure 3.9. Full Alternative Text

Continuous Attributes

For continuous attributes, the attribute test condition can be expressed as a

comparison test (e.g., A<v) producing a binary split, or as a range query of

the form vi≤A<vi+1, for i=1, …, k, producing a multiway split. The

difference between these approaches is shown in Figure 3.10. For the binary

split, any possible value *v *between the minimum and maximum attribute

values in the training data can be used for constructing the comparison test

A<v. However, it is sufficient to only consider distinct attribute values in the

training set as candidate split positions. For the multiway split, any possible

collection of attribute value ranges can be used, as long as they are mutually

exclusive and cover the entire range of attribute values between the minimum

and maximum values observed in the training set. One approach for

constructing multiway splits is to apply the discretization strategies described

in Section 2.3.6 on page 63. After discretization, a new ordinal value is

assigned to each discretized interval, and the attribute test condition is then

defined using this newly constructed ordinal attribute.

Figure 3.10.

Test condition for continuous attributes.

Figure 3.10. Full Alternative Text

3.3.3 Measures for Selecting an

Attribute Test Condition

There are many measures that can be used to determine the goodness of an

attribute test condition. These measures try to give preference to attribute test

conditions that partition the training instances into *purer *subsets in the child

nodes, which mostly have the same class labels. Having purer nodes is useful

since a node that has all of its training instances from the same class does not

need to be expanded further. In contrast, an impure node containing training

instances from multiple classes is likely to require several levels of node

expansions, thereby increasing the depth of the tree considerably. Larger trees

are less desirable as they are more susceptible to model overfitting, a

condition that may degrade the classification performance on unseen

instances, as will be discussed in Section 3.4. They are also difficult to

interpret and incur more training and test time as compared to smaller trees.

In the following, we present different ways of measuring the impurity of a

node and the collective impurity of its child nodes, both of which will be used

to identify the best attribute test condition for a node.

Impurity Measure for a Single Node

The impurity of a node measures how dissimilar the class labels are for the

data instances belonging to a common node. Following are examples of

measures that can be used to evaluate the impurity of a node *t*:

Entropy=−∑i=0c−1pi(t) log2pi(t), (3.4)

Gini index=1−∑i=0c−1pi(t)2, (3.5)

Classification error=1−maxi[pi(t)], (3.6)

where *pi*(*t*) is the relative frequency of training instances that belong to class *i*

at node *t*, *c *is the total number of classes, and 0 log2 0=0 in entropy

calculations. All three measures give a zero impurity value if a node contains

instances from a single class and maximum impurity if the node has equal

proportion of instances from multiple classes.

Figure 3.11 compares the relative magnitude of the impurity measures when

applied to binary classification problems. Since there are only two classes,

p0(t)+p1(t)=1. The horizontal axis *p *refers to the fraction of instances that

belong to one of the two classes. Observe that all three measures attain their

maximum value when the class distribution is uniform (i.e., p0(t)+p1(t)=0.5)

and minimum value when all the instances belong to a single class (i.e., either

p0(t) or p1(t) equals to 1). The following examples illustrate how the values

of the impurity measures vary as we alter the class distribution.

Figure 3.11.

Comparison among the impurity measures for binary classification

problems.

Figure 3.11. Full Alternative Text

Node N1 Count Gini=1−(0/6)2−(6/6)2=0

Class=0 0 Entropy=−(0/6) log2(0/6)−(6/6) log2(6/6)=0

Class=1 6 Error=1−max[0/6, 6/6]=0

Node N2 Count Gini=1−(1/6)2−(5/6)2=0.278

Class=0 1 Entropy=−(1/6) log2(1/6)−(5/6) log2(5/6)=0.650

Class=1 5 Error=1−max[1/6, 5/6]=0.167

Node N3 Count Gini=1−(3/6)2−(3/6)2=0.5

Class=0 3 Entropy=−(3/6) log2(3/6)−(3/6) log2(3/6)=1

Class=1 3 Error=1−max[6/6, 3/6]=0.5

Based on these calculations, node N1 has the lowest impurity value, followed

by N2 and N3. This example, along with Figure 3.11, shows the consistency

among the impurity measures, i.e., if a node N1 has lower entropy than node

N2, then the Gini index and error rate of N1 will also be lower than that of

N2. Despite their agreement, the attribute chosen as splitting criterion by the

impurity measures can still be different (see Exercise 6 on page 187).

Collective Impurity of Child Nodes

Consider an attribute test condition that splits a node containing *N *training

instances into *k *children, {v1, v2, ⋯ ,vk}, where every child node represents a

partition of the data resulting from one of the *k *outcomes of the attribute test

condition. Let N(vj) be the number of training instances associated with a

child node vj, whose impurity value is I(vj). Since a training instance in the

parent node reaches node vj for a fraction of N(vj)/N times, the collective

impurity of the child nodes can be computed by taking a weighted sum of the

impurities of the child nodes, as follows:

I(children)=∑j=1kN(vj)NI(vj), (3.7)

3.3. Example Weighted Entropy

Consider the candidate attribute test condition shown in Figures 3.12(a) and

(b) for the loan borrower classification problem. Splitting on the Home

Owner attribute will generate two child nodes

Figure 3.12.

Examples of candidate attribute test conditions.

Figure 3.12. Full Alternative Text

whose weighted entropy can be calculated as follows:

I(Home Owner=yes)=03log203−33log233=0I(Home Owner=no)=

−37log237−47log247=0.985I(Home Owner=310×0+710×0.985=0.690

Splitting on Marital Status, on the other hand, leads to three child nodes with

a weighted entropy given by

I(Marital Status=Single)=

−25log225−35log235=0.971I(Marital Status=Married)=

−03log203−33log233=0I(Marital Status=Divorced)=

−12log212−12log212=1.000I(Marital Status)=510×0.971+310×0+210×1=0.686

Thus, Marital Status has a lower weighted entropy than Home Owner.

Identifying the best attribute test

condition

To determine the goodness of an attribute test condition, we need to compare

the degree of impurity of the parent node (before splitting) with the weighted

degree of impurity of the child nodes (after splitting). The larger their

difference, the better the test condition. This difference, Δ, also termed as the

gain in purity of an attribute test condition, can be defined as follows:

Δ=I(parent)−I(children), (3.8)

Figure 3.13.

Splitting criteria for the loan borrower classification problem using

Gini index.

Figure 3.13. Full Alternative Text

where *I*(parent) is the impurity of a node before splitting and *I*(children) is the

weighted impurity measure after splitting. It can be shown that the gain is

non-negative since I(parent)≥I(children) for any reasonable measure such as

those presented above. The higher the gain, the purer are the classes in the

child nodes relative to the parent node. The splitting criterion in the decision

tree learning algorithm selects the attribute test condition that shows the

maximum gain. Note that maximizing the gain at a given node is equivalent

to minimizing the weighted impurity measure of its children since *I*(parent) is

the same for all candidate attribute test conditions. Finally, when entropy is

used as the impurity measure, the difference in entropy is commonly known

as information gain, Δinfo.

In the following, we present illustrative approaches for identifying the best

attribute test condition given qualitative or quantitative attributes.

Splitting of Qualitative Attributes

Consider the first two candidate splits shown in Figure 3.12 involving

qualitative attributes Home Owner and Marital Status. The initial class

distribution at the parent node is (0.3, 0.7), since there are 3 instances of class

Yes and 7 instances of class No in the training data. Thus,

I(parent)=−310log2310−710log2710=0.881

The information gains for Home Owner and Marital Status are each given by

Δinfo(Home Owner)=0.881−0.690=0.191Δinfo(Marital Status)=0.881−0.686=

The information gain for Marital Status is thus higher due to its lower

weighted entropy, which will thus be considered for splitting.

Binary Splitting of Qualitative

Attributes

Consider building a decision tree using only binary splits and the Gini index

as the impurity measure. Figure 3.13 shows examples of four candidate

splitting criteria for the Home Owner and Marital Status attributes. Since

there are 3 borrowers in the training set who defaulted and 7 others who

repaid their loan (see Table in Figure 3.13), the Gini index of the parent node

before splitting is

1−(310)2−(710)2=0.420.

If Home Owner is chosen as the splitting attribute, the Gini index for the child

nodes N1 and N2 are 0 and 0.490, respectively. The weighted average Gini

index for the children is

(3/10)×0+(7/10)×0.490=0.343,

where the weights represent the proportion of training instances assigned to

each child. The gain using Home Owner as splitting attribute is

0.420−0.343=0.077. Similarly, we can apply a binary split on the Marital

Status attribute. However, since Marital Status is a nominal attribute with

three outcomes, there are three possible ways to group the attribute values

into a binary split. The weighted average Gini index of the children for each

candidate binary split is shown in Figure 3.13. Based on these results, Home

Owner and the last binary split using Marital Status are clearly the best

candidates, since they both produce the lowest weighted average Gini index.

Binary splits can also be used for ordinal attributes, if the binary partitioning

of the attribute values does not violate the ordering property of the values.

Binary Splitting of Quantitative

Attributes

Consider the problem of identifying the best binary split Annual Income≤τ

for the preceding loan approval classification problem. As discussed

previously, even though τ can take any value between the minimum and

maximum values of annual income in the training set, it is sufficient to only

consider the annual income values observed in the training set as candidate

split positions. For each candidate τ, the training set is scanned once to count

the number of borrowers with annual income less than or greater than τ along

with their class proportions. We can then compute the Gini index at each

candidate split position and choose the τ that produces the lowest value.

Computing the Gini index at each candidate split position requires *O*(*N*)

operations, where *N *is the number of training instances. Since there are at

most *N *possible candidates, the overall complexity of this brute-force method

is O(N2). It is possible to reduce the complexity of this problem to *O*(*N *log

*N*) by using a method described as follows (see illustration in Figure 3.14). In

this method, we first sort the training instances based on their annual income,

a one-time cost that requires *O*(*N *log *N*) operations. The candidate split

positions are given by the midpoints between every two adjacent sorted

values: $55,000, $65,000, $72,500, and so on. For the first candidate, since

none of the instances has an annual income less than or equal to $55,000, the

Gini index for the child node with Annual Income< $55,000 is equal to zero.

In contrast, there are 3 training instances of class Yes and 7 instances of class

No with annual income greater than $55,000. The Gini index for this node is

0.420. The weighted average Gini index for the first candidate split position,

τ=$55,000, is equal to 0×0+1×0.420=0.420.

Figure 3.14.

Splitting continuous attributes.

Figure 3.14. Full Alternative Text

For the next candidate, τ=$65,000, the class distribution of its child nodes can

be obtained with a simple update of the distribution for the previous

candidate. This is because, as τ increases from $55,000 to $65,000, there is

only one training instance affected by the change. By examining the class

label of the affected training instance, the new class distribution is obtained.

For example, as τ increases to $65,000, there is only one borrower in the

training set, with an annual income of $60,000, affected by this change. Since

the class label for the borrower is No, the count for class No increases from 0

to 1 (for Annual Income≤$65,000) and decreases from 7 to 6 (for

Annual Income>$65,000), as shown in Figure 3.14. The distribution for the

Yes class remains unaffected. The updated Gini index for this candidate split

position is 0.400.

This procedure is repeated until the Gini index for all candidates are found.

The best split position corresponds to the one that produces the lowest Gini

index, which occurs at τ=$97,500. Since the Gini index at each candidate

split position can be computed in *O*(1) time, the complexity of finding the

best split position is *O*(*N*) once all the values are kept sorted, a one-time

operation that takes *O*(*N *log *N*) time. The overall complexity of this method

is thus *O*(*N *log *N*), which is much smaller than the O(N2) time taken by the

brute-force method. The amount of computation can be further reduced by

considering only candidate split positions located between two adjacent

sorted instances with different class labels. For example, we do not need to

consider candidate split positions located between $60,000 and $75,000

because all three instances with annual income in this range ($60,000,

$70,000, and $75,000) have the same class labels. Choosing a split position

within this range only increases the degree of impurity, compared to a split

position located outside this range. Therefore, the candidate split positions at

τ=$65,000 and τ=$72,500 can be ignored. Similarly, we do not need to

consider the candidate split positions at $87,500, $92,500, $110,000,

$122,500, and $172,500 because they are located between two adjacent

instances with the same labels. This strategy reduces the number of candidate

split positions to consider from 9 to 2 (excluding the two boundary cases

τ=$55,000 and τ=$230,000).

Gain Ratio

One potential limitation of impurity measures such as entropy and Gini index

is that they tend to favor qualitative attributes with large number of distinct

values. Figure 3.12 shows three candidate attributes for partitioning the data

set given in Table 3.3. As previously mentioned, the attribute Marital

Status is a better choice than the attribute Home Owner, because it provides a

larger information gain. However, if we compare them against Customer ID,

the latter produces the purest partitions with the maximum information gain,

since the weighted entropy and Gini index is equal to zero for its children.

Yet, Customer ID is not a good attribute for splitting because it has a unique

value for each instance. Even though a test condition involving Customer ID

will accurately classify every instance in the training data, we cannot use

such a test condition on new test instances with Customer ID values that

haven’t been seen before during training. This example suggests having a low

impurity value alone is insufficient to find a good attribute test condition for a

node. As we will see later in Section 3.4, having more number of child nodes

can make a decision tree more complex and consequently more susceptible to

overfitting. Hence, the number of children produced by the splitting attribute

should also be taken into consideration while deciding the best attribute test

condition.

There are two ways to overcome this problem. One way is to generate only

binary decision trees, thus avoiding the difficulty of handling attributes with

varying number of partitions. This strategy is employed by decision tree

classifiers such as CART. Another way is to modify the splitting criterion to

take into account the number of partitions produced by the attribute. For

example, in the C4.5 decision tree algorithm, a measure known as gain ratio

is used to compensate for attributes that produce a large number of child

nodes. This measure is computed as follows:

Gain ratio=ΔinfoSplit Info=Entropy(Parent)−∑i=1kN(vi)NEntropy(vi)

−∑i=1kN(vi)Nlog2N(vi)N (3.9)

where N(vi) is the number of instances assigned to node vi and *k *is the total

number of splits. The split information measures the entropy of splitting a

node into its child nodes and evaluates if the split results in a larger number

of equally-sized child nodes or not. For example, if every partition has the

same number of instances, then ∀i:N(vi)/N=1/k and the split information

would be equal to log2 *k*. Thus, if an attribute produces a large number of

splits, its split information is also large, which in turn, reduces the gain ratio.

3.4. Example Gain Ratio

Consider the data set given in Exercise 2 on page 185. We want to select the

best attribute test condition among the following three attributes: Gender, Car

Type, and Customer ID. The entropy before splitting is

Entropy(parent)=−1020log21020−1020log21020=1.

If Gender is used as attribute test condition:

Entropy(children)=1020[−610log2610−410log2410

]×2=0.971Gain Ratio=1−0.971−1020log21020−1020log21020=0.0291=0.029

If Car Type is used as attribute test condition:

Entropy(children)=420[−14log214−34log234

]+820×0+820[−18log218−78log278

]=0.380Gain Ratio=1−0.380−420log2420−820log2820−820log2820=0.6201.52

Finally, if Customer ID is used as attribute test condition:

Entropy(children)=120[−11log211−01log201

]×20=0Gain Ratio=1−0−120log2120×20=14.32=0.23

Thus, even though Customer ID has the highest information gain, its gain

ratio is lower than Car Type since it produces a larger number of splits.

3.3.4 Algorithm for Decision Tree

Induction

Algorithm 3.1 presents a pseudocode for decision tree induction algorithm.

The input to this algorithm is a set of training instances *E *along with the

attribute set *F *. The algorithm works by recursively selecting the best

attribute to split the data (Step 7) and expanding the nodes of the tree (Steps

11 and 12) until the stopping criterion is met (Step 1). The details of this

algorithm are explained below.

1. The createNode() function extends the decision tree by creating a new

node. A node in the decision tree either has a test condition, denoted as

*node.test cond*, or a class label, denoted as *node.label*.

2. The find best split() function determines the attribute test condition

for partitioning the training instances associated with a node. The

splitting attribute chosen depends on the impurity measure used. The

popular measures include entropy and the Gini index.

3. The Classify() function determines the class label to be assigned to a

leaf node. For each leaf node *t*, let p(i|t) denote the fraction of training

instances from class *i *associated with the node *t*. The label assigned to

the leaf node is typically the one that occurs most frequently in the

training instances that are associated with this node.

Algorithm 3.1 A skeleton decision

tree induction algorithm.

TreeGrowth (*E*, *F*)

1: if stopping cond(*E*,*F *) = *true *then

2: *leaf *= createNode().

3: *leaf.label *= Classify(*E*).

4: return *leaf*.

5: else

6: *root *= createNode().

7: *root.test cond *= find best split(*E*, *F *).

8: let *V *= *{v|v *is a possible outcome of *root.test cond }*

9: for each *v *∈ *V *do

10: *E*

*v *= *{e | root.test cond*(*e*) = *v *and *e *∈ *E}*.

11: *child *= TreeGrowth(*Ev*, *F *).

12: add *child *as descendent of *root *and label the edge (

13: end for

14: end if

15: return *root*.

leaf.label=argmaxi p(i|t), (3.10)

where the argmax operator returns the class *i *that maximizes p(i|t).

Besides providing the information needed to determine the class label of

a leaf node, p(i|t) can also be used as a rough estimate of the probability

that an instance assigned to the leaf node *t *belongs to class *i*. Sections

4.11.2 and 4.11.4 in the next chapter describe how such probability

estimates can be used to determine the performance of a decision tree

under different cost functions.

4. The stopping cond() function is used to terminate the tree-growing

process by checking whether all the instances have identical class label

or attribute values. Since decision tree classifiers employ a top-down,

recursive partitioning approach for building a model, the number of

training instances associated with a node decreases as the depth of the

tree increases. As a result, a leaf node may contain too few training

instances to make a statistically significant decision about its class label.

This is known as the data fragmentation problem. One way to avoid

this problem is to disallow splitting of a node when the number of

instances associated with the node fall below a certain threshold. A more

systematic way to control the size of a decision tree (number of leaf

nodes) will be discussed in Section 3.5.4.

3.3.5 Example Application: Web

Robot Detection

Consider the task of distinguishing the access patterns of web robots from

those generated by human users. A web robot (also known as a web crawler)

is a software program that automatically retrieves files from one or more

websites by following the hyperlinks extracted from an initial set of seed

URLs. These programs have been deployed for various purposes, from

gathering web pages on behalf of search engines to more malicious activities

such as spamming and committing click frauds in online advertisements.

Figure 3.15.

Input data for web robot detection.

Figure 3.15. Full Alternative Text

The web robot detection problem can be cast as a binary classification task.

The input data for the classification task is a web server log, a sample of

which is shown in Figure 3.15(a). Each line in the log file corresponds to a

request made by a client (i.e., a human user or a web robot) to the web server.

The fields recorded in the web log include the client’s IP address, timestamp

of the request, URL of the requested file, size of the file, and user agent,

which is a field that contains identifying information about the client. For

human users, the user agent field specifies the type of web browser or mobile

device used to fetch the files, whereas for web robots, it should technically

contain the name of the crawler program. However, web robots may conceal

their true identities by declaring their user agent fields to be identical to

known browsers. Therefore, user agent is not a reliable field to detect web

robots.

The first step toward building a classification model is to precisely define a

data instance and associated attributes. A simple approach is to consider each

log entry as a data instance and use the appropriate fields in the log file as its

attribute set. This approach, however, is inadequate for several reasons. First,

many of the attributes are nominal-valued and have a wide range of domain

values. For example, the number of unique client IP addresses, URLs, and

referrers in a log file can be very large. These attributes are undesirable for

building a decision tree because their split information is extremely high (see

Equation (3.9)). In addition, it might not be possible to classify test instances

containing IP addresses, URLs, or referrers that are not present in the training

data. Finally, by considering each log entry as a separate data instance, we

disregard the sequence of web pages retrieved by the client—a critical piece

of information that can help distinguish web robot accesses from those of a

human user.

A better alternative is to consider each web session as a data instance. A web

session is a sequence of requests made by a client during a given visit to the

website. Each web session can be modeled as a directed graph, in which the

nodes correspond to web pages and the edges correspond to hyperlinks

connecting one web page to another. Figure 3.15(b) shows a graphical

representation of the first web session given in the log file. Every web session

can be characterized using some meaningful attributes about the graph that

contain discriminatory information. Figure 3.15(c) shows some of the

attributes extracted from the graph, including the depth and breadth of its

corresponding tree rooted at the entry point to the website. For example, the

depth and breadth of the tree shown in Figure 3.15(b) are both equal to two.

The derived attributes shown in Figure 3.15(c) are more informative than the

original attributes given in the log file because they characterize the behavior

of the client at the website. Using this approach, a data set containing 2916

instances was created, with equal numbers of sessions due to web robots

(class 1) and human users (class 0). 10% of the data were reserved for

training while the remaining 90% were used for testing. The induced decision

tree is shown in Figure 3.16, which has an error rate equal to 3.8% on the

training set and 5.3% on the test set. In addition to its low error rate, the tree

also reveals some interesting properties that can help discriminate web robots

from human users:

1. Accesses by web robots tend to be broad but shallow, whereas accesses

by human users tend to be more focused (narrow but deep).

2. Web robots seldom retrieve the image pages associated with a web page.

3. Sessions due to web robots tend to be long and contain a large number

of requested pages.

4. Web robots are more likely to make repeated requests for the same web

page than human users since the web pages retrieved by human users are

often cached by the browser.

3.3.6 Characteristics of Decision

Tree Classifiers

The following is a summary of the important characteristics of decision tree

induction algorithms.

1. Applicability: Decision trees are a nonparametric approach for building

classification models. This approach does not require any prior

assumption about the probability distribution governing the class and

attributes of the data, and thus, is applicable to a wide variety of data

sets. It is also applicable to both categorical and continuous data without

requiring the attributes to be transformed into a common representation

via binarization, normalization, or standardization. Unlike some binary

classifiers described in Chapter 4, it can also deal with multiclass

problems without the need to decompose them into multiple binary

classification tasks. Another appealing feature of decision tree classifiers

is that the induced trees, especially the shorter ones, are relatively easy

to interpret. The accuracies of the trees are also quite comparable to

other classification techniques for many simple data sets.

2. Expressiveness: A decision tree provides a universal representation for

discrete-valued functions. In other words, it can encode any function of

discrete-valued attributes. This is because every discrete-valued function

can be represented as an assignment table, where every unique

combination of discrete attributes is assigned a class label. Since every

combination of attributes can be represented as a leaf in the decision

tree, we can always find a decision tree whose label assignments at the

leaf nodes matches with the assignment table of the original function.

Decision trees can also help in providing compact representations of

functions when some of the unique combinations of attributes can be

represented by the same leaf node. For example, Figure 3.17 shows the

assignment table of the Boolean function (A∧B)∨(C∧D) involving

four binary attributes, resulting in a total of 24=16 possible assignments.

The tree shown in Figure 3.17 shows a compressed encoding of this

assignment table. Instead of requiring a fully-grown tree with 16 leaf

nodes, it is possible to encode the function using a simpler tree with only

7 leaf nodes. Nevertheless, not all decision trees for discrete-valued

attributes can be simplified. One notable example is the parity function,

whose value is 1 when there is an even number of true values among its

Boolean attributes, and 0 otherwise. Accurate modeling of such a

function requires a full decision tree with 2d nodes, where *d *is the

number of Boolean attributes (see Exercise 1 on page 185).

Figure 3.16.

Decision tree model for web robot detection.

Figure 3.16. Full Alternative Text

Figure 3.17.

Decision tree for the Boolean function (A∧B)∨(C∧D).

Figure 3.17. Full Alternative Text

3. Computational Efficiency: Since the number of possible decision trees

can be very large, many decision tree algorithms employ a heuristicbased approach to guide their search in the vast hypothesis space. For

example, the algorithm presented in Section 3.3.4 uses a greedy, topdown, recursive partitioning strategy for growing a decision tree. For

many data sets, such techniques quickly construct a reasonably good

decision tree even when the training set size is very large. Furthermore,

once a decision tree has been built, classifying a test record is extremely

fast, with a worst-case complexity of *O*(*w*), where *w *is the maximum

depth of the tree.

4. Handling Missing Values: A decision tree classifier can handle missing

attribute values in a number of ways, both in the training and the test

sets. When there are missing values in the test set, the classifier must

decide which branch to follow if the value of a splitting node attribute is

missing for a given test instance. One approach, known as the

probabilistic split method, which is employed by the C4.5 decision

tree classifier, distributes the data instance to every child of the splitting

node according to the probability that the missing attribute has a

particular value. In contrast, the CART algorithm uses the surrogate

split method, where the instance whose splitting attribute value is

missing is assigned to one of the child nodes based on the value of

another non-missing surrogate attribute whose splits most resemble the

partitions made by the missing attribute. Another approach, known as

the separate class method is used by the CHAID algorithm, where the

missing value is treated as a separate categorical value distinct from

other values of the splitting attribute. Figure 3.18 shows an example of

the three different ways for handling missing values in a decision tree

classifier. Other strategies for dealing with missing values are based on

data preprocessing, where the instance with missing value is either

imputed with the mode (for categorical attribute) or mean (for

continuous attribute) value or discarded before the classifier is trained.

Figure 3.18.

Methods for handling missing attribute values in decision tree

classifier.

Figure 3.18. Full Alternative Text

During training, if an attribute *v *has missing values in some of the

training instances associated with a node, we need a way to measure the

gain in purity if *v *is used for splitting. One simple way is to exclude

instances with missing values of *v *in the counting of instances

associated with every child node, generated for every possible outcome

of *v*.Further, if *v *is chosen as the attribute test condition at a node,

training instances with missing values of *v *can be propagated to the

child nodes using any of the methods described above for handling

missing values in test instances.

5. Handling Interactions among Attributes: Attributes are considered

interacting if they are able to distinguish between classes when used

together, but individually they provide little or no information. Due to

the greedy nature of the splitting criteria in decision trees, such attributes

could be passed over in favor of other attributes that are not as useful.

This could result in more complex decision trees than necessary. Hence,

decision trees can perform poorly when there are interactions among

attributes.

To illustrate this point, consider the three-dimensional data shown in

Figure 3.19(a), which contains 2000 data points from one of two classes,

denoted as + and ∘ in the diagram. Figure 3.19(b) shows the

distribution of the two classes in the two-dimensional space involving

attributes *X *and *Y *, which is a noisy version of the XOR Boolean

function. We can see that even though the two classes are well-separated

in this two-dimensional space, neither of the two attributes contain

sufficient information to distinguish between the two classes when used

alone. For example, the entropies of the following attribute test

conditions: X≤10 and Y≤10, are close to 1, indicating that neither *X *nor

*Y *provide any reduction in the impurity measure when used individually.

*X *and *Y *thus represent a case of interaction among attributes. The data

set also contains a third attribute, *Z*, in which both classes are distributed

uniformly, as shown in Figures 3.19(c) and 3.19(d), and hence, the

entropy of any split involving *Z *is close to 1. As a result, *Z *is as likely to

be chosen for splitting as the interacting but useful attributes, *X *and *Y *.

For further illustration of this issue, readers are referred to Example 3.7

in Section 3.4.1 and Exercise 7 at the end of this chapter.

Figure 3.19.

Example of a XOR data involving *X *and *Y *, along with an

irrelevant attribute *Z*.

Figure 3.19. Full Alternative Text

6. Handling Irrelevant Attributes: An attribute is irrelevant if it is not

useful for the classification task. Since irrelevant attributes are poorly

associated with the target class labels, they will provide little or no gain

in purity and thus will be passed over by other more relevant features.

Hence, the presence of a small number of irrelevant attributes will not

impact the decision tree construction process. However, not all attributes

that provide little to no gain are irrelevant (see Figure 3.19). Hence, if

the classification problem is complex (e.g., involving interactions among

attributes) and there are a large number of irrelevant attributes, then

some of these attributes may be accidentally chosen during the treegrowing process, since they may provide a better gain than a relevant

attribute just by random chance. Feature selection techniques can help to

improve the accuracy of decision trees by eliminating the irrelevant

attributes during preprocessing. We will investigate the issue of too

many irrelevant attributes in Section 3.4.1.

7. Handling Redundant Attributes: An attribute is redundant if it is strongly

correlated with another attribute in the data. Since redundant attributes

show similar gains in purity if they are selected for splitting, only one of

them will be selected as an attribute test condition in the decision tree

algorithm. Decision trees can thus handle the presence of redundant

attributes.

8. Using Rectilinear Splits: The test conditions described so far in this

chapter involve using only a single attribute at a time. As a consequence,

the tree-growing procedure can be viewed as the process of partitioning

the attribute space into disjoint regions until each region contains

records of the same class. The border between two neighboring regions

of different classes is known as a decision boundary. Figure 3.20 shows

the decision tree as well as the decision boundary for a binary

classification problem. Since the test condition involves only a single

attribute, the decision boundaries are rectilinear; i.e., parallel to the

coordinate axes. This limits the expressiveness of decision trees in

representing decision boundaries of data sets with continuous attributes.

Figure 3.21 shows a two-dimensional data set involving binary classes

that cannot be perfectly classified by a decision tree whose attribute test

conditions are defined based on single attributes. The binary classes in

the data set are generated from two skewed Gaussian distributions,

centered at (8,8) and (12,12), respectively. The true decision boundary is

represented by the diagonal dashed line, whereas the rectilinear decision

boundary produced by the decision tree classifier is shown by the thick

solid line. In contrast, an oblique decision tree may overcome this

limitation by allowing the test condition to be specified using more than

one attribute. For example, the binary classification data shown in

Figure 3.21 can be easily represented by an oblique decision tree with a

single root node with test condition

x+y<20.

Figure 3.20.

Example of a decision tree and its decision boundaries for a

two-dimensional data set.

Figure 3.20. Full Alternative Text

Figure 3.21.

Example of data set that cannot be partitioned optimally using

a decision tree with single attribute test conditions. The true

decision boundary is shown by the dashed line.

Although an oblique decision tree is more expressive and can produce

more compact trees, finding the optimal test condition is

computationally more expensive.

9. Choice of Impurity Measure: It should be noted that the choice of

impurity measure often has little effect on the performance of decision

tree classifiers since many of the impurity measures are quite consistent

with each other, as shown in Figure 3.11 on page 129. Instead, the

strategy used to prune the tree has a greater impact on the final tree than

the choice of impurity measure.

3.4 Model Overfitting

Methods presented so far try to learn classification models that show the

lowest error on the training set. However, as we will show in the following

example, even if a model fits well over the training data, it can still show

poor generalization performance, a phenomenon known as model overfitting.

Figure 3.22.

Examples of training and test sets of a two-dimensional

classification problem.

Figure 3.22. Full Alternative Text

Figure 3.23.

Effect of varying tree size (number of leaf nodes) on training and

test errors.

Figure 3.23. Full Alternative Text

3.5. Example Overfitting and

Underfitting of Decision Trees

Consider the two-dimensional data set shown in Figure 3.22(a). The data set

contains instances that belong to two separate classes, represented as + and

∘ , respectively, where each class has 5400 instances. All instances

belonging to the ∘ class were generated from a uniform distribution. For the

+ class, 5000 instances were generated from a Gaussian distribution centered

at (10,10) with unit variance, while the remaining 400 instances were

sampled from the same uniform distribution as the ∘ class. We can see from

Figure 3.22(a) that the + class can be largely distinguished from the ∘ class

by drawing a circle of appropriate size centered at (10,10). To learn a

classifier using this two-dimensional data set, we randomly sampled 10% of

the data for training and used the remaining 90% for testing. The training set,

shown in Figure 3.22(b), looks quite representative of the overall data. We

used Gini index as the impurity measure to construct decision trees of

increasing sizes (number of leaf nodes), by recursively expanding a node into

child nodes till every leaf node was pure, as described in Section 3.3.4.

Figure 3.23(a) shows changes in the training and test error rates as the size of

the tree varies from 1 to 8. Both error rates are initially large when the tree

has only one or two leaf nodes. This situation is known as model

underfitting. Underfitting occurs when the learned decision tree is too

simplistic, and thus, incapable of fully representing the true relationship

between the attributes and the class labels. As we increase the tree size from

1 to 8, we can observe two effects. First, both the error rates decrease since

larger trees are able to represent more complex decision boundaries. Second,

the training and test error rates are quite close to each other, which indicates

that the performance on the training set is fairly representative of the

generalization performance. As we further increase the size of the tree from 8

to 150, the training error continues to steadily decrease till it eventually

reaches zero, as shown in Figure 3.23(b). However, in a striking contrast, the

test error rate ceases to decrease any further beyond a certain tree size, and

then it begins to increase. The training error rate thus grossly under-estimates

the test error rate once the tree becomes too large. Further, the gap between

the training and test error rates keeps on widening as we increase the tree

size. This behavior, which may seem counter-intuitive at first, can be

attributed to the phenomena of model overfitting.

3.4.1 Reasons for Model Overfitting

Model overfitting is the phenomena where, in the pursuit of minimizing the

training error rate, an overly complex model is selected that captures specific

patterns in the training data but fails to learn the *true *nature of relationships

between attributes and class labels in the overall data. To illustrate this,

Figure 3.24 shows decision trees and their corresponding decision boundaries

(shaded rectangles represent regions assigned to the + class) for two trees of

sizes 5 and 50. We can see that the decision tree of size 5 appears quite

simple and its decision boundaries provide a reasonable approximation to the

ideal decision boundary, which in this case corresponds to a circle centered

around the Gaussian distribution at (10, 10). Although its training and test

error rates are non-zero, they are very close to each other, which indicates

that the patterns learned in the training set should generalize well over the test

set. On the other hand, the decision tree of size 50 appears much more

complex than the tree of size 5, with complicated decision boundaries. For

example, some of its shaded rectangles (assigned the + class) attempt to cover

narrow regions in the input space that contain only one or two + training

instances. Note that the prevalence of + instances in such regions is highly

specific to the training set, as these regions are mostly dominated by –

instances in the overall data. Hence, in an attempt to perfectly fit the training

data, the decision tree of size 50 starts fine tuning itself to specific patterns in

the training data, leading to poor performance on an independently chosen

test set.

Figure 3.24.

Decision trees with different model complexities.

Figure 3.24. Full Alternative Text

Figure 3.25.

Performance of decision trees using 20% data for training (twice

the original training size).

Figure 3.25. Full Alternative Text

There are a number of factors that influence model overfitting. In the

following, we provide brief descriptions of two of the major factors: limited

training size and high model complexity. Though they are not exhaustive, the

interplay between them can help explain most of the common model

overfitting phenomena in real-world applications.

Limited Training Size

Note that a training set consisting of a finite number of instances can only

provide a limited representation of the overall data. Hence, it is possible that

the patterns learned from a training set do not fully represent the true patterns

in the overall data, leading to model overfitting. In general, as we increase the

size of a training set (number of training instances), the patterns learned from

the training set start resembling the true patterns in the overall data. Hence,

the effect of overfitting can be reduced by increasing the training size, as

illustrated in the following example.

3.6 Example Effect of Training Size

Suppose that we use twice the number of training instances than what we had

used in the experiments conducted in Example 3.5. Specifically, we use 20%

data for training and use the remainder for testing. Figure 3.25(b) shows the

training and test error rates as the size of the tree is varied from 1 to 150.

There are two major differences in the trends shown in this figure and those

shown in Figure 3.23(b) (using only 10% of the data for training). First, even

though the training error rate decreases with increasing tree size in both

figures, its rate of decrease is much smaller when we use twice the training

size. Second, for a given tree size, the gap between the training and test error

rates is much smaller when we use twice the training size. These differences

suggest that the patterns learned using 20% of data for training are more

generalizable than those learned using 10% of data for training.

Figure 3.25(a) shows the decision boundaries for the tree of size 50, learned

using 20% of data for training. In contrast to the tree of the same size learned

using 10% data for training (see Figure 3.24(d)), we can see that the decision

tree is not capturing specific patterns of noisy + instances in the training set.

Instead, the high model complexity of 50 leaf nodes is being effectively used

to learn the boundaries of the + instances centered at (10, 10).

High Model Complexity

Generally, a more complex model has a better ability to represent complex

patterns in the data. For example, decision trees with larger number of leaf

nodes can represent more complex decision boundaries than decision trees

with fewer leaf nodes. However, an overly complex model also has a

tendency to learn specific patterns in the training set that do not generalize

well over unseen instances. Models with high complexity should thus be

judiciously used to avoid overfitting.

One measure of model complexity is the number of “parameters” that need to

be inferred from the training set. For example, in the case of decision tree

induction, the attribute test conditions at internal nodes correspond to the

parameters of the model that need to be inferred from the training set. A

decision tree with larger number of attribute test conditions (and

consequently more leaf nodes) thus involves more “parameters” and hence is

more complex.

Given a class of models with a certain number of parameters, a learning

algorithm attempts to select the best combination of parameter values that

maximizes an evaluation metric (e.g., accuracy) over the training set. If the

number of parameter value combinations (and hence the complexity) is large,

the learning algorithm has to select the best combination from a large number

of possibilities, using a limited training set. In such cases, there is a high

chance for the learning algorithm to pick a *spurious *combination of

parameters that maximizes the evaluation metric just by random chance. This

is similar to the multiple comparisons problem (also referred as multiple

testing problem) in statistics.

As an illustration of the multiple comparisons problem, consider the task of

predicting whether the stock market will rise or fall in the next ten trading

days. If a stock analyst simply makes random guesses, the probability that her

prediction is correct on any trading day is 0.5. However, the probability that

she will predict correctly at least nine out of ten times is

(109)+(1010)210=0.0107,

which is extremely low.

Suppose we are interested in choosing an investment advisor from a pool of

200 stock analysts. Our strategy is to select the analyst who makes the most

number of correct predictions in the next ten trading days. The flaw in this

strategy is that even if all the analysts make their predictions in a random

fashion, the probability that at least one of them makes at least nine correct

predictions is

1−(1−0.0107)200=0.8847,

which is very high. Although each analyst has a low probability of predicting

at least nine times correctly, considered together, we have a high probability

of finding at least one analyst who can do so. However, there is no guarantee

in the future that such an analyst will continue to make accurate predictions

by random guessing.

How does the multiple comparisons problem relate to model overfitting? In

the context of learning a classification model, each combination of parameter

values corresponds to an analyst, while the number of training instances

corresponds to the number of days. Analogous to the task of selecting the

best analyst who makes the most accurate predictions on consecutive days,

the task of a learning algorithm is to select the best combination of

parameters that results in the highest accuracy on the training set. If the

number of parameter combinations is large but the training size is small, it is

highly likely for the learning algorithm to choose a spurious parameter

combination that provides high training accuracy just by random chance. In

the following example, we illustrate the phenomena of overfitting due to

multiple comparisons in the context of decision tree induction.

Figure 3.26.

Example of a two-dimensional (X-Y) data set.

Figure 3.27.

Training and test error rates illustrating the effect of multiple

comparisons problem on model overfitting.

Figure 3.27. Full Alternative Text

3.7. Example Multiple Comparisons

and Overfitting

Consider the two-dimensional data set shown in Figure 3.26 containing 500 +

and 500 ∘ instances, which is similar to the data shown in Figure 3.19. In

this data set, the distributions of both classes are well-separated in the twodimensional (*XY*) attribute space, but none of the two attributes (*X *or *Y*) are

sufficiently informative to be used alone for separating the two classes.

Hence, splitting the data set based on any value of an X or Y attribute will

provide close to zero reduction in an impurity measure. However, if X and Y

attributes are used together in the splitting criterion (e.g., splitting X at 10

*and *Y at 10), the two classes can be effectively separated.

Figure 3.28.

Decision tree with 6 leaf nodes using X and Y as attributes. Splits

have been numbered from 1 to 5 in order of other occurrence in the

tree.

Figure 3.28. Full Alternative Text

Figure 3.27(a) shows the training and test error rates for learning decision

trees of varying sizes, when 30% of the data is used for training and the

remainder of the data for testing. We can see that the two classes can be

separated using a small number of leaf nodes. Figure 3.28 shows the decision

boundaries for the tree with six leaf nodes, where the splits have been

numbered according to their order of appearance in the tree. Note that the

even though splits 1 and 3 provide trivial gains, their consequent splits (2, 4,

and 5) provide large gains, resulting in effective discrimination of the two

classes.

Assume we add 100 irrelevant attributes to the two-dimensional X-Y data.

Learning a decision tree from this resultant data will be challenging because

the number of candidate attributes to choose for splitting at every internal

node will increase from two to 102. With such a large number of candidate

attribute test conditions to choose from, it is quite likely that spurious

attribute test conditions will be selected at internal nodes because of the

multiple comparisons problem. Figure 3.27(b) shows the training and test

error rates after adding 100 irrelevant attributes to the training set. We can see

that the test error rate remains close to 0.5 even after using 50 leaf nodes,

while the training error rate keeps on declining and eventually becomes 0.

3.5 Model Selection

There are many possible classification models with varying levels of model

complexity that can be used to capture patterns in the training data. Among

these possibilities, we want to select the model that shows lowest

generalization error rate. The process of selecting a model with the right level

of complexity, which is expected to generalize well over unseen test

instances, is known as model selection. As described in the previous section,

the training error rate cannot be reliably used as the sole criterion for model

selection. In the following, we present three generic approaches to estimate

the generalization performance of a model that can be used for model

selection. We conclude this section by presenting specific strategies for using

these approaches in the context of decision tree induction.

3.5.1 Using a Validation Set

Note that we can always estimate the generalization error rate of a model by

using “out-of-sample” estimates, i.e. by evaluating the model on a separate

validation set that is not used for training the model. The error rate on the

validation set, termed as the validation error rate, is a better indicator of

generalization performance than the training error rate, since the validation

set has not been used for training the model. The validation error rate can be

used for model selection as follows.

Given a training set *D.train*, we can partition *D.train *into two smaller

subsets, *D.tr *and *D.val*, such that *D.tr *is used for training while *D.val *is used

as the validation set. For example, two-thirds of *D.train *can be reserved as

*D.tr *for training, while the remaining one-third is used as *D.val *for

computing validation error rate. For any choice of classification model *m *that

is trained on *D.tr*, we can estimate its validation error rate on *D.val*,

errval(m). The model that shows the lowest value of errval(m) can then be

selected as the preferred choice of model.

The use of validation set provides a generic approach for model selection.

However, one limitation of this approach is that it is sensitive to the sizes of

*D.tr *and *D.val*, obtained by partitioning *D.train*. If the size of *D.tr *is too

small, it may result in the learning of a poor classification model with substandard performance, since a smaller training set will be less representative

of the overall data. On the other hand, if the size of *D.val *is too small, the

validation error rate might not be reliable for selecting models, as it would be

computed over a small number of instances.

Figure 3.29.

Class distribution of validation data for the two decision trees

shown in Figure 3.30.

Figure 3.29. Full Alternative Text

3.8. Example Validation Error

In the following example, we illustrate one possible approach for using a

validation set in decision tree induction. Figure 3.29 shows the predicted

labels at the leaf nodes of the decision trees generated in Figure 3.30. The

counts given beneath the leaf nodes represent the proportion of data instances

in the validation set that reach each of the nodes. Based on the predicted

labels of the nodes, the validation error rate for the left tree is

errval(TL)=6/16=0.375, while the validation error rate for the right tree is

errval(TR)=4/16=0.25. Based on their validation error rates, the right tree is

preferred over the left one.

3.5.2 Incorporating Model

Complexity

Since the chance for model overfitting increases as the model becomes more

complex, a model selection approach should not only consider the training

error rate but also the model complexity. This strategy is inspired by a wellknown principle known as Occam’s razor or the principle of parsimony,

which suggests that given two models with the same errors, the simpler

model is preferred over the more complex model. A generic approach to

account for model complexity while estimating generalization performance is

formally described as follows.

Given a training set *D.train*, let us consider learning a classification model *m*

that belongs to a certain class of models, M. For example, if M represents the

set of all possible decision trees, then *m *can correspond to a specific decision

tree learned from the training set. We are interested in estimating the

generalization error rate of *m*, *gen.error*(*m*). As discussed previously, the

training error rate of *m*, *train.error*(*m, D.train*), can under-estimate

*gen.error*(*m*) when the model complexity is high. Hence, we represent

*gen.error*(*m*) as a function of not just the training error rate but also the

model complexity of M, complexity(M), as follows:

gen.error(m)=train.error(m, D.train)+α×complexity(M), (3.11)

where α is a hyper-parameter that strikes a balance between minimizing

training error and reducing model complexity. A higher value of α gives more

emphasis to the model complexity in the estimation of generalization

performance. To choose the right value of α, we can make use of the

validation set in a similar way as described in 3.5.1. For example, we can

iterate through a range of values of α and for every possible value, we can

learn a model on a subset of the training set, *D.tr*, and compute its validation

error rate on a separate subset, *D.val*. We can then select the value of α that

provides the lowest validation error rate.

Equation 3.11 provides one possible approach for incorporating model

complexity into the estimate of generalization performance. This approach is

at the heart of a number of techniques for estimating generalization

performance, such as the structural risk minimization principle, the Akaike’s

Information Criterion (AIC), and the Bayesian Information Criterion (BIC).

The structural risk minimization principle serves as the building block for

learning support vector machines, which will be discussed later in Chapter 4.

For more details on AIC and BIC, see the Bibliographic Notes.

In the following, we present two different approaches for estimating the

complexity of a model, complexity(M). While the former is specific to

decision trees, the latter is more generic and can be used with any class of

models.

Estimating the Complexity of

Decision Trees

In the context of decision trees, the complexity of a decision tree can be

estimated as the ratio of the number of leaf nodes to the number of training

instances. Let *k *be the number of leaf nodes and Ntrain be the number of

training instances. The complexity of a decision tree can then be described as

k/Ntrain. This reflects the intuition that for a larger training size, we can learn

a decision tree with larger number of leaf nodes without it becoming overly

complex. The generalization error rate of a decision tree *T *can then be

computed using Equation 3.11 as follows:

errgen(T)=err(T)+Ω×kNtrain,

where *err*(*T*) is the training error of the decision tree and Ω is a hyperparameter that makes a trade-off between reducing training error and

minimizing model complexity, similar to the use of α in Equation 3.11. Ω can

be viewed as the relative cost of adding a leaf node relative to incurring a

training error. In the literature on decision tree induction, the above approach

for estimating generalization error rate is also termed as the pessimistic error

estimate. It is called pessimistic as it assumes the generalization error rate to

be worse than the training error rate (by adding a penalty term for model

complexity). On the other hand, simply using the training error rate as an

estimate of the generalization error rate is called the optimistic error

estimate or the resubstitution estimate.

3.9. Example Generalization Error

Estimates

Consider the two binary decision trees, TL and TR, shown in Figure 3.30.

Both trees are generated from the same training data and TL is generated by

expanding three leaf nodes of TR. The counts shown in the leaf nodes of the

trees represent the class distribution of the training instances. If each leaf

node is labeled according to the majority class of training instances that reach

the node, the training error rate for the left tree will be err(TL)=4/24=0.167,

while the training error rate for the right tree will be err(TR)=6/24=0.25.

Based on their training error rates alone, TL would preferred over TR, even

though TL is more complex (contains larger number of leaf nodes) than TR.

Figure 3.30.

Example of two decision trees generated from the same training

data.

Figure 3.30. Full Alternative Text

Now, assume that the cost associated with each leaf node is Ω=0.5. Then, the

generalization error estimate for TL will be

errgen(TL)=424+0.5×724=7.524=0.3125

and the generalization error estimate for TR will be

errgen(TR) =624+0.5×424=824=0.3333.

Since TL has a lower generalization error rate, it will still be preferred over

TR. Note that Ω=0.5 implies that a node should always be expanded into its

two child nodes if it improves the prediction of at least one training instance,

since expanding a node is less costly than misclassifying a training instance.

On the other hand, if Ω=1, then the generalization error rate for TL is

errgen(TL)=11/24=0.458 and for TR is errgen(TR)=10/24=0.417. In this

case, TR will be preferred over TL because it has a lower generalization error

rate. This example illustrates that different choices of Ω can change our

preference of decision trees based on their generalization error estimates.

However, for a given choice of Ω, the pessimistic error estimate provides an

approach for modeling the generalization performance on unseen test

instances. The value of Ω can be selected with the help of a validation set.

Minimum Description Length

Principle

Another way to incorporate model complexity is based on an informationtheoretic approach known as the minimum description length or MDL

principle. To illustrate this approach, consider the example shown in Figure

3.31. In this example, both person A and person B are given a set of instances

with known attribute values x. Assume person A knows the class label *y *for

every instance, while person B has no such information. A would like to share

the class information with B by sending a message containing the labels. The

message would contain Θ(N) bits of information, where *N *is the number of

instances.

Figure 3.31.

An illustration of the minimum description length principle.

Figure 3.31. Full Alternative Text

Alternatively, instead of sending the class labels explicitly, A can build a

classification model from the instances and transmit it to B. B can then apply

the model to determine the class labels of the instances. If the model is 100%

accurate, then the cost for transmission is equal to the number of bits required

to encode the model. Otherwise, A must also transmit information about

which instances are misclassified by the model so that B can reproduce the

same class labels. Thus, the overall transmission cost, which is equal to the

total description length of the message, is

Cost(model, data)=Cost(data|model)+α×Cost(model), (3.12)

where the first term on the right-hand side is the number of bits needed to

encode the misclassified instances, while the second term is the number of

bits required to encode the model. There is also a hyper-parameter α that

trades-off the relative costs of the misclassified instances and the model.

Notice the familiarity between this equation and the generic equation for

generalization error rate presented in Equation 3.11. A good model must have

a total description length less than the number of bits required to encode the

entire sequence of class labels. Furthermore, given two competing models,

the model with lower total description length is preferred. An example

showing how to compute the total description length of a decision tree is

given in Exercise 10 on page 189.

3.5.3 Estimating Statistical Bounds

Instead of using Equation 3.11 to estimate the generalization error rate of a

model, an alternative way is to apply a statistical correction to the training

error rate of the model that is indicative of its model complexity. This can be

done if the probability distribution of training error is available or can be

assumed. For example, the number of errors committed by a leaf node in a

decision tree can be assumed to follow a binomial distribution. We can thus

compute an upper bound limit to the observed training error rate that can be

used for model selection, as illustrated in the following example.

3.10. Example Statistical Bounds on

Training Error

Consider the left-most branch of the binary decision trees shown in Figure

3.30. Observe that the left-most leaf node of TR has been expanded into two

child nodes in TL. Before splitting, the training error rate of the node is

2/7=0.286. By approximating a binomial distribution with a normal

distribution, the following upper bound of the training error rate *e *can be

derived:

eupper(N, e, α)=e+zα/222N+zα/2e(1−e)N+zα/224N21+zα/22N, (3.13)

where α is the confidence level, zα/2 is the standardized value from a

standard normal distribution, and *N *is the total number of training instances

used to compute *e*. By replacing α=25%,N=7, and e=2/7, the upper bound for

the error rate is eupper(7, 2/7, 0.25)=0.503, which corresponds to

7×0.503=3.521 errors. If we expand the node into its child nodes as shown in

TL, the training error rates for the child nodes are 1/4=0.250 and 1/3=0.333,

respectively. Using Equation (3.13), the upper bounds of these error rates are

eupper(4, 1/4,0.25)=0.537 and eupper(3, 1/3, 0.25)=0.650, respectively. The

overall training error of the child nodes is 4×0.537+3×0.650=4.098, which is

larger than the estimated error for the corresponding node in TR, suggesting

that it should not be split.

3.5.4 Model Selection for Decision

Trees

Building on the generic approaches presented above, we present two

commonly used model selection strategies for decision tree induction.

Prepruning (Early Stopping Rule)

In this approach, the tree-growing algorithm is halted before generating a

fully grown tree that perfectly fits the entire training data. To do this, a more

restrictive stopping condition must be used; e.g., stop expanding a leaf node

when the observed gain in the generalization error estimate falls below a

certain threshold. This estimate of the generalization error rate can be

computed using any of the approaches presented in the preceding three

subsections, e.g., by using pessimistic error estimates, by using validation

error estimates, or by using statistical bounds. The advantage of prepruning is

that it avoids the computations associated with generating overly complex

subtrees that overfit the training data. However, one major drawback of this

method is that, even if no significant gain is obtained using one of the

existing splitting criterion, subsequent splitting may result in better subtrees.

Such subtrees would not be reached if prepruning is used because of the

greedy nature of decision tree induction.

Post-pruning

In this approach, the decision tree is initially grown to its maximum size. This

is followed by a tree-pruning step, which proceeds to trim the fully grown

tree in a bottom-up fashion. Trimming can be done by replacing a subtree

with (1) a new leaf node whose class label is determined from the majority

class of instances affiliated with the subtree (approach known as subtree

replacement), or (2) the most frequently used branch of the subtree

(approach known as subtree raising). The tree-pruning step terminates when

no further improvement in the generalization error estimate is observed

beyond a certain threshold. Again, the estimates of generalization error rate

can be computed using any of the approaches presented in the previous three

subsections. Post-pruning tends to give better results than prepruning because

it makes pruning decisions based on a fully grown tree, unlike prepruning,

which can suffer from premature termination of the tree-growing process.

However, for post-pruning, the additional computations needed to grow the

full tree may be wasted when the subtree is pruned.

Figure 3.32 illustrates the simplified decision tree model for the web robot

detection example given in Section 3.3.5. Notice that the subtree rooted at

depth=1 has been replaced by one of its branches corresponding to

breadth<=7, width>3, and MultiP=1, using subtree raising. On the other

hand, the subtree corresponding to depth>1 and MultiAgent=0 has been

replaced by a leaf node assigned to class 0, using subtree replacement. The

subtree for depth>1 and MultiAgent=1 remains intact.

Figure 3.32.

Post-pruning of the decision tree for web robot detection.

Figure 3.32. Full Alternative Text

3.6 Model Evaluation

The previous section discussed several approaches for model selection that

can be used to learn a classification model from a training set *D.train*. Here

we discuss methods for estimating its generalization performance, i.e. its

performance on unseen instances outside of *D.train*. This process is known as

model evaluation.

Note that model selection approaches discussed in Section 3.5 also compute

an estimate of the generalization performance using the training set *D.train*.

However, these estimates are *biased *indicators of the performance on unseen

instances, since they were used to guide the selection of classification model.

For example, if we use the validation error rate for model selection (as

described in Section 3.5.1), the resulting model would be deliberately chosen

to minimize the errors on the validation set. The validation error rate may

thus under-estimate the true generalization error rate, and hence cannot be

reliably used for model evaluation.

A correct approach for model evaluation would be to assess the performance

of a learned model on a labeled test set has not been used at any stage of

model selection. This can be achieved by partitioning the entire set of labeled

instances *D*, into two disjoint subsets, *D.train*, which is used for model

selection and *D.test*, which is used for computing the test error rate, errtest. In

the following, we present two different approaches for partitioning *D *into

*D.train *and *D.test*, and computing the test error rate, errtest.

3.6.1 Holdout Method

The most basic technique for partitioning a labeled data set is the holdout

method, where the labeled set *D *is randomly partitioned into two disjoint

sets, called the training set *D.train *and the test set *D.test*. A classification

model is then induced from *D.train *using the model selection approaches

presented in Section 3.5, and its error rate on *D.test*, errtest, is used as an

estimate of the generalization error rate. The proportion of data reserved for

training and for testing is typically at the discretion of the analysts, e.g., twothirds for training and one-third for testing.

Similar to the trade-off faced while partitioning *D.train *into *D.tr *and *D.val *in

Section 3.5.1, choosing the right fraction of labeled data to be used for

training and testing is non-trivial. If the size of *D.train *is small, the learned

classification model may be improperly learned using an insufficient number

of training instances, resulting in a biased estimation of generalization

performance. On the other hand, if the size of *D.test *is small, errtest may be

less reliable as it would be computed over a small number of test instances.

Moreover, errtest can have a high variance as we change the random

partitioning of *D *into *D.train *and *D.test*.

The holdout method can be repeated several times to obtain a distribution of

the test error rates, an approach known as random subsampling or repeated

holdout method. This method produces a distribution of the error rates that

can be used to understand the variance of errtest.

3.6.2 Cross-Validation

Cross-validation is a widely-used model evaluation method that aims to make

effective use of all labeled instances in *D *for both training and testing. To

illustrate this method, suppose that we are given a labeled set that we have

randomly partitioned into three equal-sized subsets, S1, S2, and S3, as shown

in Figure 3.33. For the first run, we train a model using subsets S2 and *S*3

(shown as empty blocks) and test the model on subset S1. The test error rate

on S1, denoted as err(S1), is thus computed in the first run. Similarly, for the

second run, we use S1 and S3 as the training set and S2 as the test set, to

compute the test error rate, err(S2), on S2. Finally, we use S1 and S3 for

training in the third run, while S3 is used for testing, thus resulting in the test

error rate err(S3) for S3. The overall test error rate is obtained by summing

up the number of errors committed in each test subset across all runs and

dividing it by the total number of instances. This approach is called three-fold

cross-validation.

Figure 3.33.

Example demonstrating the technique of 3-fold cross-validation.

Figure 3.33. Full Alternative Text

The *k*-fold cross-validation method generalizes this approach by segmenting

the labeled data *D *(of size *N*) into *k *equal-sized partitions (or folds). During

the *i*th run, one of the partitions of *D *is chosen as *D.test*(*i*) for testing, while

the rest of the partitions are used as *D.train*(*i*) for training. A model *m*(*i*) is

learned using *D.train*(*i*) and applied on *D.test*(*i*) to obtain the sum of test

errors, errsum(i). This procedure is repeated *k *times. The total test error rate,

errtest, is then computed as

errtest=∑i=1kerrsum(i)N.(3.14)

Every instance in the data is thus used for testing exactly once and for

training exactly (k−1) times. Also, every run uses (k−1)/k fraction of the data

for training and 1/*k *fraction for testing.

The right choice of *k *in *k*-fold cross-validation depends on a number of

characteristics of the problem. A small value of *k *will result in a smaller

training set at every run, which will result in a larger estimate of

generalization error rate than what is expected of a model trained over the

entire labeled set. On the other hand, a high value of *k *results in a larger

training set at every run, which reduces the bias in the estimate of

generalization error rate. In the extreme case, when k=N, every run uses

exactly one data instance for testing and the remainder of the data for testing.

This special case of *k*-fold cross-validation is called the leave-one-out

approach. This approach has the advantage of utilizing as much data as

possible for training. However, leave-one-out can produce quite misleading

results in some special scenarios, as illustrated in Exercise 11. Furthermore,

leave-one-out can be computationally expensive for large data sets as the

cross-validation procedure needs to be repeated *N *times. For most practical

applications, the choice of *k *between 5 and 10 provides a reasonable

approach for estimating the generalization error rate, because each fold is

able to make use of 80% to 90% of the labeled data for training.

The *k*-fold cross-validation method, as described above, produces a single

estimate of the generalization error rate, without providing any information

about the variance of the estimate. To obtain this information, we can run *k*fold cross-validation for every possible partitioning of the data into *k*

partitions, and obtain a distribution of test error rates computed for every

such partitioning. The average test error rate across all possible partitionings

serves as a more robust estimate of generalization error rate. This approach of

estimating the generalization error rate and its variance is known as the

complete cross-validation approach. Even though such an estimate is quite

robust, it is usually too expensive to consider all possible partitionings of a

large data set into *k *partitions. A more practical solution is to repeat the

cross-validation approach multiple times, using a different random

partitioning of the data into *k *partitions at every time, and use the average test

error rate as the estimate of generalization error rate. Note that since there is

only one possible partitioning for the leave-one-out approach, it is not

possible to estimate the variance of generalization error rate, which is another

limitation of this method.

The *k*-fold cross-validation does not guarantee that the fraction of positive

and negative instances in every partition of the data is equal to the fraction

observed in the overall data. A simple solution to this problem is to perform a

stratified sampling of the positive and negative instances into *k *partitions, an

approach called stratified cross-validation.

In *k*-fold cross-validation, a different model is learned at every run and the

performance of every one of the *k *models on their respective test folds is then

aggregated to compute the overall test error rate, errtest. Hence, errtest does

not reflect the generalization error rate of any of the *k *models. Instead, it

reflects the *expected *generalization error rate of the *model selection*

*approach*, when applied on a training set of the same size as one of the

training folds (N(k−1)/k). This is different than the errtest computed in the

holdout method, which exactly corresponds to the specific model learned

over *D.train*. Hence, although effectively utilizing every data instance in *D*

for training and testing, the errtest computed in the cross-validation method

does not represent the performance of a single model learned over a specific

*D.train*.

Nonetheless, in practice, errtest is typically used as an estimate of the

generalization error of a model built on *D*. One motivation for this is that

when the size of the training folds is closer to the size of the overall data

(when *k *is large), then errtest resembles the expected performance of a model

learned over a data set of the same size as *D*. For example, when *k *is 10,

every training fold is 90% of the overall data. The errtest then should

approach the expected performance of a model learned over 90% of the

overall data, which will be close to the expected performance of a model

learned over *D*.

3.7 Presence of Hyper-parameters

Hyper-parameters are parameters of learning algorithms that need to be

determined before learning the classification model. For instance, consider

the hyper-parameter α that appeared in Equation 3.11, which is repeated here

for convenience. This equation was used for estimating the generalization

error for a model selection approach that used an explicit representations of

model complexity. (See Section 3.5.2.)

gen.error(m)=train.error(m, D.train)+α×complexity(M)

For other examples of hyper-parameters, see Chapter 4.

Unlike regular model parameters, such as the test conditions in the internal

nodes of a decision tree, hyper-parameters such as α do not appear in the final

classification model that is used to classify unlabeled instances. However, the

values of hyper-parameters need to be determined during model selection—a

process known as hyper-parameter selection—and must be taken into

account during model evaluation. Fortunately, both tasks can be effectively

accomplished via slight modifications of the cross-validation approach

described in the previous section.

3.7.1 Hyper-parameter Selection

In Section 3.5.2, a validation set was used to select α and this approach is

generally applicable for hyper-parameter section. Let *p *be the hyperparameter that needs to be selected from a finite range of values, P=

{p1, p2, … pn }. Partition *D.train *into *D.tr *and *D.val*. For every choice of

hyper-parameter value pi, we can learn a model mi on *D.tr*, and apply this

model on *D.val *to obtain the validation error rate errval(pi). Let p* be the

hyper-parameter value that provides the lowest validation error rate. We can

then use the model m* corresponding to p* as the final choice of

classification model.

The above approach, although useful, uses only a subset of the data, *D.train*,

for training and a subset, *D.val*, for validation. The framework of crossvalidation, presented in Section 3.6.2, addresses both of those issues, albeit in

the context of model evaluation. Here we indicate how to use a crossvalidation approach for hyper-parameter selection. To illustrate this approach,

let us partition *D.train *into three folds as shown in Figure 3.34. At every run,

one of the folds is used as *D.val *for validation, and the remaining two folds

are used as *D.tr *for learning a model, for every choice of hyper-parameter

value pi. The overall validation error rate corresponding to each pi is

computed by summing the errors across all the three folds. We then select the

hyper-parameter value p* that provides the lowest validation error rate, and

use it to learn a model m* on the entire training set *D.train*.

Figure 3.34.

Example demonstrating the 3-fold cross-validation framework for

hyper-parameter selection using *D.train*.

Figure 3.34. Full Alternative Text

Algorithm 3.2 generalizes the above approach using a *k*-fold cross-validation

framework for hyper-parameter selection. At the *i*th run of cross-validation,

the data in the *i*th fold is used as *D.val*(*i*) for validation (Step 4), while the

remainder of the data in *D.train *is used as *D.tr*(*i*) for training (Step 5). Then

for every choice of hyper-parameter value pi, a model is learned on *D.tr*(*i*)

(Step 7), which is applied on *D.val*(*i*) to compute its validation error (Step 8).

This is used to compute the validation error rate corresponding to models

learning using pi over all the folds (Step 11). The hyper-parameter value p*

that provides the lowest validation error rate (Step 12) is now used to learn

the final model m* on the entire training set *D.train *(Step 13). Hence, at the

end of this algorithm, we obtain the best choice of the hyper-parameter value

as well as the final classification model (Step 14), both of which are obtained

by making an effective use of every data instance in *D.train*.

Algorithm 3.2 Procedure modelselect(*k*, P, *D.train*)

1: *N*

*train *= *|D.train| *{Size of *D.train*.}

2: Divide *D.train *into *k *partitions, *D.train*1 to *D.train**k*.

3: for each run *i *= 1 to *k *do

4: *D.val*(*i*) = *D.train**i*. {Partition used for validation.}

5: *D.tr*(*i*) = *D.train \ D.train**i*. {Partitions used for training.}

6: for each parameter *p *∈ P do

7: *m *= model-train(*p*, *D.tr*(*i*)). {Train model}

8: *err*

*sum*(*p*, *i*) = model-test(*m*, *D.val*(*i*)). {Sum of validation errors.}

9: end for

10: end for

11: errval(p)=∑ik errsum(p, i)/Ntrain. {Compute validation error rate.}

12: *p** = argmin*p **err**val*(*p*). {Select best hyper-parameter value.}

13: *m** = model-train(*p**, *D.train*). {Learn final model on *D.train*}

14: return (*p**, *m**).

3.7.2 Nested Cross-Validation

The approach of the previous section provides a way to effectively use all the

instances in *D.train *to learn a classification model when hyper-parameter

selection is required. This approach can be applied over the entire data set *D*

to learn the final classification model. However, applying Algorithm 3.2 on *D*

would only return the final classification model m* but not an estimate of its

generalization performance, errtest. Recall that the validation error rates used

in Algorithm 3.2 cannot be used as estimates of generalization performance,

since they are used to guide the selection of the final model m*. However, to

compute errtest, we can again use a cross-validation framework for

evaluating the performance on the entire data set *D*, as described originally in

Section 3.6.2. In this approach, *D *is partitioned into *D.train *(for training) and

*D.test *(for testing) at every run of cross-validation. When hyper-parameters

are involved, we can use Algorithm 3.2 to train a model using *D.train *at

every run, thus “internally” using cross-validation for model selection. This

approach is called nested cross-validation or double cross-validation.

Algorithm 3.3 describes the complete approach for estimating errtest using

nested cross-validation in the presence of hyper-parameters.

As an illustration of this approach, see Figure 3.35 where the labeled set *D *is

partitioned into *D.train *and *D.test*, using a 3-fold cross-validation method.

Figure 3.35.

Example demonstrating 3-fold nested cross-validation for

computing errtest.

Figure 3.35. Full Alternative Text

At the *i*th run of this method, one of the folds is used as the test set, *D.test*(*i*),

while the remaining two folds are used as the training set, *D.train*(*i*). This is

represented in Figure 3.35 as the *i*th “outer” run. In order to select a model

using *D.train*(*i*), we again use an “inner” 3-fold cross-validation framework

that partitions *D.train*(*i*) into *D.tr *and *D.val *at every one of the three inner

runs (iterations). As described in Section 3.7, we can use the inner crossvalidation framework to select the best hyper-parameter value p*(i) as well as

its resulting classification model m*(i) learned over *D.train*(*i*). We can then

apply m*(i) on *D.test*(*i*) to obtain the test error at the *i*th outer run. By

repeating this process for every outer run, we can compute the average test

error rate, errtest, over the entire labeled set *D*. Note that in the above

approach, the inner cross-validation framework is being used for model

selection while the outer cross-validation framework is being used for model

evaluation.

Algorithm 3.3 The nested crossvalidation approach for computing

errtest.

1: Divide *D *into *k *partitions, *D*1 to *D**k*.

2: for each outer run *i *= 1 to *k *do

3: *D.test*(*i*) = *D**i*. {Partition used for testing.}

4: *D.train*(*i*) = *D *\ *D**i*. {Partitions used for model selection.}

5: (*p**, *m**(*i*)) = model-select(*k*, *P*, *D.train*(*i*)). {Inner cross-validation.}

6: *err*

*sum*(*i*) = model-test(*m**(*i*), *D.test*(*i*)). {Sum of test errors.}

7: end for

8: errtest=∑ikerrtest(i)/N. {Compute test error rate.}

3.8 Pitfalls of Model Selection and

Evaluation

Model selection and evaluation, when used effectively, serve as excellent

tools for learning classification models and assessing their generalization

performance. However, when using them effectively in practical settings,

there are several pitfalls that can result in improper and often misleading

conclusions. Some of these pitfalls are simple to understand and easy to

avoid, while others are quite subtle in nature and difficult to catch. In the

following, we present two of these pitfalls and discuss best practices to avoid

them.

3.8.1 Overlap between Training and

Test Sets

One of the basic requirements of a *clean *model selection and evaluation

setup is that the data used for model selection (*D.train*) must be kept separate

from the data used for model evaluation (*D.test*). If there is any overlap

between the two, the test error rate errtest computed over *D.test *cannot be

considered representative of the performance on *unseen *instances.

Comparing the effectiveness of classification models using errtest can then be

quite misleading, as an overly complex model can show an inaccurately low

value of errtest due to model overfitting (see Exercise 12 at the end of this

chapter).

To illustrate the importance of ensuring no overlap between *D.train *and

*D.test*, consider a labeled data set where all the attributes are irrelevant, i.e.

they have no relationship with the class labels. Using such attributes, we

should expect no classification model to perform better than random

guessing. However, if the test set involves even a small number of data

instances that were used for training, there is a possibility for an overly

complex model to show better performance than random, even though the

attributes are completely irrelevant. As we will see later in Chapter 10, this

scenario can actually be used as a criterion to detect overfitting due to

improper setup of experiment. If a model shows better performance than a

random classifier even when the attributes are irrelevant, it is an indication of

a potential feedback between the training and test sets.

3.8.2 Use of Validation Error as

Generalization Error

The validation error rate errval serves an important role during model

selection, as it provides “out-of-sample” error estimates of models on *D.val*,

which is not used for training the models. Hence, errval serves as a better

metric than the training error rate for selecting models and hyper-parameter

values, as described in Sections 3.5.1 and 3.7, respectively. However, once

the validation set has been used for selecting a classification model m*, errval

no longer reflects the performance of m* on *unseen *instances.

To realize the pitfall in using validation error rate as an estimate of

generalization performance, consider the problem of selecting a hyperparameter value *p *from a range of values P, using a validation set *D.val*. If

the number of possible values in P is quite large and the size of *D.val *is

small, it is possible to select a hyper-parameter value p* that shows favorable

performance on *D.val *just by random chance. Notice the similarity of this

problem with the multiple comparisons problem discussed in Section 3.4.1.

Even though the classification model m* learned using p* would show a low

validation error rate, it would lack generalizability on unseen test instances.

The correct approach for estimating the generalization error rate of a model

m* is to use an independently chosen test set *D.test *that hasn’t been used in

any way to influence the selection of m*. As a rule of thumb, the test set

should never be examined during model selection, to ensure the absence of

any form of overfitting. If the insights gained from any portion of a labeled

data set help in improving the classification model even in an indirect way,

then that portion of data must be discarded during testing.

3.9 Model Comparison*

One difficulty when comparing the performance of different classification

models is whether the observed difference in their performance is statistically

significant. For example, consider a pair of classification models, MA and

MB. Suppose MA achieves 85% accuracy when evaluated on a test set

containing 30 instances, while MB achieves 75% accuracy on a different test

set containing 5000 instances. Based on this information, is MA a better

model than MB? This example raises two key questions regarding the

statistical significance of a performance metric:

1. Although MA has a higher accuracy than MB, it was tested on a smaller

test set. How much confidence do we have that the accuracy for MA is

actually 85%?

2. Is it possible to explain the difference in accuracies between MA and

MB as a result of variations in the composition of their test sets?

The first question relates to the issue of estimating the confidence interval of

model accuracy. The second question relates to the issue of testing the

statistical significance of the observed deviation. These issues are

investigated in the remainder of this section.

3.9.1 Estimating the Confidence

Interval for Accuracy

To determine its confidence interval, we need to establish the probability

distribution for sample accuracy. This section describes an approach for

deriving the confidence interval by modeling the classification task as a

binomial random experiment. The following describes characteristics of such

an experiment:

1. The random experiment consists of *N *independent trials, where each

trial has two possible outcomes: success or failure.

2. The probability of success, *p*, in each trial is constant.

An example of a binomial experiment is counting the number of heads that

turn up when a coin is flipped *N *times. If *X *is the number of successes

observed in *N *trials, then the probability that *X *takes a particular value is

given by a binomial distribution with mean Np and variance Np(1−p):

P(X=υ)=(Nυ)pυ(1−p)N−υ.

For example, if the coin is fair (p=0.5) and is flipped fifty times, then the

probability that the head shows up 20 times is

P(X=20)=(5020)0.520(1−0.5)30=0.0419.

If the experiment is repeated many times, then the average number of heads

expected to show up is 50×0.5=25, while its variance is 50×0.5×0.5=12.5.

The task of predicting the class labels of test instances can also be considered

as a binomial experiment. Given a test set that contains *N *instances, let *X *be

the number of instances correctly predicted by a model and *p *be the true

accuracy of the model. If the prediction task is modeled as a binomial

experiment, then *X *has a binomial distribution with mean Np and variance

Np(1−p). It can be shown that the empirical accuracy, acc=X/N, also has a

binomial distribution with mean *p *and variance p(1−p)/N (see Exercise 14).

The binomial distribution can be approximated by a normal distribution when

*N *is sufficiently large. Based on the normal distribution, the confidence

interval for *acc *can be derived as follows:

P(−Zα/2≤acc−pp(1−p)/N≤Z1−α/2)=1−α, (3.15)

where Zα/2 and Z1−α/2 are the upper and lower bounds obtained from a

standard normal distribution at confidence level (1−α). Since a standard

normal distribution is symmetric around Z=0, it follows that Zα/2=Z1−α/2.

Rearranging this inequality leads to the following confidence interval for *p*:

2×N×acc×Zα/22±Zα/2Zα/22+4Nacc−4Nacc22(N+Zα/22). (3.16)

The following table shows the values of Zα/2 at different confidence levels:

1−α 0.99 0.98 0.95 0.9 0.8 0.7 0.5

Zα/2 2.58 2.33 1.96 1.65 1.28 1.04 0.67

3.11. Example Confidence Interval

for Accuracy

Consider a model that has an accuracy of 80% when evaluated on 100 test

instances. What is the confidence interval for its true accuracy at a 95%

confidence level? The confidence level of 95% corresponds to Za/2=1.96

according to the table given above. Inserting this term into Equation 3.16

yields a confidence interval between 71.1% and 86.7%. The following table

shows the confidence interval when the number of instances, *N*, increases:

*N *20 50 100 500 1000 5000

Confidence 0.584 0.670 0.711 0.763 0.774 0.789

Interval −0.919 −0.888 −0.867 −0.833 −0.824 −0.811

Note that the confidence interval becomes tighter when *N *increases.

3.9.2 Comparing the Performance

of Two Models

Consider a pair of models, M1 and M2, which are evaluated on two

independent test sets, D1 and D2. Let n1 denote the number of instances in

D1 and n2 denote the number of instances in D2. In addition, suppose the

error rate for M1 on D1 is e1 and the error rate for M2 on D2 is e2. Our goal

is to test whether the observed difference between e1 and e2 is statistically

significant.

Assuming that n1 and n2 are sufficiently large, the error rates e1 and e2 can

be approximated using normal distributions. If the observed difference in the

error rate is denoted as d=e1−e2, then *d *is also normally distributed with

mean dt, its true difference, and variance, σd2. The variance of *d *can be

computed as follows:

σd2≃σ^d2=e1(1−e1)n1+e2(1−e2)n2, (3.17)

where e1(1−e1)/n1 and e2(1−e1)/n2 are the variances of the error rates.

Finally, at the (1−α)% confidence level, it can be shown that the confidence

interval for the true difference *dt *is given by the following equation:

dt=d±zα/2σ^d. (3.18)

3.12. Example Significance Testing

Consider the problem described at the beginning of this section. Model MA

has an error rate of e1=0.15 when applied to N1=30 test instances, while

model MB has an error rate of e2=0.25 when applied to N2=5000 test

instances. The observed difference in their error rates is d=|0.15−0.25|=0.1. In

this example, we are performing a two-sided test to check whether dt=0 or

dt≠0. The estimated variance of the observed difference in error rates can be

computed as follows:

σ^d2=0.15(1−0.15)30+0.25(1−0.25)5000=0.0043

or σ^d=0.0655. Inserting this value into Equation 3.18, we obtain the

following confidence interval for dt at 95% confidence level:

dt=0.1±1.96×0.0655=0.1±0.128.

As the interval spans the value zero, we can conclude that the observed

difference is not statistically significant at a 95% confidence level.

At what confidence level can we reject the hypothesis that dt=0? To do this,

we need to determine the value of Zα/2 such that the confidence interval for

dt does not span the value zero. We can reverse the preceding computation

and look for the value Zα/2 such that d>Zσ/2σ^d. Replacing the values of *d*

and σ^d gives Zσ/2<1.527. This value first occurs when (1−α)<~0.936 (for a

two-sided test). The result suggests that the null hypothesis can be rejected at

confidence level of 93.6% or lower.

3.10 Bibliographic Notes

Early classification systems were developed to organize various collections

of objects, from living organisms to inanimate ones. Examples abound, from

Aristotle’s cataloguing of species to the Dewey Decimal and Library of

Congress classification systems for books. Such a task typically requires

considerable human efforts, both to identify properties of the objects to be

classified and to organize them into well distinguished categories.

With the development of statistics and computing, automated classification

has been a subject of intensive research. The study of classification in

classical statistics is sometimes known as discriminant analysis, where the

objective is to predict the group membership of an object based on its

corresponding features. A well-known classical method is Fisher’s linear

discriminant analysis [142], which seeks to find a linear projection of the data

that produces the best separation between objects from different classes.

Many pattern recognition problems also require the discrimination of objects

from different classes. Examples include speech recognition, handwritten

character identification, and image classification. Readers who are interested

in the application of classification techniques for pattern recognition may

refer to the survey articles by Jain et al. [150] and Kulkarni et al. [157] or

classic pattern recognition books by Bishop [125], Duda et al. [137], and

Fukunaga [143]. The subject of classification is also a major research topic in

neural networks, statistical learning, and machine learning. An in-depth

treatment on the topic of classification from the statistical and machine

learning perspectives can be found in the books by Bishop [126], Cherkassky

and Mulier [132], Hastie et al. [148], Michie et al. [162], Murphy [167], and

Mitchell [165]. Recent years have also seen the release of many publicly

available software packages for classification, which can be embedded in

programming languages such as Java (Weka [147]) and Python (scikit-learn

[174]).

An overview of decision tree induction algorithms can be found in the survey

articles by Buntine [129], Moret [166], Murthy [168], and Safavian et al.

[179]. Examples of some well-known decision tree algorithms include CART

[127], ID3 [175], C4.5 [177], and CHAID [153]. Both ID3 and C4.5 employ

the entropy measure as their splitting function. An in-depth discussion of the

C4.5 decision tree algorithm is given by Quinlan [177]. The CART algorithm

was developed by Breiman et al. [127] and uses the Gini index as its splitting

function. CHAID [153] uses the statistical χ2 test to determine the best split

during the tree-growing process.

The decision tree algorithm presented in this chapter assumes that the

splitting condition at each internal node contains only one attribute. An

oblique decision tree can use multiple attributes to form the attribute test

condition in a single node [149, 187]. Breiman et al. [127] provide an option

for using linear combinations of attributes in their CART implementation.

Other approaches for inducing oblique decision trees were proposed by Heath

et al. [149], Murthy et al. [169], Cant*ú*-Paz and Kamath [130], and Utgoff

and Brodley [187]. Although an oblique decision tree helps to improve the

expressiveness of the model representation, the tree induction process

becomes computationally challenging. Another way to improve the

expressiveness of a decision tree without using oblique decision trees is to

apply a method known as constructive induction [161]. This method

simplifies the task of learning complex splitting functions by creating

compound features from the original data.

Besides the top-down approach, other strategies for growing a decision tree

include the bottom-up approach by Landeweerd et al. [159] and Pattipati and

Alexandridis [173], as well as the bidirectional approach by Kim and

Landgrebe [154]. Schuermann and Doster [181] and Wang and Suen [193]

proposed using a soft splitting criterion to address the data fragmentation

problem. In this approach, each instance is assigned to different branches of

the decision tree with different probabilities.

Model overfitting is an important issue that must be addressed to ensure that

a decision tree classifier performs equally well on previously unlabeled data

instances. The model overfitting problem has been investigated by many

authors including Breiman et al. [127], Schaffer [180], Mingers [164], and

Jensen and Cohen [151]. While the presence of noise is often regarded as one

of the primary reasons for overfitting [164, 170], Jensen and Cohen [151]

viewed overfitting as an artifact of failure to compensate for the multiple

comparisons problem.

Bishop [126] and Hastie et al. [148] provide an excellent discussion of model

overfitting, relating it to a well-known framework of theoretical analysis,

known as bias-variance decomposition [146]. In this framework, the

prediction of a learning algorithm is considered to be a function of the

training set, which varies as the training set is changed. The generalization

error of a model is then described in terms of its *bias *(the error of the average

prediction obtained using different training sets), its *variance *(how different

are the predictions obtained using different training sets), and *noise *(the

irreducible error inherent to the problem). An underfit model is considered to

have high bias but low variance, while an overfit model is considered to have

low bias but high variance. Although the bias-variance decomposition was

originally proposed for regression problems (where the target attribute is a

continuous variable), a unified analysis that is applicable for classification

has been proposed by Domingos [136]. The bias variance decomposition will

be discussed in more detail while introducing ensemble learning methods in

Chapter 4.

Various learning principles, such as the Probably Approximately Correct

(PAC) learning framework [188], have been developed to provide a

theoretical framework for explaining the generalization performance of

learning algorithms. In the field of statistics, a number of performance

estimation methods have been proposed that make a trade-off between the

goodness of fit of a model and the model complexity. Most noteworthy

among them are the Akaike’s Information Criterion [120] and the Bayesian

Information Criterion [182]. They both apply corrective terms to the training

error rate of a model, so as to penalize more complex models. Another

widely-used approach for measuring the complexity of any general model is

the VapnikChervonenkis (VC) Dimension [190]. The VC dimension of a

class of functions *C *is defined as the maximum number of points that can be

*shattered *(every point can be distinguished from the rest) by functions

belonging to *C*, for any possible configuration of points. The VC dimension

lays the foundation of the structural risk minimization principle [189], which

is extensively used in many learning algorithms, e.g., support vector

machines, which will be discussed in detail in Chapter 4.

The Occam’s razor principle is often attributed to the philosopher William of

Occam. Domingos [135] cautioned against the pitfall of misinterpreting

Occam’s razor as comparing models with similar training errors, instead of

generalization errors. A survey on decision tree-pruning methods to avoid

overfitting is given by Breslow and Aha [128] and Esposito et al. [141].

Some of the typical pruning methods include reduced error pruning [176],

pessimistic error pruning [176], minimum error pruning [171], critical value

pruning [163], cost-complexity pruning [127], and error-based pruning [177].

Quinlan and Rivest proposed using the minimum description length principle

for decision tree pruning in [178].

The discussions in this chapter on the significance of cross-validation error

estimates is inspired from Chapter 7 in Hastie et al. [148]. It is also an

excellent resource for understanding “the right and wrong ways to do crossvalidation”, which is similar to the discussion on pitfalls in Section 3.8 of this

chapter. A comprehensive discussion of some of the common pitfalls in using

cross-validation for model selection and evaluation is provided in Krstajic et

al. [156].

The original cross-validation method was proposed independently by Allen

[121], Stone [184], and Geisser [145] for model assessment (evaluation).

Even though cross-validation can be used for model selection [194], its usage

for model selection is quite different than when it is used for model

evaluation, as emphasized by Stone [184]. Over the years, the distinction

between the two usages has often been ignored, resulting in incorrect

findings. One of the common mistakes while using cross-validation is to

perform pre-processing operations (e.g., hyper-parameter tuning or feature

selection) using the entire data set and not “within” the training fold of every

cross-validation run. Ambroise et al., using a number of gene expression

studies as examples, [124] provide an extensive discussion of the *selection*

*bias *that arises when feature selection is performed outside cross-validation.

Useful guidelines for evaluating models on microarray data have also been

provided by Allison et al. [122].

The use of the cross-validation protocol for hyper-parameter tuning has been

described in detail by Dudoit and van der Laan [138]. This approach has been

called “grid-search cross-validation.” The correct approach in using cross-

validation for both hyper-parameter selection and model evaluation, as

discussed in Section 3.7 of this chapter, is extensively covered by Varma and

Simon [191]. This combined approach has been referred to as “nested crossvalidation” or “double cross-validation” in the existing literature. Recently,

Tibshirani and Tibshirani [185] have proposed a new approach for hyperparameter selection and model evaluation. Tsamardinos et al. [186] compared

this approach to nested cross-validation. The experiments they performed

found that, on average, both approaches provide conservative estimates of

model performance with the Tibshirani and Tibshirani approach being more

computationally efficient.

Kohavi [155] has performed an extensive empirical study to compare the

performance metrics obtained using different estimation methods such as

random subsampling and *k*-fold cross-validation. Their results suggest that

the best estimation method is ten-fold, stratified cross-validation.

An alternative approach for model evaluation is the bootstrap method, which

was presented by Efron in 1979 [139]. In this method, training instances are

sampled *with replacement *from the labeled set, i.e., an instance previously

selected to be part of the training set is equally likely to be drawn again. If the

original data has *N *instances, it can be shown that, on average, a bootstrap

sample of size *N *contains about 63.2% of the instances in the original data.

Instances that are not included in the bootstrap sample become part of the test

set. The bootstrap procedure for obtaining training and test sets is repeated *b*

times, resulting in a different error rate on the test set, *err*(*i*), at the *i*th run. To

obtain the overall error rate, errboot, the .632 bootstrap approach combines

*err*(*i*) with the error rate obtained from a training set containing all the

labeled examples, errs, as follows:

errboot=1b∑i=1b(0.632)×err(i)+0.386×errs). (3.19)

Efron and Tibshirani [140] provided a theoretical and empirical comparison

between cross-validation and a bootstrap method known as the 632+ rule.

While the .632 bootstrap method presented above provides a robust estimate

of the generalization performance with low variance in its estimate, it may

produce misleading results for highly complex models in certain conditions,

as demonstrated by Kohavi [155]. This is because the overall error rate is not

truly an out-of-sample error estimate as it depends on the training error rate,

errs, which can be quite small if there is overfitting.

Current techniques such as C4.5 require that the entire training data set fit

into main memory. There has been considerable effort to develop parallel and

scalable versions of decision tree induction algorithms. Some of the proposed

algorithms include SLIQ by Mehta et al. [160], SPRINT by Shafer et al.

[183], CMP by Wang and Zaniolo [192], CLOUDS by Alsabti et al. [123],

RainForest by Gehrke et al. [144], and ScalParC by Joshi et al. [152]. A

survey of parallel algorithms for classification and other data mining tasks is

given in [158]. More recently, there has been extensive research to implement

large-scale classifiers on the compute unified device architecture (CUDA)

[131, 134] and MapReduce [133, 172] platforms.

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3.11 Exercises

1. 1. Draw the full decision tree for the parity function of four Boolean

attributes, *A*, *B*, *C*, and *D*. Is it possible to simplify the tree?

2. 2. Consider the training examples shown in Table 3.5 for a binary

classification problem.

Table 3.5. Data set for

Exercise 2.

Customer ID Gender Car Type Shirt Size Class

1 M Family Small C0

2 M Sports Medium C0

3 M Sports Medium C0

4 M Sports Large C0

5 M Sports Extra Large C0

6 M Sports Extra Large C0

7 F Sports Small C0

8 F Sports Small C0

9 F Sports Medium C0

10 F Luxury Large C0

11 M Family Large C1

12 M Family Extra Large C1

13 M Family Medium C1

14 M Luxury Extra Large C1

15 F Luxury Small C1

16 F Luxury Small C1

17 F Luxury Medium C1

18 F Luxury Medium C1

19 F Luxury Medium C1

20 F Luxury Large C1

1. Compute the Gini index for the overall collection of training

examples.

2. Compute the Gini index for the Customer ID attribute.

3. Compute the Gini index for the Gender attribute.

4. Compute the Gini index for the Car Type attribute using multiway

split.

5. Compute the Gini index for the Shirt Size attribute using

multiway split.

6. Which attribute is better, Gender, Car Type, or Shirt Size?

7. Explain why Customer ID should not be used as the attribute test

condition even though it has the lowest Gini.

3. 3. Consider the training examples shown in Table 3.6 for a binary

classification problem.

Table 3.6. Data set for

Exercise 3.

Instance a1 a2 a3 Target Class

1 T T 1.0 +

2 T T 6.0 +

3 T F 5.0 −

4 F F 4.0 +

5 F T 7.0 −

6 F T 3.0 −

7 F F 8.0 −

8 T F 7.0 +

9 F T 5.0 −

1. What is the entropy of this collection of training examples with

respect to the class attribute?

2. What are the information gains of a1 and a2 relative to these

training examples?

3. For a3, which is a continuous attribute, compute the information

gain for every possible split.

4. What is the best split (among a1, a2, and a3) according to the

information gain?

5. What is the best split (between a1 and a2) according to the

misclassification error rate?

6. What is the best split (between a1 and a2) according to the Gini

index?

4. 4. Show that the entropy of a node never increases after splitting it into

smaller successor nodes.

5. 5. Consider the following data set for a binary class problem.

A B Class Label

T F +

T T +

T T +

T F −

T T +

F F −

F F −

F F −

T T −

T F −

1. Calculate the information gain when splitting on *A *and *B*. Which

attribute would the decision tree induction algorithm choose?

2. Calculate the gain in the Gini index when splitting on *A *and *B*.

Which attribute would the decision tree induction algorithm

choose?

3. Figure 3.11 shows that entropy and the Gini index are both

monotonically increasing on the range [0, 0.5] and they are both

monotonically decreasing on the range [0.5, 1]. Is it possible that

information gain and the gain in the Gini index favor different

attributes? Explain.

6. 6. Consider splitting a parent node *P *into two child nodes, C1 and C2,

using some attribute test condition. The composition of labeled training

instances at every node is summarized in the Table below.

*P *C1 C2

Class 0 7 3 4

Class 1 3 0 3

1. Calculate the Gini index and misclassification error rate of the

parent node *P *.

2. Calculate the weighted Gini index of the child nodes. Would you

consider this attribute test condition if Gini is used as the impurity

measure?

3. Calculate the weighted misclassification rate of the child nodes.

Would you consider this attribute test condition if misclassification

rate is used as the impurity measure?

7. 7. Consider the following set of training examples.

*X Y Z *No. of Class C1 Examples No. of Class C2 Examples

0 0 0 5 40

0 0 1 0 15

0 1 0 10 5

0 1 1 45 0

1 0 0 10 5

1 0 1 25 0

1 1 0 5 20

1 1 1 0 15

1. Compute a two-level decision tree using the greedy approach

described in this chapter. Use the classification error rate as the

criterion for splitting. What is the overall error rate of the induced

tree?

2. Repeat part (a) using *X *as the first splitting attribute and then

choose the best remaining attribute for splitting at each of the two

successor nodes. What is the error rate of the induced tree?

3. Compare the results of parts (a) and (b). Comment on the suitability

of the greedy heuristic used for splitting attribute selection.

8. 8. The following table summarizes a data set with three attributes *A*, *B*,

*C *and two class labels +, −. Build a two-level decision tree.

A B C Number of Instances

+ −

T T T 5 0

F T T 0 20

T F T 20 0

F F T 0 5

T T F 0 0

F T F 25 0

T F F 0 0

F F F 0 25

1. According to the classification error rate, which attribute would be

chosen as the first splitting attribute? For each attribute, show the

contingency table and the gains in classification error rate.

2. Repeat for the two children of the root node.

3. How many instances are misclassified by the resulting decision

tree?

4. Repeat parts (a), (b), and (c) using *C *as the splitting attribute.

5. Use the results in parts (c) and (d) to conclude about the greedy

nature of the decision tree induction algorithm.

9. 9. Consider the decision tree shown in Figure 3.36.

Figure 3.36.

Decision tree and data sets for Exercise 9.

Figure 3.36. Full Alternative Text

1. Compute the generalization error rate of the tree using the

optimistic approach.

2. Compute the generalization error rate of the tree using the

pessimistic approach. (For simplicity, use the strategy of adding a

factor of 0.5 to each leaf node.)

3. Compute the generalization error rate of the tree using the

validation set shown above. This approach is known as reduced

error pruning.

10. 10. Consider the decision trees shown in Figure 3.37. Assume they are

generated from a data set that contains 16 binary attributes and 3 classes,

C1, C2, and C3.

Compute the total description length of each decision tree according to

the following formulation of the minimum description length principle.

The total description length of a tree is given by

Cost(tree,data)=Cost(tree)+Cost(data|tree).

Each internal node of the tree is encoded by the ID of the splitting

attribute. If there are *m *attributes, the cost of encoding each

attribute is log2m bits.

Figure 3.37.

Decision trees for Exercise 10.

Figure 3.37. Full Alternative Text

Each leaf is encoded using the ID of the class it is associated with.

If there are *k *classes, the cost of encoding a class is log2 k bits.

*Cost*(*tree*) is the cost of encoding all the nodes in the tree. To

simplify the computation, you can assume that the total cost of the

tree is obtained by adding up the costs of encoding each internal

node and each leaf node.

Cost(data|tree) is encoded using the classification errors the tree

commits on the training set. Each error is encoded by log2 n bits,

where *n *is the total number of training instances.

Which decision tree is better, according to the MDL principle?

11. 11. This exercise, inspired by the discussions in [155], highlights one of

the known limitations of the leave-one-out model evaluation procedure.

Let us consider a data set containing 50 positive and 50 negative

instances, where the attributes are purely random and contain no

information about the class labels. Hence, the generalization error rate of

any classification model learned over this data is expected to be 0.5. Let

us consider a classifier that assigns the majority class label of training

instances (ties resolved by using the positive label as the default class) to

any test instance, irrespective of its attribute values. We can call this

approach as the *majority inducer *classifier. Determine the error rate of

this classifier using the following methods.

1. Leave-one-out.

2. 2-fold stratified cross-validation, where the proportion of class

labels at every fold is kept same as that of the overall data.

3. From the results above, which method provides a more reliable

evaluation of the classifier’s generalization error rate?

12. 12. Consider a labeled data set containing 100 data instances, which is

randomly partitioned into two sets *A *and *B*, each containing 50

instances. We use *A *as the training set to learn two decision trees, T10

with 10 leaf nodes and T100 with 100 leaf nodes. The accuracies of the

two decision trees on data sets *A *and *B *are shown in Table 3.7.

Table 3.7. Comparing the

test accuracy of decision

trees T10 and T100.

Accuracy

Data Set T10 T100

*A *0.86 0.97

*B *0.84 0.77

1. Based on the accuracies shown in Table 3.7, which classification

model would you expect to have better performance on unseen

instances?

2. Now, you tested T10 and T100 on the entire data set (A+B) and

found that the classification accuracy of T10 on data set (A+B) is

0.85, whereas the classification accuracy of T100 on the data set

(A+B) is 0.87. Based on this new information and your

observations from Table 3.7, which classification model would you

finally choose for classification?

13. 13. Consider the following approach for testing whether a classifier A

beats another classifier B. Let *N *be the size of a given dataset, pA be the

accuracy of classifier A, pB be the accuracy of classifier *B*, and p=

(pA+pB)/2 be the average accuracy for both classifiers. To test whether

classifier A is significantly better than B, the following Z-statistic is

used:

Z=pA−pB2p(1−p)N.

Classifier A is assumed to be better than classifier B if Z>1.96.

Table 3.8 compares the accuracies of three different classifiers, decision

tree classifiers, naïve Bayes classifiers, and support vector machines, on

various data sets. (The latter two classifiers are described in Chapter 4.)

Summarize the performance of the classifiers given in Table 3.8 using

the following 3×3 table:

win-loss-draw Decision

tree

Naïve

Bayes

Support vector

machine

Decision tree 0 – 0 – 23

Naïve Bayes 0 – 0 – 23

Support vector

machine 0 – 0 – 23

Table 3.8. Comparing the

accuracy of various

classification methods.

Data Set Size(*N*) Decision

Tree (%)

naïve

Bayes

(%)

Support vector

machine (%)

Anneal 898 92.09 79.62 87.19

Australia 690 85.51 76.81 84.78

Auto 205 81.95 58.05 70.73

Breast 699 95.14 95.99 96.42

Cleve 303 76.24 83.50 84.49

Credit 690 85.80 77.54 85.07

Diabetes 768 72.40 75.91 76.82

German 1000 70.90 74.70 74.40

Glass 214 67.29 48.59 59.81

Heart 270 80.00 84.07 83.70

Hepatitis 155 81.94 83.23 87.10

Horse 368 85.33 78.80 82.61

Ionosphere 351 89.17 82.34 88.89

Iris 150 94.67 95.33 96.00

Labor 57 78.95 94.74 92.98

Led7 3200 73.34 73.16 73.56

Lymphography 148 77.03 83.11 86.49

Pima 768 74.35 76.04 76.95

Sonar 208 78.85 69.71 76.92

Tic-tac-toe 958 83.72 70.04 98.33

Vehicle 846 71.04 45.04 74.94

Wine 178 94.38 96.63 98.88

Zoo 101 93.07 93.07 96.04

Each cell in the table contains the number of wins, losses, and draws

when comparing the classifier in a given row to the classifier in a given

column.

14. 14. Let *X *be a binomial random variable with mean Np and variance

Np(1−p). Show that the ratio *X/N *also has a binomial distribution with

mean *p *and variance p(1−p)N.

4 Classification: Alternative

Techniques

The previous chapter introduced the classification problem and presented a

technique known as the decision tree classifier. Issues such as model

overfitting and model evaluation were also discussed. This chapter presents

alternative techniques for building classification models—from simple

techniques such as rule-based and nearest neighbor classifiers to more

sophisticated techniques such as artificial neural networks, deep learning,

support vector machines, and ensemble methods. Other practical issues such

as the class imbalance and multiclass problems are also discussed at the end

of the chapter.

4.1 Types of Classifiers

Before presenting specific techniques, we first categorize the different types

of classifiers available. One way to distinguish classifiers is by considering

the characteristics of their output.

Binary versus Multiclass

Binary classifiers assign each data instance to one of two possible labels,

typically denoted as +1 and −1. The positive class usually refers to the

category we are more interested in predicting correctly compared to the

negative class (e.g., the spam category in email classification problems). If

there are more than two possible labels available, then the technique is known

as a multiclass classifier. As some classifiers were designed for binary classes

only, they must be adapted to deal with multiclass problems. Techniques for

transforming binary classifiers into multiclass classifiers are described in

Section 4.12.

Deterministic versus Probabilistic

A deterministic classifier produces a discrete-valued label to each data

instance it classifies whereas a probabilistic classifier assigns a continuous

score between 0 and 1 to indicate how likely it is that an instance belongs to a

particular class, where the probability scores for all the classes sum up to 1.

Some examples of probabilistic classifiers include the naïve Bayes classifier,

Bayesian networks, and logistic regression. Probabilistic classifiers provide

additional information about the confidence in assigning an instance to a

class than deterministic classifiers. A data instance is typically assigned to the

class with the highest probability score, except when the cost of

misclassifying the class with lower probability is significantly higher. We

will discuss the topic of cost-sensitive classification with probabilistic outputs

in Section 4.11.2.

Another way to distinguish the different types of classifiers is based on their

technique for discriminating instances from different classes.

Linear versus Nonlinear

A linear classifier uses a linear separating hyperplane to discriminate

instances from different classes whereas a nonlinear classifier enables the

construction of more complex, nonlinear decision surfaces. We illustrate an

example of a linear classifier (perceptron) and its nonlinear counterpart

(multi-layer neural network) in Section 4.7. Although the linearity

assumption makes the model less flexible in terms of fitting complex data,

linear classifiers are thus less susceptible to model overfitting compared to

nonlinear classifiers. Furthermore, one can transform the original set of

attributes, x=(x1, x2, ⋯ ,xd), into a more complex feature set, e.g., Φ(x)=

(x1, x2, x1x2, x12, x22, ⋯), before applying the linear classifier. Such feature

transformation allows the linear classifier to fit data sets with nonlinear

decision surfaces (see Section 4.9.4).

Global versus Local

A global classifier fits a single model to the entire data set. Unless the model

is highly nonlinear, this one-size-fits-all strategy may not be effective when

the relationship between the attributes and the class labels varies over the

input space. In contrast, a local classifier partitions the input space into

smaller regions and fits a distinct model to training instances in each region.

The *k*-nearest neighbor classifier (see Section 4.3) is a classic example of

local classifiers. While local classifiers are more flexible in terms of fitting

complex decision boundaries, they are also more susceptible to the model

overfitting problem, especially when the local regions contain few training

examples.

Generative versus Discriminative

Given a data instance x, the primary objective of any classifier is to predict

the class label, *y*, of the data instance. However, apart from predicting the

class label, we may also be interested in describing the underlying

mechanism that *generates *the instances belonging to every class label. For

example, in the process of classifying spam email messages, it may be useful

to understand the typical characteristics of email messages that are labeled as

spam, e.g., specific usage of keywords in the subject or the body of the email.

Classifiers that learn a generative model of every class in the process of

predicting class labels are known as generative classifiers. Some examples of

generative classifiers include the naïve Bayes classifier and Bayesian

networks. In contrast, discriminative classifiers directly predict the class

labels without explicitly describing the distribution of every class label. They

solve a simpler problem than generative models since they do not have the

onus of deriving insights about the generative mechanism of data instances.

They are thus sometimes preferred over generative models, especially when it

is not crucial to obtain information about the properties of every class. Some

examples of discriminative classifiers include decision trees, rule-based

classifier, nearest neighbor classifier, artificial neural networks, and support

vector machines.

4.2 Rule-Based Classifier

A rule-based classifier uses a collection of “if …then…” rules (also known as

a rule set) to classify data instances. Table 4.1 shows an example of a rule set

generated for the vertebrate classification problem described in the previous

chapter. Each classification rule in the rule set can be expressed in the

following way:

ri:(Conditioni)→yi. (4.1)

The left-hand side of the rule is called the rule antecedent or precondition.

It contains a conjunction of attribute test conditions:

Conditioni=(A1 op v1)∧(A2 op v2)…(Ak op vk), (4.2)

where (Aj, vj) is an attribute-value pair and *op *is a comparison operator

chosen from the set {=, ≠, <, >, ≤, ≥}. Each attribute test (Aj op vj) is also

known as a conjunct. The right-hand side of the rule is called the rule

consequent, which contains the predicted class yi.

A rule *r *covers a data instance *x *if the precondition of *r *matches the attributes

of *x*. *r *is also said to be fired or triggered whenever it covers a given instance.

For an illustration, consider the rule r1 given in Table 4.1 and the following

attributes for two vertebrates: hawk and grizzly bear.

Table 4.1. Example of a rule set

for the vertebrate classification

problem.

r1:(Gives Birth=no)∧(Aerial Creature=yes)→Birdsr2:(Gives Birth=no)∧

blooded)→Mammalsr4:(Gives Birth=no)∧(Aerial Creature=no)→Reptiles

Name Body

Temperature

Skin

Cover

Gives

Birth

Aquatic

Creature

Aerial

Creature

Has

Legs Hibernates

hawk warmblooded feather no no yes yes no

grizzly

bear

warmblooded fur yes no no yes yes

r1 covers the first vertebrate because its precondition is satisfied by the

hawk’s attributes. The rule does not cover the second vertebrate because

grizzly bears give birth to their young and cannot fly, thus violating the

precondition of r1.

The quality of a classification rule can be evaluated using measures such as

coverage and accuracy. Given a data set *D *and a classification rule *r *: A→y,

the coverage of the rule is the fraction of instances in *D *that trigger the rule *r*.

On the other hand, its accuracy or confidence factor is the fraction of

instances triggered by *r *whose class labels are equal to *y*. The formal

definitions of these measures are

Coverage(r)=| A || D |Coverage(r)=|A∩y || A |, (4.3)

where |A| is the number of instances that satisfy the rule antecedent, |A∩y| is

the number of instances that satisfy both the antecedent and consequent, and

|D| is the total number of instances.

Example 4.1.

Consider the data set shown in Table 4.2. The rule

(Gives Birth=yes)∧(Body Temperature=warm-blooded)→Mammals

has a coverage of 33% since five of the fifteen instances support the rule

antecedent. The rule accuracy is 100% because all five vertebrates covered by

the rule are mammals.

Table 4.2. The vertebrate data

set.

Name Body

Temperature

Skin

Cover

Gives

Birth

Aquatic

Creature

Aerial

Creature

Has

Legs Hibernates

human warmblooded hair yes no no yes no

python cold-blooded scales no no no no yes

salmon cold-blooded scales no yes no no no

whale warmblooded hair yes yes no no no

frog cold-blooded none no semi no yes yes

komodo

dragon cold-blooded scales no no no yes no

bat warmblooded hair yes no yes yes yes

pigeon warmblooded feathers no no yes yes no

cat warmblooded fur yes no no yes no

guppy cold-blooded scales yes yes no no no

alligator cold-blooded scales no semi no yes no

penguin warmblooded feathers no semi no yes no

porcupine warmblooded quills yes no no yes yes

eel cold-blooded scales no yes no no no

salamander cold-blooded none no semi no yes yes

4.2.1 How a Rule-Based Classifier

Works

A rule-based classifier classifies a test instance based on the rule triggered by

the instance. To illustrate how a rule-based classifier works, consider the rule

set shown in Table 4.1 and the following vertebrates:

Name Body

Temperature

Skin

Cover

Gives

Birth

Aquatic

Creature

Aerial

Creature

Has

Legs Hibernates

lemur warmblooded fur yes no no yes yes

turtle cold-blooded scales no semi no yes no

dogfish

shark cold-blooded scales yes yes no no no

The first vertebrate, which is a lemur, is warm-blooded and gives birth

to its young. It triggers the rule r3, and thus, is classified as a mammal.

The second vertebrate, which is a turtle, triggers the rules r4 and r5.

Since the classes predicted by the rules are contradictory (reptiles versus

amphibians), their conflicting classes must be resolved.

None of the rules are applicable to a dogfish shark. In this case, we need

to determine what class to assign to such a test instance.

4.2.2 Properties of a Rule Set

The rule set generated by a rule-based classifier can be characterized by the

following two properties.

Definition 4.1 (Mutually Exclusive

Rule Set).

The rules in a rule set *R *are mutually exclusive if no two rules in *R *are

triggered by the same instance. This property ensures that every instance is

covered by at most one rule in *R*.

Definition 4.2 (Exhaustive Rule Set).

A rule set *R *has exhaustive coverage if there is a rule for each combination of

attribute values. This property ensures that every instance is covered by at

least one rule in *R*.

Table 4.3. Example of a

mutually exclusive and

exhaustive rule set.

r1: (Body Temperature=cold-blooded)→Nonmammalsr2: (Body Temperature=warmblooded)∧(Gives Birth=yes)→Mammalsr3: (Body Temperature=warmblooded)∧(Gives Birth=no)→Non-mammals

Together, these two properties ensure that every instance is covered by

exactly one rule. An example of a mutually exclusive and exhaustive rule set

is shown in Table 4.3. Unfortunately, many rule-based classifiers, including

the one shown in Table 4.1, do not have such properties. If the rule set is not

exhaustive, then a default rule, rd: ()→yd, must be added to cover the

remaining cases. A default rule has an empty antecedent and is triggered

when all other rules have failed. yd is known as the default class and is

typically assigned to the majority class of training instances not covered by

the existing rules. If the rule set is not mutually exclusive, then an instance

can be covered by more than one rule, some of which may predict conflicting

classes.

Definition 4.3 (Ordered Rule Set).

The rules in an ordered rule set *R *are ranked in decreasing order of their

priority. An ordered rule set is also known as a decision list.

The rank of a rule can be defined in many ways, e.g., based on its accuracy or

total description length. When a test instance is presented, it will be classified

by the highest-ranked rule that covers the instance. This avoids the problem

of having conflicting classes predicted by multiple classification rules if the

rule set is not mutually exclusive.

An alternative way to handle a non-mutually exclusive rule set without

ordering the rules is to consider the consequent of each rule triggered by a

test instance as a vote for a particular class. The votes are then tallied to

determine the class label of the test instance. The instance is usually assigned

to the class that receives the highest number of votes. The vote may also be

weighted by the rule’s accuracy. Using unordered rules to build a rule-based

classifier has both advantages and disadvantages. Unordered rules are less

susceptible to errors caused by the wrong rule being selected to classify a test

instance unlike classifiers based on ordered rules, which are sensitive to the

choice of rule-ordering criteria. Model building is also less expensive because

the rules do not need to be kept in sorted order. Nevertheless, classifying a

test instance can be quite expensive because the attributes of the test instance

must be compared against the precondition of every rule in the rule set.

In the next two sections, we present techniques for extracting an ordered rule

set from data. A rule-based classifier can be constructed using (1) direct

methods, which extract classification rules directly from data, and (2) indirect

methods, which extract classification rules from more complex classification

models, such as decision trees and neural networks. Detailed discussions of

these methods are presented in Sections 4.2.3 and 4.2.4, respectively.

4.2.3 Direct Methods for Rule

Extraction

To illustrate the direct method, we consider a widely-used rule induction

algorithm called RIPPER. This algorithm scales almost linearly with the

number of training instances and is particularly suited for building models

from data sets with imbalanced class distributions. RIPPER also works well

with noisy data because it uses a validation set to prevent model overfitting.

RIPPER uses the sequential covering algorithm to extract rules directly from

data. Rules are grown in a greedy fashion one class at a time. For binary class

problems, RIPPER chooses the majority class as its default class and learns

the rules to detect instances from the minority class. For multiclass problems,

the classes are ordered according to their prevalence in the training set. Let

(y1, y2, … ,yc) be the ordered list of classes, where y1 is the least prevalent

class and yc is the most prevalent class. All training instances that belong to

y1 are initially labeled as positive examples, while those that belong to other

classes are labeled as negative examples. The sequential covering algorithm

learns a set of rules to discriminate the positive from negative examples.

Next, all training instances from y2 are labeled as positive, while those from

classes y3, y4, ⋯, yc are labeled as negative. The sequential covering

algorithm would learn the next set of rules to distinguish y2 from other

remaining classes. This process is repeated until we are left with only one

class, yc, which is designated as the default class.

Example 4.1. Sequential covering

algorithm.

1: Let *E *be the training instances and *A *be the set of attribute-value pairs, {(A

2: Let *Y*

*o *be an ordered set of classes {*y*1, *y*2, … ,*y**k*.

3: Let *R *= { } be the initial rule list.

4: for each class *y *∈ *Y**o *−{*y**k*} do

5: while stopping condition is not met do

6: *r*← Learn-One-Rule (*E*, *A*, *y*).

7: Remove training instances from *E *that are covered by *r*.

8: Add *r *to the bottom of the rule list: *R *← *R *∨ *r*.

9: end while

10: end for

11: Insert the default rule, {} → *y**k*, to the bottom of the rule list

A summary of the sequential covering algorithm is shown in Algorithm 4.1.

The algorithm starts with an empty decision list, *R*, and extracts rules for each

class based on the ordering specified by the class prevalence. It iteratively

extracts the rules for a given class *y *using the Learn-One-Rule function. Once

such a rule is found, all the training instances covered by the rule are

eliminated. The new rule is added to the bottom of the decision list *R*. This

procedure is repeated until the stopping criterion is met. The algorithm then

proceeds to generate rules for the next class.

Figure 4.1 demonstrates how the sequential covering algorithm works for a

data set that contains a collection of positive and negative examples. The rule

*R*1, whose coverage is shown in Figure 4.1(b), is extracted first because it

covers the largest fraction of positive examples. All the training instances

covered by *R*1 are subsequently removed and the algorithm proceeds to look

for the next best rule, which is *R*2.

Learn-One-Rule Function

Finding an optimal rule is computationally expensive due to the exponential

search space to explore. The Learn-One-Rule function addresses this problem

by growing the rules in a greedy fashion. It generates an initial rule r: {}→+,

where the left-hand side is an empty set and the right-hand side corresponds

to the positive class. It then refines the rule until a certain stopping criterion is

met. The accuracy of the initial rule may be poor because some of the

training instances covered by the rule belong to the negative class. A new

conjunct must be added to the rule antecedent to improve its accuracy.

Figure 4.1.

An example of the sequential covering algorithm.

Figure 4.1. Full Alternative Text

RIPPER uses the FOIL’s information gain measure to choose the best

conjunct to be added into the rule antecedent. The measure takes into

consideration both the gain in accuracy and support of a candidate rule,

where support is defined as the number of positive examples covered by the

rule. For example, suppose the rule r: A→+ initially covers p0 positive

examples and n0 negative examples. After adding a new conjunct *B*, the

extended rule r′: A∧B→+ covers p1 positive examples and n1 negative

examples. The FOIL’s information gain of the extended rule is computed as

follows:

FOIL’s information gain=p1×(log2p1p1+n1−log2p0p0+n0). (4.4)

RIPPER chooses the conjunct with highest FOIL’s information gain to extend

the rule, as illustrated in the next example.

Example 4.2. [Foil’s Information

Gain]

Consider the training set for the vertebrate classification problem shown in

Table 4.2. Suppose the target class for the Learn-One-Rule function is

mammals. Initially, the antecedent of the rule {}→Mammals covers 5

positive and 10 negative examples. Thus, the accuracy of the rule is only

0.333. Next, consider the following three candidate conjuncts to be added to

the left-hand side of the rule: Skin cover=hair, Body temperature=warm, and

Has legs=No. The number of positive and negative examples covered by the

rule after adding each conjunct along with their respective accuracy and

FOIL’s information gain are shown in the following table.

Candidate rule p1 n1 Accuracy Info Gain

{Skin Cover=hair}→mammals 3 0 1.000 4.755

{Body temperature=wam}→mammals 5 1 0.714 5.498

{Has legs=No}→mammals 2 4 0.200 −0.737

Although Skin cover=hair has the highest accuracy among the three

candidates, the conjunct Body temperature=warm has the highest FOIL’s

information gain. Thus, it is chosen to extend the rule (see Figure 4.2). This

process continues until adding new conjuncts no longer improves the

information gain measure.

Rule Pruning

The rules generated by the Learn-One-Rule function can be pruned to

improve their generalization errors. RIPPER prunes the rules based on their

performance on the validation set. The following metric is computed to

determine whether pruning is needed: (p−n)/(p+n), where *p*(*n*) is the number

of positive (negative) examples in the validation set covered by the rule. This

metric is monotonically related to the rule’s accuracy on the validation set. If

the metric improves after pruning, then the conjunct is removed. Pruning is

done starting from the last conjunct added to the rule. For example, given a

rule ABCD→y, RIPPER checks whether *D *should be pruned first, followed

by *CD*, *BCD*, etc. While the original rule covers only positive examples, the

pruned rule may cover some of the negative examples in the training set.

Building the Rule Set

After generating a rule, all the positive and negative examples covered by the

rule are eliminated. The rule is then added into the rule set as long as it does

not violate the stopping condition, which is based on the minimum

description length principle. If the new rule increases the total description

length of the rule set by at least *d *bits, then RIPPER stops adding rules into

its rule set (by default, *d *is chosen to be 64 bits). Another stopping condition

used by RIPPER is that the error rate of the rule on the validation set must

not exceed 50%.

Figure 4.2.

General-to-specific and specific-to-general rule-growing strategies.

Figure 4.2. Full Alternative Text

RIPPER also performs additional optimization steps to determine whether

some of the existing rules in the rule set can be replaced by better alternative

rules. Readers who are interested in the details of the optimization method

may refer to the reference cited at the end of this chapter.

Instance Elimination

After a rule is extracted, RIPPER eliminates the positive and negative

examples covered by the rule. The rationale for doing this is illustrated in the

next example.

Figure 4.3 shows three possible rules, *R*1, *R*2, and *R*3, extracted from a

training set that contains 29 positive examples and 21 negative examples. The

accuracies of *R*1, *R*2, and *R*3 are 12/15 (80%), 7/10 (70%), and 8/12 (66.7%),

respectively. *R*1 is generated first because it has the highest accuracy. After

generating *R*1, the algorithm must remove the examples covered by the rule

so that the next rule generated by the algorithm is different than *R*1. The

question is, should it remove the positive examples only, negative examples

only, or both? To answer this, suppose the algorithm must choose between

generating *R*2 or *R*3 after *R*1. Even though *R*2 has a higher accuracy than *R*3

(70% versus 66.7%), observe that the region covered by *R*2 is disjoint from

*R*1, while the region covered by *R*3 overlaps with *R*1. As a result, *R*1 and *R*3

together cover 18 positive and 5 negative examples (resulting in an overall

accuracy of 78.3%), whereas *R*1 and *R*2 together cover 19 positive and 6

negative examples (resulting in a lower overall accuracy of 76%). If the

positive examples covered by *R*1 are not removed, then we may overestimate

the effective accuracy of *R*3. If the negative examples covered by *R*1 are not

removed, then we may underestimate the accuracy of *R*3. In the latter case,

we might end up preferring *R*2 over *R*3 even though half of the false positive

errors committed by *R*3 have already been accounted for by the preceding

rule, *R*1. This example shows that the effective accuracy after adding *R*2 or

*R*3 to the rule set becomes evident only when both positive and negative

examples covered by *R*1 are removed.

Figure 4.3.

Elimination of training instances by the sequential covering

algorithm. *R*1, *R*2, and *R*3 represent regions covered by three

different rules.

Figure 4.3. Full Alternative Text

4.2.4 Indirect Methods for Rule

Extraction

This section presents a method for generating a rule set from a decision tree.

In principle, every path from the root node to the leaf node of a decision tree

can be expressed as a classification rule. The test conditions encountered

along the path form the conjuncts of the rule antecedent, while the class label

at the leaf node is assigned to the rule consequent. Figure 4.4 shows an

example of a rule set generated from a decision tree. Notice that the rule set is

exhaustive and contains mutually exclusive rules. However, some of the rules

can be simplified as shown in the next example.

Figure 4.4.

Converting a decision tree into classification rules.

Figure 4.4. Full Alternative Text

Example 4.3.

Consider the following three rules from Figure 4.4:

r2:(P=No)∧(Q=Yes)→+r3:(P=Yes)∧(R=No)→+r5:

(P=Yes)∧(R=Yes)∧(Q=Yes)→+.

Observe that the rule set always predicts a positive class when the value of *Q*

is Yes. Therefore, we may simplify the rules as follows:

r2′:(Q=Yes)→+r3:(P=Yes)∧(R=No)→+.

r3 is retained to cover the remaining instances of the positive class. Although

the rules obtained after simplification are no longer mutually exclusive, they

are less complex and are easier to interpret.

In the following, we describe an approach used by the C4.5rules algorithm to

generate a rule set from a decision tree. Figure 4.5 shows the decision tree

and resulting classification rules obtained for the data set given in Table 4.2.

Rule Generation

Classification rules are extracted for every path from the root to one of the

leaf nodes in the decision tree. Given a classification rule r:A→y, we

consider a simplified rule, r′:A′→y where A′ is obtained by removing one of

the conjuncts in *A*. The simplified rule with the lowest pessimistic error rate

is retained provided its error rate is less than that of the original rule. The

rule-pruning step is repeated until the pessimistic error of the rule cannot be

improved further. Because some of the rules may become identical after

pruning, the duplicate rules are discarded.

Figure 4.5.

Classification rules extracted from a decision tree for the vertebrate

classification problem.

Figure 4.5. Full Alternative Text

Rule Ordering

After generating the rule set, C4.5rules uses the class-based ordering scheme

to order the extracted rules. Rules that predict the same class are grouped

together into the same subset. The total description length for each subset is

computed, and the classes are arranged in increasing order of their total

description length. The class that has the smallest description length is given

the highest priority because it is expected to contain the best set of rules. The

total description length for a class is given by Lexception+g×Lmodel, where

Lexception is the number of bits needed to encode the misclassified

examples, *L*model is the number of bits needed to encode the model, and *g *is

a tuning parameter whose default value is 0.5. The tuning parameter depends

on the number of redundant attributes present in the model. The value of the

tuning parameter is small if the model contains many redundant attributes.

4.2.5 Characteristics of Rule-Based

Classifiers

1. Rule-based classifiers have very similar characteristics as decision trees.

The expressiveness of a rule set is almost equivalent to that of a decision

tree because a decision tree can be represented by a set of mutually

exclusive and exhaustive rules. Both rule-based and decision tree

classifiers create rectilinear partitions of the attribute space and assign a

class to each partition. However, a rule-based classifier can allow

multiple rules to be triggered for a given instance, thus enabling the

learning of more complex models than decision trees.

2. Like decision trees, rule-based classifiers can handle varying types of

categorical and continuous attributes and can easily work in multiclass

classification scenarios. Rule-based classifiers are generally used to

produce descriptive models that are easier to interpret but give

comparable performance to the decision tree classifier.

3. Rule-based classifiers can easily handle the presence of redundant

attributes that are highly correlated with one other. This is because once

an attribute has been used as a conjunct in a rule antecedent, the

remaining redundant attributes would show little to no FOIL’s

information gain and would thus be ignored.

4. Since irrelevant attributes show poor information gain, rule-based

classifiers can avoid selecting irrelevant attributes if there are other

relevant attributes that show better information gain. However, if the

problem is complex and there are interacting attributes that can

collectively distinguish between the classes but individually show poor

information gain, it is likely for an irrelevant attribute to be accidentally

favored over a relevant attribute just by random chance. Hence, rulebased classifiers can show poor performance in the presence of

interacting attributes, when the number of irrelevant attributes is large.

5. The class-based ordering strategy adopted by RIPPER, which

emphasizes giving higher priority to rare classes, is well suited for

handling training data sets with imbalanced class distributions.

6. Rule-based classifiers are not well-suited for handling missing values in

the test set. This is because the position of rules in a rule set follows a

certain ordering strategy and even if a test instance is covered by

multiple rules, they can assign different class labels depending on their

position in the rule set. Hence, if a certain rule involves an attribute that

is missing in a test instance, it is difficult to ignore the rule and proceed

to the subsequent rules in the rule set, as such a strategy can result in

incorrect class assignments.

4.3 Nearest Neighbor Classifiers

The classification framework shown in Figure 3.3 involves a two-step

process:

(1) an inductive step for constructing a classification model from data, and

(2) a deductive step for applying the model to test examples. Decision tree

and rule-based classifiers are examples of eager learners because they are

designed to learn a model that maps the input attributes to the class label as

soon as the training data becomes available. An opposite strategy would be to

delay the process of modeling the training data until it is needed to classify

the test instances. Techniques that employ this strategy are known as lazy

learners. An example of a lazy learner is the Rote classifier, which

memorizes the entire training data and performs classification only if the

attributes of a test instance match one of the training examples exactly. An

obvious drawback of this approach is that some test instances may not be

classified because they do not match any training example.

One way to make this approach more flexible is to find all the training

examples that are relatively similar to the attributes of the test instances.

These examples, which are known as nearest neighbors, can be used to

determine the class label of the test instance. The justification for using

nearest neighbors is best exemplified by the following saying: *“If it walks*

*like a duck, quacks like a duck, and looks like a duck, then it’s probably a*

*duck.” *A nearest neighbor classifier represents each example as a data point

in a *d*-dimensional space, where *d *is the number of attributes. Given a test

instance, we compute its proximity to the training instances according to one

of the proximity measures described in Section 2.4 on page 71. The *k*-nearest

neighbors of a given test instance *z *refer to the *k *training examples that are

closest to *z*.

Figure 4.6 illustrates the 1-, 2-, and 3-nearest neighbors of a test instance

located at the center of each circle. The instance is classified based on the

class labels of its neighbors. In the case where the neighbors have more than

one label, the test instance is assigned to the majority class of its nearest

neighbors. In Figure 4.6(a), the 1-nearest neighbor of the instance is a

negative example. Therefore the instance is assigned to the negative class. If

the number of nearest neighbors is three, as shown in Figure 4.6(c), then the

neighborhood contains two positive examples and one negative example.

Using the majority voting scheme, the instance is assigned to the positive

class. In the case where there is a tie between the classes (see Figure 4.6(b)),

we may randomly choose one of them to classify the data point.

Figure 4.6.

The 1-, 2-, and 3-nearest neighbors of an instance.

Figure 4.6. Full Alternative Text

The preceding discussion underscores the importance of choosing the right

value for *k*. If *k *is too small, then the nearest neighbor classifier may be

susceptible to overfitting due to noise, i.e., mislabeled examples in the

training data. On the other hand, if *k *is too large, the nearest neighbor

classifier may misclassify the test instance because its list of nearest

neighbors includes training examples that are located far away from its

neighborhood (see Figure 4.7).

Figure 4.7.

*k*-nearest neighbor classification with large *k*.

4.3.1 Algorithm

A high-level summary of the nearest neighbor classification method is given

in Algorithm 4.2. The algorithm computes the distance (or similarity)

between each test instance z=(x′, y′) and all the training examples (x, y)∈D

to determine its nearest neighbor list, Dz. Such computation can be costly if

the number of training examples is large. However, efficient indexing

techniques are available to reduce the computation needed to find the nearest

neighbors of a test instance.

Algorithm 4.2 The *k*-nearest

neighbor classifier.

1: Let *k *be the number of nearest neighbors and *D *be the set of training examples.

2: for each test instance z=(x′, y′) do

3: Compute z=(x′, x), the distance between *z *and every example,

4: Select Dz⊆D, the set of *k *closest training examples to *z*.

5: y′=argmaxv∑(xi, yi)∈DzI(v=yi)

6: end for

Once the nearest neighbor list is obtained, the test instance is classified based

on the majority class of its nearest neighbors:

Majority Voting: y′=argmaxv∑(xi, yi)∈DzI(v=yi), (4.5)

where *v *is a class label, yi is the class label for one of the nearest neighbors,

and I(⋅) is an indicator function that returns the value 1 if its argument is true

and 0 otherwise.

In the majority voting approach, every neighbor has the same impact on the

classification. This makes the algorithm sensitive to the choice of *k*, as shown

in Figure 4.6. One way to reduce the impact of *k *is to weight the influence of

each nearest neighbor xi according to its distance: wi=1/d(x′, xi)2. As a

result, training examples that are located far away from *z *have a weaker

impact on the classification compared to those that are located close to *z*.

Using the distance-weighted voting scheme, the class label can be determined

as follows:

Distance-Weighted Voting: y′=argmaxv∑(xi, yi)∈Dzwi×I(v=yi). (4.6)

4.3.2 Characteristics of Nearest

Neighbor Classifiers

1. Nearest neighbor classification is part of a more general technique

known as instance-based learning, which does not build a global model,

but rather uses the training examples to make predictions for a test

instance. (Thus, such classifiers are often said to be “model free.”) Such

algorithms require a proximity measure to determine the similarity or

distance between instances and a classification function that returns the

predicted class of a test instance based on its proximity to other

instances.

2. Although lazy learners, such as nearest neighbor classifiers, do not

require model building, classifying a test instance can be quite expensive

because we need to compute the proximity values individually between

the test and training examples. In contrast, eager learners often spend the

bulk of their computing resources for model building. Once a model has

been built, classifying a test instance is extremely fast.

3. Nearest neighbor classifiers make their predictions based on local

information. (This is equivalent to building a local model for each test

instance.) By contrast, decision tree and rule-based classifiers attempt to

find a global model that fits the entire input space. Because the

classification decisions are made locally, nearest neighbor classifiers

(with small values of *k*) are quite susceptible to noise.

4. Nearest neighbor classifiers can produce decision boundaries of

arbitrary shape. Such boundaries provide a more flexible model

representation compared to decision tree and rule-based classifiers that

are often constrained to rectilinear decision boundaries. The decision

boundaries of nearest neighbor classifiers also have high variability

because they depend on the composition of training examples in the

local neighborhood. Increasing the number of nearest neighbors may

reduce such variability.

5. Nearest neighbor classifiers have difficulty handling missing values in

both the training and test sets since proximity computations normally

require the presence of all attributes. Although, the subset of attributes

present in two instances can be used to compute a proximity, such an

approach may not produce good results since the proximity measures

may be different for each pair of instances and thus hard to compare.

6. Nearest neighbor classifiers can handle the presence of interacting

attributes, i.e., attributes that have more predictive power taken in

combination then by themselves, by using appropriate proximity

measures that can incorporate the effects of multiple attributes together.

7. The presence of irrelevant attributes can distort commonly used

proximity measures, especially when the number of irrelevant attributes

is large. Furthermore, if there are a large number of redundant attributes

that are highly correlated with each other, then the proximity measure

can be overly biased toward such attributes, resulting in improper

estimates of distance. Hence, the presence of irrelevant and redundant

attributes can adversely affect the performance of nearest neighbor

classifiers.

8. Nearest neighbor classifiers can produce wrong predictions unless the

appropriate proximity measure and data preprocessing steps are taken.

For example, suppose we want to classify a group of people based on

attributes such as height (measured in meters) and weight (measured in

pounds). The height attribute has a low variability, ranging from 1.5 m

to 1.85 m, whereas the weight attribute may vary from 90 lb. to 250 lb.

If the scale of the attributes are not taken into consideration, the

proximity measure may be dominated by differences in the weights of a

person.

4.4 Naïve Bayes Classifier

Many classification problems involve uncertainty. First, the observed

attributes and class labels may be unreliable due to imperfections in the

measurement process, e.g., due to the limited preciseness of sensor devices.

Second, the set of attributes chosen for classification may not be fully

representative of the target class, resulting in uncertain predictions. To

illustrate this, consider the problem of predicting a person’s risk for heart

disease based on a model that uses their diet and workout frequency as

attributes. Although most people who eat healthily and exercise regularly

have less chance of developing heart disease, they may still be at risk due to

other *latent factors*, such as heredity, excessive smoking, and alcohol abuse,

that are not captured in the model. Third, a classification model learned over

a finite training set may not be able to fully capture the true relationships in

the overall data, as discussed in the context of model overfitting in the

previous chapter. Finally, uncertainty in predictions may arise due to the

inherent *random nature *of real-world systems, such as those encountered in

weather forecasting problems.

In the presence of uncertainty, there is a need to not only make predictions of

class labels but also provide a measure of confidence associated with every

prediction. Probability theory offers a systematic way for quantifying and

manipulating uncertainty in data, and thus, is an appealing framework for

assessing the confidence of predictions. Classification models that make use

of probability theory to represent the relationship between attributes and class

labels are known as probabilistic classification models. In this section, we

present the naïve Bayes classifier, which is one of the simplest and most

widely-used probabilistic classification models.

4.4.1 Basics of Probability Theory

Before we discuss how the naïve Bayes classifier works, we first introduce

some basics of probability theory that will be useful in understanding the

probabilistic classification models presented in this chapter. This involves

defining the notion of probability and introducing some common approaches

for manipulating probability values.

Consider a variable *X*, which can take any discrete value from the set

{x1, …, xk}. When we have multiple observations of that variable, such as in

a data set where the variable describes some characteristic of data objects,

then we can compute the relative frequency with which each value occurs.

Specifically, suppose that *X *has the value xi for ni data objects. The relative

frequency with which we observe the event X=xi is then ni/N, where *N*

denotes the total number of occurrences (N=∑i=1kni). These relative

frequencies characterize the uncertainty that we have with respect to what

value *X *may take for an unseen observation and motivates the notion of

probability.

More formally, the probability of an event *e*, e.g., P(X=xi), measures how

likely it is for the event *e *to occur. The most traditional view of probability is

based on relative frequency of events (frequentist), while the Bayesian

viewpoint (described later) takes a more flexible view of probabilities. In

either case, a probability is always a number between 0 and 1. Further, the

sum of probability values of all possible events, e.g., outcomes of a variable

*X *is equal to 1. Variables that have probabilities associated with each possible

outcome (values) are known as random variables.

Now, let us consider two random variables, *X *and *Y *, that can each take *k*

discrete values. Let nij be the number of times we observe X=xi and Y=yj,

out of a total number of *N *occurrences. The joint probability of observing

X=xi and Y=yj together can be estimated as

P(X=xi, Y=yi)=nijN. (4.7)

(This is an estimate since we typically have only a finite subset of all possible

observations.) Joint probabilities can be used to answer questions such as

“what is the probability that there will be a surprise quiz today AND I will be

late for the class.” Joint probabilities are symmetric, i.e.,

P(X=x, Y=y)=P(Y=y, X=x). For joint probabilities, it is to useful to consider

their sum with respect to one of the random variables, as described in the

following equation:

∑j=1kP(X=xi,Y=yj)=∑j=1knijN=niN=P(X=xi), (4.8)

where ni is the total number of times we observe X=xi irrespective of the

value of *Y*. Notice that ni/N is essentially the probability of observing X=xi.

Hence, by summing out the joint probabilities with respect to a random

variable *Y *, we obtain the probability of observing the remaining variable *X*.

This operation is called marginalization and the probability value P(X=xi)

obtained by marginalizing out *Y *is sometimes called the marginal

probability of *X*. As we will see later, joint probability and marginal

probability form the basic building blocks of a number of probabilistic

classification models discussed in this chapter.

Notice that in the previous discussions, we used P(X=xi) to denote the

probability of a particular outcome of a random variable *X*. This notation can

easily become cumbersome when a number of random variables are involved.

Hence, in the remainder of this section, we will use *P*(*X*) to denote the

probability of any generic outcome of the random variable *X*, while P(xi) will

be used to represent the probability of the specific outcome xi.

Bayes Theorem

*Suppose you have invited two of your friends Alex and Martha to a*

*dinner party. You know that Alex attends *40% *of the parties he is invited*

*to. Further, if Alex is going to a party, there is an *80% *chance of*

*Martha coming along. On the other hand, if Alex is not going to the*

*party, the chance of Martha coming to the party is reduced to *30%*. If*

*Martha has responded that she will be coming to your party, what is the*

*probability that Alex will also be coming?*

Bayes theorem presents the statistical principle for answering questions like

the previous one, where evidence from multiple sources has to be combined

with prior beliefs to arrive at predictions. Bayes theorem can be briefly

described as follows.

Let P(Y|X) denotethe conditional probability of observing the random

variable *Y *whenever the random variable *X *takes a particular value. P(Y|X) is

often read as the probability of observing *Y conditioned *on the outcome of *X*.

Conditional probabilities can be used for answering questions such as “given

that it is going to rain today, what will be the probability that I will go to the

class.” Conditional probabilities of *X *and *Y *are related to their joint

probability in the following way:

P(Y|X)=P(X, Y)P(X), which implies (4.9)

P(X, Y)=P(Y|X)×P(X)=P(X|Y)×P(Y). (4.10)

Rearranging the last two expressions in Equation 4.10 leads to Equation 4.11,

which is known as Bayes theorem:

P(Y|X)=P(X|Y)P(Y)P(X). (4.11)

Bayes theorem provides a relationship between the conditional probabilities

P(Y|X) and P(X|Y). Note that the denominator in Equation 4.11 involves the

marginal probability of *X*, which can also be represented as

P(X)=∑i=1kP(X, yi)=∑i=1kP(X|yi)×P(yi).

Using the previous expression for *P*(*X*), we can obtain the following equation

for P(Y|X) solely in terms of P(X|Y) and *P*(*Y*):

P(Y|X)=P(X|Y)P(Y)∑i−1kP(X|yi)P(yi). (4.12)

Example 4.4. [Bayes Theorem]

Bayes theorem can be used to solve a number of inferential questions about

random variables. For example, consider the problem stated at the beginning

on inferring whether Alex will come to the party. Let P(A=1) denote the

probability of Alex going to a party, while P(A=0) denotes the probability of

him not going to a party. We know that

P(A=1)=0.4,andP(A=0)=1−P(A=1)=0.6.

Further, let P(M=1|A) denote the conditional probability of Martha going to a

party conditioned on whether Alex is going to the party. P(M=1|A) takes the

following values:

P(M=1|A=1)=0.8,andP(M=1|A=0)=0.3.

We can use the above values of P(M|A) and *P*(*A*) to compute the probability

of Alex going to the party given Martha is going to the party, P(A=1|M=1),

as follows:

P(A=1|M=1)=P(M=1|A=1)P(A=1)P(M=1|A=0)P(A=0)+P(M=1|A=1)P(A=1),=

(4.13)

Notice that even though the prior probability *P*(*A*) of Alex going to the party

is low, the observation that Martha is going, M=1, affects the conditional

probability P(A=1|M=1). This shows the value of Bayes theorem in

combining prior assumptions with observed outcomes to make predictions.

Since P(A=1|M=1)>0.5, it is more likely for Alex to join if Martha is going to

the party.

Using Bayes Theorem for

Classification

For the purpose of classification, we are interested in computing the

probability of observing a class label *y *for a data instance given its set of

attribute values x. This can be represented as P(y|x), which is known as the

posterior probability of the target class. Using the Bayes Theorem, we can

represent the posterior probability as

P(y|x)=P(x|y)P(y)P(x) (4.14)

Note that the numerator of the previous equation involves two terms, P(x|y)

and *P*(*y*), both of which contribute to the posterior probability P(y|x). We

describe both of these terms in the following.

The first term P(x|y) is known as the class-conditional probability of the

attributes given the class label. P(x|y) measures the likelihood of observing x

from the distribution of instances belonging to *y*. If x indeed belongs to class

*y*, then we should expect P(x|y) to be high. From this point of view, the use of

class-conditional probabilities attempts to capture the process from which the

data instances were generated. Because of this interpretation, probabilistic

classification models that involve computing class-conditional probabilities

are known as generative classification models. Apart from their use in

computing posterior probabilities and making predictions, class-conditional

probabilities also provide insights about the underlying mechanism behind

the generation of attribute values.

The second term in the numerator of Equation 4.14 is the prior probability

*P*(*y*). The prior probability captures our prior beliefs about the distribution of

class labels, independent of the observed attribute values. (This is the

Bayesian viewpoint.) For example, we may have a prior belief that the

likelihood of any person to suffer from a heart disease is α, irrespective of

their diagnosis reports. The prior probability can either be obtained using

expert knowledge, or inferred from historical distribution of class labels.

The denominator in Equation 4.14 involves the probability of evidence, *P *(x).

Note that this term does not depend on the class label and thus can be treated

as a normalization constant in the computation of posterior probabilities.

Further, the value of *P*(x) can be calculated as P(x)=∑iP(x|yi)P(yi).

Bayes theorem provides a convenient way to combine our prior beliefs with

the likelihood of obtaining the observed attribute values. During the training

phase, we are required to learn the parameters for *P*(*y*) and P(x|y). The prior

probability *P*(*y*) can be easily estimated from the training set by computing

the fraction of training instances that belong to each class. To compute the

class-conditional probabilities, one approach is to consider the fraction of

training instances of a given class for every possible combination of attribute

values. For example, suppose that there are two attributes X1 and X2 that can

each take a discrete value from c1 to ck. Let n0 denote the number of training

instances belonging to class 0, out of which nij0 number of training instances

have X1=ci and X2=cj. The class-conditional probability can then be given as

P(X1=ci, X2=cj|Y=0)=nij0n0.

This approach can easily become computationally prohibitive as the number

of attributes increase, due to the exponential growth in the number of

attribute value combinations. For example, if every attribute can take *k*

discrete values, then the number of attribute value combinations is equal to

kd, where *d *is the number of attributes. The large number of attribute value

combinations can also result in poor estimates of class-conditional

probabilities, since every combination will have fewer training instances

when the size of training set is small.

In the following, we present the naïve Bayes classifier, which makes a

simplifying assumption about the class-conditional probabilities, known as

the naïve Bayes assumption. The use of this assumption significantly helps

in obtaining reliable estimates of class-conditional probabilities, even when

the number of attributes are large.

4.4.2 Naïve Bayes Assumption

The naïve Bayes classifier assumes that the class-conditional probability of

all attributes x can be factored as a product of class-conditional probabilities

of every attribute xi, as described in the following equation:

P(x|y)=∏i=1dP(xi|y), (4.15)

where every data instance x consists of *d *attributes, {x1, x2, …, xd}. The

basic assumption behind the previous equation is that the attribute values xi

are conditionally independent of each other, given the class label *y*. This

means that the attributes are influenced only by the target class and if we

know the class label, then we can consider the attributes to be independent of

each other. The concept of conditional independence can be formally stated

as follows.

Conditional Independence

Let X1, X2,, and Y denote three sets of random variables. The variables in

X1 are said to be conditionally independent of X2, given Y, if the following

condition holds:

P(X1|X2, Y)=P(X1|Y). (4.16)

This means that conditioned on Y, the distribution of X1 is not influenced by

the outcomes of X2, and hence is conditionally independent of X2. To

illustrate the notion of conditional independence, consider the relationship

between a person’s arm length (X1) and his or her reading skills (X2). One

might observe that people with longer arms tend to have higher levels of

reading skills, and thus consider X1 and X2 to be related to each other.

However, this relationship can be explained by another factor, which is the

age of the person (Y). A young child tends to have short arms and lacks the

reading skills of an adult. If the age of a person is fixed, then the observed

relationship between arm length and reading skills disappears. Thus, we can

conclude that arm length and reading skills are not directly related to each

other and are conditionally independent when the age variable is fixed.

Another way of describing conditional independence is to consider the joint

conditional probability, P(X1, X2|Y), as follows:

P(X1, X2|Y)=P(X1, X2, Y)P(Y)=P(X1, X2, Y)P(X2, Y)×P(X2, Y)P(Y)=P(X1

(4.17)

where Equation 4.16 was used to obtain the last line of Equation 4.17. The

previous description of conditional independence is quite useful from an

operational perspective. It states that the joint conditional probability of X1

and X2 given *Y *can be factored as the product of conditional probabilities of

X1 and X2 considered separately. This forms the basis of the naïve Bayes

assumption stated in Equation 4.15.

How a Naïve Bayes Classifier

Works

Using the naïve Bayes assumption, we only need to estimate the conditional

probability of each xi given *Y *separately, instead of computing the class-

conditional probability for every combination of attribute values. For

example, if ni0 and nj0 denote the number of training instances belonging to

class 0 with X1=ci and X2=cj, respectively, then the class-conditional

probability can be estimated as

P(X1=ci, X2=xj|Y=0)=ni0n0×nj0n0.

In the previous equation, we only need to count the number of training

instances for every one of the *k *values of an attribute *X*, irrespective of the

values of other attributes. Hence, the number of parameters needed to learn

class-conditional probabilities is reduced from dk to *dk*. This greatly

simplifies the expression for the class-conditional probability and makes it

more amenable to learning parameters and making predictions, even in highdimensional settings.

The naïve Bayes classifier computes the posterior probability for a test

instance x by using the following equation:

P(y|x)=P(y)∏i=1dP(xi|y)P(x) (4.18)

Since *P *(x) is fixed for every *y *and only acts as a normalizing constant to

ensure that P(y|x)∈[0, 1], we can write

P(y|x)∝P(y)∏i=1dP(xi|y).

Hence, it is sufficient to choose the class that maximizes P(y)∏i=1dP(xi|y).

One of the useful properties of the naïve Bayes classifier is that it can easily

work with incomplete information about data instances, when only a subset

of attributes are observed at every instance. For example, if we only observe

*p *out of *d *attributes at a data instance, then we can still compute

P(y)∏i=1pP(xi|y) using those *p *attributes and choose the class with the

maximum value. The naïve Bayes classifier can thus naturally handle missing

values in test instances. In fact, in the extreme case where no attributes are

observed, we can still use the prior probability *P*(*y*) as an estimate of the

posterior probability. As we observe more attributes, we can keep refining the

posterior probability to better reflect the likelihood of observing the data

instance.

In the next two subsections, we describe several approaches for estimating

the conditional probabilities P(xi|y) for categorical and continuous attributes

from the training set.

Estimating Conditional

Probabilities for Categorical

Attributes

For a categorical attribute Xi, the conditional probability P(Xi=c|y) *is*

*estimated according to the fraction of training instances in class y where *Xi

takes on a particular categorical value *c*.

P(Xi=c|y)=ncn,

where *n *is the number of training instances belonging to class *y*, out of which

nc number of instances have Xi=c. For example, in the training set given in

Figure 4.8, seven people have the class label Defaulted Borrower=No, out of

which three people have Home Owner=Yes while the remaining four have

Home Owner=No. As a result, the conditional probability for

P(Home Owner=Yes|Defaulted Borrower=No) is equal to 3/7. Similarly, the

conditional probability for defaulted borrowers with Marital Status=Single is

given by P(Marital Status=Single|Defaulted Borrower=Yes)=2/3. Note that

the sum of conditional probabilities over all possible outcomes of Xi is equal

to one, i.e., ∑cP(Xi=c|y)=1,.

Figure 4.8.

Training set for predicting the loan default problem.

Estimating Conditional

Probabilities for Continuous

Attributes

There are two ways to estimate the class-conditional probabilities for

continuous attributes:

1. We can discretize each continuous attribute and then replace the

continuous values with their corresponding discrete intervals. This

approach transforms the continuous attributes into ordinal attributes, and

the simple method described previously for computing the conditional

probabilities of categorical attributes can be employed. Note that the

estimation error of this method depends on the discretization strategy (as

described in Section 2.3.6 on page 63), as well as the number of discrete

intervals. If the number of intervals is too large, every interval may have

an insufficient number of training instances to provide a reliable

estimate of P(Xi|Y). On the other hand, if the number of intervals is too

small, then the discretization process may loose information about the

true distribution of continuous values, and thus result in poor

predictions.

2. We can assume a certain form of probability distribution for the

continuous variable and estimate the parameters of the distribution using

the training data. For example, we can use a Gaussian distribution to

represent the conditional probability of continuous attributes. The

Gaussian distribution is characterized by two parameters, the mean, μ,

and the variance, σ2. For each class yj, the class-conditional probability

for attribute Xi is

P(Xi=xi|Y=yj)=12πσijexp[−(xi−μij)22σij2 ]. (4.19)

The parameter μij can be estimated using the sample mean of Xi(x¯) for

all training instances that belong to yj. Similarly, σij2 can be estimated

from the sample variance (s2) of such training instances. For example,

consider the annual income attribute shown in Figure 4.8. The sample

mean and variance for this attribute with respect to the class No are

x¯=125+100+70+…+757=100s2=(125−110)2+(100−110)2+…

(75−110)26=2975s=2975=54.54.

Given a test instance with taxable income equal to $120K, we can use

the following value as its conditional probability given class No:

P(Income=120|No)=12π(54.54)exp−(120−110)22×2975=0.0072.

Example 4.5. [Naïve Bayes

Classifier]

Consider the data set shown in Figure 4.9(a), where the target class is

Defaulted Borrower, which can take two values Yes and No. We can

compute the class-conditional probability for each categorical attribute and

the sample mean and variance for the continuous attribute, as summarized in

Figure 4.9(b).

We are interested in predicting the class label of a test instance x=

(Home Owner=No, Marital Status=Married, Annual Income=$120K). To do

this, we first compute the prior probabilities by counting the number of

training instances belonging to every class. We thus obtain P(yes)=0.3 and

P(No)=0.7. Next, we can compute the class-conditional probability as

follows:

Figure 4.9.

The naïve Bayes classifier for the loan classification problem.

Figure 4.9. Full Alternative Text

P(x|NO)=P(Home Owner=No|No)×P(Status=Married|No)×P(Annual Income=

Notice that the class-conditional probability for class Yes has become 0

because there are no instances belonging to class Yes with Status=Married in

the training set. Using these class-conditional probabilities, we can estimate

the posterior probabilities as

P(No|x)=0.7×0.0024P(x).=0.0016α.P(Yes|x)=0.3×0P(x)=0.

where α=1/P(x) is a normalizing constant. Since P(No|x)>P(Yes|x), the

instance is classified as No.

Handling Zero Conditional

Probabilities

The preceding example illustrates a potential problem with using the naïve

Bayes assumption in estimating class-conditional probabilities. If the

conditional probability for any of the attributes is zero, then the entire

expression for the class-conditional probability becomes zero. Note that zero

conditional probabilities arise when the number of training instances is small

and the number of possible values of an attribute is large. In such cases, it

may happen that a combination of attribute values and class labels are never

observed, resulting in a zero conditional probability.

In a more extreme case, if the training instances do not cover some

combinations of attribute values and class labels, then we may not be able to

even classify some of the test instances. For example, if

P(Marital Status=Divorced|No) is zero instead of 1/7, then a data instance

with attribute set x=

(Home Owner=Yes, Marital Status=Divorced, Income=$120K) has the

following class-conditional probabilities:

P(x|No)=3/7×0×0.0072=0.P(x|Yes)=0×1/3×1.2×10−9=0.

Since both the class-conditional probabilities are 0, the naïve Bayes classifier

will not be able to classify the instance. To address this problem, it is

important to adjust the conditional probability estimates so that they are not

as brittle as simply using fractions of training instances. This can be achieved

by using the following alternate estimates of conditional probability:

Laplace estimate:P(Xi=c|y)=nc+1n+v, (4.20)

m-estimate:P(Xi=c|y)=nc+mpn+m, (4.21)

where *n *is the number of training instances belonging to class *y*, nc is the

number of training instances with Xi=c and Y=y, *v *is the total number of

attribute values that Xi can take, *p *is some initial estimate of P(Xi=c|y) that is

known a priori, and *m *is a hyper-parameter that indicates our confidence in

using *p *when the fraction of training instances is too brittle. Note that even if

nc=0, both Laplace and m-estimate provide non-zero values of conditional

probabilities. Hence, they avoid the problem of vanishing class-conditional

probabilities and thus generally provide more robust estimates of posterior

probabilities.

Characteristics of Naïve Bayes

Classifiers

1. Naïve Bayes classifiers are probabilistic classification models that are

able to quantify the uncertainty in predictions by providing posterior

probability estimates. They are also generative classification models as

they treat the target class as the causative factor for generating the data

instances. Hence, apart from computing posterior probabilities, naïve

Bayes classifiers also attempt to capture the underlying mechanism

behind the generation of data instances belonging to every class. They

are thus useful for gaining predictive as well as descriptive insights.

2. By using the naïve Bayes assumption, they can easily compute classconditional probabilities even in high-dimensional settings, provided

that the attributes are conditionally independent of each other given the

class labels. This property makes naïve Bayes classifier a simple and

effective classification technique that is commonly used in diverse

application problems, such as text classification.

3. Naïve Bayes classifiers are robust to isolated noise points because such

points are not able to significantly impact the conditional probability

estimates, as they are often averaged out during training.

4. Naïve Bayes classifiers can handle missing values in the training set by

ignoring the missing values of every attribute while computing its

conditional probability estimates. Further, naïve Bayes classifiers can

effectively handle missing values in a test instance, by using only the

non-missing attribute values while computing posterior probabilities. If

the frequency of missing values for a particular attribute value depends

on class label, then this approach will not accurately estimate posterior

probabilities.

5. Naïve Bayes classifiers are robust to irrelevant attributes. If Xi is an

irrelevant attribute, then P(Xi|Y) becomes almost uniformly distributed

for every class *y*. The class-conditional probabilities for every class thus

receive similar contributions of P(Xi|Y), resulting in negligible impact

on the posterior probability estimates.

6. Correlated attributes can degrade the performance of naïve Bayes

classifiers because the naïve Bayes assumption of conditional

independence no longer holds for such attributes. For example, consider

the following probabilities:

P(A=0|Y=0)=0.4,P(A=1|Y=0)=0.6,P(A=0|Y=1)=0.6,P(A=1|Y=1)=0.4,

where *A *is a binary attribute and *Y *is a binary class variable. Suppose

there is another binary attribute *B *that is perfectly correlated with *A*

when Y=0, but is independent of *A *when Y=1. For simplicity, assume

that the conditional probabilities for *B *are the same as for *A*. Given an

instance with attributes A=0, B=0, and assuming conditional

independence, we can compute its posterior probabilities as follows:

P(Y=0|A=0, B=0)=P(A=0|Y=0)P(B=0|Y=0)P(Y=0)P(A=0, B=0)=0.16×P

If P(Y=0)=P(Y=1), then the naïve Bayes classifier would assign the instance

to class 1. However, the truth is,

P(A=0, B=0|Y=0)=P(A=0|Y=0)=0.4,

because *A *and *B *are perfectly correlated when Y=0. As a result, the posterior

probability for Y=0 is

P(Y=0|A=0, B=0)=P(A=0, B=0|Y=0)P(Y=0)P(A=0, B=0)=0.4×P(Y=0)P(A=0

which is larger than that for Y=1. The instance should have been classified as

class 0. Hence, the naïve Bayes classifier can produce incorrect results when

the attributes are not conditionally independent given the class labels. Naïve

Bayes classifiers are thus not well-suited for handling redundant or

interacting attributes.

4.5 Bayesian Networks

The conditional independence assumption made by naïve Bayes classifiers

may seem too rigid, especially for classification problems where the

attributes are dependent on each other even after conditioning on the class

labels. We thus need an approach to relax the naïve Bayes assumption so that

we can capture more generic representations of conditional independence

among attributes.

In this section, we present a flexible framework for modeling probabilistic

relationships between attributes and class labels, known as Bayesian

Networks. By building on concepts from probability theory and graph

theory, Bayesian networks are able to capture more generic forms of

conditional independence using simple schematic representations. They also

provide the necessary computational structure to perform inferences over

random variables in an efficient way. In the following, we first describe the

basic representation of a Bayesian network, and then discuss methods for

performing inference and learning model parameters in the context of

classification.

4.5.1 Graphical Representation

Bayesian networks belong to a broader family of models for capturing

probabilistic relationships among random variables, known as probabilistic

graphical models. The basic concept behind these models is to use graphical

representations where the nodes of the graph correspond to random variables

and the edges between the nodes express probabilistic relationships. Figures

4.10(a) and 4.10(b) show examples of probabilistic graphical models using

directed edges (with arrows) and undirected edges (without arrows),

respectively. Directed graphical models are also known as Bayesian networks

while undirected graphical models are known as Markov random fields.

The two approaches use different semantics for expressing relationships

among random variables and are thus useful in different contexts. In the

following, we briefly describe Bayesian networks that are useful in the

context of classification.

A Bayesian network (also referred to as a belief network) involves directed

edges between nodes, where every edge represents a direction of influence

among random variables. For example, Figure 4.10(a) shows a Bayesian

network where variable *C *depends upon the values of variables *A *and *B*, as

indicated by the arrows pointing toward *C *from *A *and *B*. Consequently, the

variable *C *influences the values of variables *D *and *E*. Every edge in a

Bayesian network thus encodes a dependence relationship between random

variables with a particular directionality.

Figure 4.10.

Illustrations of two basic types of graphical models.

Figure 4.10. Full Alternative Text

Bayesian networks are directed acyclic graphs (DAG) because they do not

contain any directed cycles such that the influence of a node loops back to the

same node. Figure 4.11 shows some examples of Bayesian networks that

capture different types of dependence structures among random variables. In

a directed acyclic graph, if there is a directed edge from *X *to *Y *,then *X *is

called the parent of *Y *and *Y *is called the child of *X*. Note that a node can

have multiple parents in a Bayesian network, e.g., node *D *has two parent

nodes, *B *and *C*, in Figure 4.11(a). Furthermore, if there is a directed path in

the network from *X *to *Z*, then *X *is an ancestor of *Z*, while *Z *is a descendant

of *X*. For example, in the diagram shown in Figure 4.11(b), *A *is a descendant

of *D *and *D *is an ancestor of *B*. Note that there can be multiple directed paths

between two nodes of a directed acyclic graph, as is the case for nodes *A *and

*D *in Figure 4.11(a).

Figure 4.11.

Examples of Bayesian networks.

Conditional Independence

An important property of a Bayesian network is its ability to represent

varying forms of conditional independence among random variables. There

are several ways of describing the conditional independence assumptions

captured by Bayesian networks. One of the most generic ways of expressing

conditional independence is the concept of *d*-separation, which can be used

to determine if any two sets of nodes *A *and *B *are conditionally independent

given another set of nodes *C*. Another useful concept is that of the Markov

blanket of a node *Y *, which denotes the minimal set of nodes *X *that makes *Y*

independent of the other nodes in the graph, when conditioned on *X*. (See

Bibliographic Notes for more details on *d*-separation and Markov blanket.)

However, for the purpose of classification, it is sufficient to describe a

simpler expression of conditional independence in Bayesian networks, known

as the local Markov property.

Property 1 (Local Markov

Property).

A node in a Bayesian network is conditionally independent of its nondescendants, if its parents are known.

To illustrate the local Markov property, consider the Bayes network shown in

Figure 4.11(b). We can state that *A *is conditionally independent of both *B*

and *D *given *C*, because *C *is the parent of *A *and nodes *B *and *D *are nondescendants of *A*. The local Markov property helps in interpreting parentchild relationships in Bayesian networks as representations of conditional

probabilities. Since a node is conditionally independent of its nondescendants given it parents, the conditional independence assumptions

imposed by a Bayesian network is often sparse in structure. Nonetheless,

Bayesian networks are able to express a richer class of conditional

independence statements among attributes and class labels than the naïve

Bayes classifier. In fact, the naïve Bayes classifier can be viewed as a special

type of Bayesian network, where the target class *Y *is at the root of a tree and

every attribute Xi is connected to the root node by a directed edge, as shown

in Figure 4.12(a).

Figure 4.12.

Comparing the graphical representation of a naïve Bayes classifier

with that of a generic Bayesian network.

Figure 4.12. Full Alternative Text

Note that in a naïve Bayes classifier, every directed edge points from the

target class to the observed attributes, suggesting that the class label is a

factor behind the generation of attributes. Inferring the class label can thus be

viewed as *diagnosing *the root cause behind the observed attributes. On the

other hand, Bayesian networks provide a more generic structure of

probabilistic relationships, since the target class is not required to be at the

root of a tree but can appear anywhere in the graph, as shown in Figure

4.12(b). In this diagram, inferring *Y *not only helps in diagnosing the factors

influencing X3 and X4, but also helps in *predicting *the influence of X1 and

X2.

Joint Probability

The local Markov property can be used to succinctly express the joint

probability of the set of random variables involved in a Bayesian network. To

realize this, let us first consider a Bayesian network consisting of *d *nodes, X1

to Xd, where the nodes have been numbered in such a way that Xi is an

ancestor of Xj only if i<j. The joint probability of X={X1, …, Xd} can be

generically factorized using the chain rule of probability as

P(X)=P(X1)P(X2|X1)P(X3|X1, X2) … P(Xd|X1, … Xd

−1)=∏i=1dP(Xi|X1, … Xi−1) (4.22)

By the way we have constructed the graph, note that the set {X1, … Xi−1 }

contains only non-descendants of Xi. Hence, by using the local Markov

property, we can write P(Xi|X1, … Xi−1) as P(Xi|pa(Xi)), where pa(Xi)

denotes the parents of Xi. The joint probability can then be represented as

P(X)=∏i=1dP(Xi|pa(Xi)) (4.23)

It is thus sufficient to represent the probability of every node Xi in terms of

its parent nodes, pa(Xi), for computing *P*(x). This is achieved with the help of

probability tables that associate every node to its parent nodes as follows:

1. The probability table for node Xi contains the conditional probability

values P(Xi|pa(Xi)) for every combination of values in Xi and pa(Xi).

2. If Xi has no parents (pa(Xi)=ϕ), then the table contains only the prior

probability P(Xi).

Example 4.6. [Probability Tables]

Figure 4.13 shows an example of a Bayesian network for modeling the

relationships between a patient’s symptoms and risk factors. The probability

tables are shown at the side of every node in the figure. The probability tables

associated with the risk factors (Exercise and Diet) contain only the prior

probabilities, whereas the tables for heart disease, heartburn, blood pressure,

and chest pain, contain the conditional probabilities.

Figure 4.13.

A Bayesian network for detecting heart disease and heartburn in

patients.

Figure 4.13. Full Alternative Text

Use of Hidden Variables

A Bayesian network typically involves two types of variables: *observed*

*variables *that are clamped to specific observed values, and *unobserved*

*variables*, whose values are not known and need to be inferred from the

network. To distinguish between these two types of variables, observed

variables are generally represented using shaded nodes while unobserved

variables are represented using empty nodes. Figure 4.14 shows an example

of a Bayesian network with observed variables (*A*, *B*, and *E *) and unobserved

variables (*C *and *D*).

Figure 4.14.

Observed and unobserved variables are represented using unshaded

and shaded circles, respectively.

In the context of classification, the observed variables correspond to the set of

attributes X, while the target class is represented using an unobserved

variable *Y *that needs to be inferred during testing. However, note that a

generic Bayesian network may contain many other unobserved variables

apart from the target class, as represented in Figure 4.15 as the set of

variables H. These unobserved variables represent hidden or confounding

factors that affect the probabilities of attributes and class labels, although they

are never directly observed. The use of hidden variables enhances the

expressive power of Bayesian networks in representing complex probabilistic

relationships between attributes and class labels. This is one of the key

distinguishing properties of Bayesian networks as compared to naïve Bayes

classifiers.

4.5.2 Inference and Learning

Given the probability tables corresponding to every node in a Bayesian

network, the problem of inference corresponds to computing the probabilities

of different sets of random variables. In the context of classification, one of

the key inference problems is to compute the probability of a target class *Y*

taking on a specific value *y*, given the set of observed attributes at a data

instance, x. This can be represented using the following conditional

probability:

P(Y=y|x)=(y, x)P(x)=(y, x)∑y′P(y′, x) (4.24)

The previous equation involves marginal probabilities of the form *P*(*y*, x).

They can be computed by marginalizing out the hidden variables H from the

joint probability as follows:

P(y, x)=∑HP(y, x, H), (4.25)

where the joint probability *P*(*y*, x, H) can be obtained by using the

factorization described in Equation 4.23. To understand the nature of

computations involved in estimating *P*(*y*, x), consider the example Bayesian

network shown in Figure 4.15, which involves a target class, *Y *, three

observed attributes, X1 to X3, and four hidden variables, H1 to H4. For this

network, we can express *P*(*y*, x) as

Figure 4.15.

An example of a Bayesian network with four hidden variables, H1

to H4, three observed attributes, X1 to X3, and one target class *Y *.

P(y, x)=∑h1∑h2∑h3∑h4P(y, x1, x2, h1, h2, h3, h4),=∑h1∑h2∑h3∑h4

[P(h1)P(h2)P(x2)P(h4)P(x1|h1, h2) ×P(h3|x2, h2)P(y|x1, h3)P(x3|h3, h4) ],

(4.26)

=∑h1∑h2∑h3∑h4f(h1, h2, h3, h4), (4.27)

where *f *is a factor that depends on the values of h1 to h4. In the previous

simplistic expression of *P*(*y*, x), a different summand is considered for every

combination of values, h1 to h4, in the hidden variables, H1 to H4. If we

assume that every variable in the network can take *k *discrete values, then the

summation has to be carried out for a total number of k4 times. The

computational complexity of this approach is thus O(k4). Moreover, the

number of computations grows exponentially with the number of hidden

variables, making it difficult to use this approach with networks that have a

large number of hidden variables. In the following, we present different

computational techniques for efficiently performing inferences in Bayesian

networks.

Variable Elimination

To reduce the number of computations involved in estimating *P*(*y*, x), let us

closely examine the expressions in Equations 4.26 and 4.27. Notice that

although f(h1, h2, h3, h4) depends on the values of all four hidden variables,

it can be decomposed as a product of several smaller factors, where every

factor involves only a small number of hidden variables. For example, the

factor P(h4) depends only on the value of h4, and thus acts as a constant

multiplicative term when summations are performed over h1, h2, or h3.

Hence, if we place P(h4) outside the summations of h1 to h3, we can save

some repeated multiplications occurring inside every summand.

In general, we can push every summation as far inside as possible, so that the

factors that do not depend on the summing variable are placed outside the

summation. This will help reduce the number of wasteful computations by

using smaller factors at every summation. To illustrate this process, consider

the following sequence of steps for computing *P*(*y*, x), by rearranging the

order

of summations in Equation 4.26.

P(y, x)=P(x2)∑h4P(h4)∑h3P(y|x1, h3)P(x3|h3, h4)×∑h2P(h2)P(h3|x2, h2)∑h1

(4.28)

=P(x2)∑h4P(h4)∑h3P(y|x1, h3)P(x3|h3, h4)×∑h2P(h2)P(h3|x2, h2)f1(h2)

(4.29)

=P(x2)∑h4P(h4)∑h3P(y|x1, h3)P(x3|h3, h4)f2(h3) (4.30)

=P(x2)∑h4P(h4)f3(h4) (4.31)

where fi represents the intermediate factor term obtained by summing out hi.

To check if the previous rearrangements provide any improvements in

computational efficiency, let us count the number of computations occurring

at every step of the process. At the first step (Equation 4.28), we perform a

summation over h1 using factors that depend on h1 and h2. This requires

considering every pair of values in h1 and h2, resulting in O(k2)

computations. Similarly, the second step (Equation 4.29) involves summing

out h2 using factors of h2 and h3, leading to O(k2) computations. The third

step (Equation 4.30) again requires O(k2) computations as it involves

summing out h3 over factors depending on h3 and h4. Finally, the fourth step

(Equation 4.31) involves summing out h4 using factors depending on h4,

resulting in *O*(*k*) computations.

The overall complexity of the previous approach is thus O(k2), which is

considerably smaller than the O(k4) complexity of the basic approach.

Hence, by merely rearranging summations and using algebraic

manipulations, we are able to improve the computational efficiency in

computing *P*(*y*, x). This procedure is known as variable elimination.

The basic concept that variable elimination exploits to reduce the number of

computations is the distributive nature of multiplication over addition

operations. For example, consider the following multiplication and addition

operations:

a.(b+c+d)=a.b+a.c+a.d

Notice that the right-hand side of the previous equation involves three

multiplications and three additions, while the left-hand side involves only one

multiplication and three additions, thus saving on two arithmetic operations.

This property is utilized by variable elimination in pushing out constant terms

outside the summation, such that they are multiplied only once.

Note that the efficiency of variable elimination depends on the order of

hidden variables used for performing summations. Hence, we would ideally

like to find the optimal order of variables that result in the smallest number of

computations. Unfortunately, finding the optimal order of summations for a

generic Bayesian network is an NP-Hard problem, i.e., there does not exist an

efficient algorithm for finding the optimal ordering that can run in

polynomial time. However, there exists efficient techniques for handling

special types of Bayesian networks, e.g., those involving *tree-like *graphs, as

described in the following.

Sum-Product Algorithm for Trees

Note that in Equations 4.28 and 4.29, whenever a variable hi is eliminated

during marginalization, it results in the creation of a factor fi that depends on

the neighboring nodes of hi. fi is then absorbed in the factors of neighboring

variables and the process is repeated until all unobserved variables have

marginalized. This phenomena of variable elimination can be viewed as

transmitting a *local message *from the variable being marginalized to its

neighboring nodes. This idea of message passing utilizes the structure of the

graph for performing computations, thus making it possible to use graphtheoretic approaches for making effective inferences. The sum-product

algorithm builds on the concept of message passing for computing marginal

and conditional probabilities on tree-based graphs.

Figure 4.16 shows an example of a tree involving five variables, X1 to X5. A

key characteristic of a tree is that every node in the tree has exactly one

parent, and there is only one directed edge between any two nodes in the tree.

For the purpose of illustration, let us consider the problem of estimating the

marginal probability of X2, P(X2). This can be obtained by marginalizing out

every variable in the graph except X2 and rearranging the summations to

obtain the following expression:

Figure 4.16.

An example of a Bayesian network with a tree structure.

P(x2)=∑x1∑x3∑x4∑x5P(x1)P(x2|x1)P(x3|x2)P(x4|x3)P(x5|x3),=

(∑x1P(x1)P(x2|x1))︸m12(x2)(∑x3P(x3|x2)(∑x4P(x4|x3))︸m43(x3)

(∑x5P(x5|x3))︸m53(x3)),︸m32(x2)

where mij(xj) has been conveniently chosen to represent the factor of xj that

is obtained by summing out xi. We can view mij(xj) as a local message

passed from node xi to node xj, as shown using arrows in Figure 4.17(a).

These local messages capture the influence of eliminating nodes on the

marginal probabilities of neighboring nodes.

Before we formally describe the formula for computing mij(xj) and P(xj), we

first define a potential function ψ(⋅) that is associated every node and edge of

the graph. We can define the potential of a node Xi as

ψ(Xi)={P(Xi),if Xi is the root node.1,otherwise. (4.32)

Figure 4.17.

Illustration of message passing in the sum-product algorithm.

Figure 4.17. Full Alternative Text

Similarly, we can define the potential of an edge between nodes Xi and Xj

(where Xi is the parent of Xj) as

ψ(Xi, Xj)=P(Xj|Xi).

Using ψ(Xi) and ψ(Xi, Xj), we can represent mij(xj) using the following

equation:

mij(xj)=∑xi(ψ(xi)ψ(xi, xj)∏k∈N(i)\imki(xi)), (4.33)

where *N*(*i*) represents the set of neighbors of node Xi. The message mij that is

transmitted from Xi to Xj can thus be recursively computed using the

messages incident on Xi from its neighboring nodes excluding Xi. Note that

the formula for mij involves taking a sum over all possible values of Xj, after

multiplying the factors obtained from the neighbors of Xj. This approach of

message passing is thus called the “sum-product” algorithm. Further, since

mij represents a notion of “belief” propagated from Xi to Xj, this algorithm is

also known as belief propagation. The marginal probability of a node Xi

is then given as

P(xi)=ψ(xi)∏j∈N(i)mji(xi). (4.34)

A useful property of the sum-product algorithm is that it allows the messages

to be reused for computing a different marginal probability in the future. For

example, if we had to compute the marginal probability for node X3, we

would require the following messages from its neighboring nodes:

m23(x3), m43(x3), and m53(x3). However, note that m43(x3), and m53(x3)

have already been computed in the process of computing the marginal

probability of X2 and thus can be reused.

Notice that the basic operations of the sum-product algorithm resemble a

message passing protocol over the edges of the network. A node sends out a

message to all its neighboring nodes only after it has received incoming

messages from all its neighbors. Hence, we can initialize the message passing

protocol from the leaf nodes, and transmit messages till we reach the root

node. We can then run a second pass of messages from the root node back to

the leaf nodes. In this way, we can compute the messages for every edge in

both directions, using just O(2|E|) operations, where |E| is the number of

edges. Once we have transmitted all possible messages as shown in Figure

4.17(b), we can easily compute the marginal probability of every node in the

graph using Equation 4.34.

In the context of classification, the sum-product algorithm can be easily

modified for computing the conditional probability of the class label *y *given

the set of observed attributes x^, i.e., P(y|x^). This basically amounts to

computing P(y, X=x^) in Equation 4.24, where X is clamped to the observed

values x^. To handle the scenario where some of the random variables are

fixed and do not need to be normalized, we consider the following

modification.

If Xi is a random variable that is fixed to a specific value x^i, then we can

simply modify ψ(Xi) and ψ(Xi, Xj) as follows:

ψ(Xi)={1,if Xi=x^i.0,otherwise. (4.35)

ψ(Xi, Xj)={P(Xi|x^i),if Xi=x^i.0,otherwise. (4.36)

We can run the sum-product algorithm using these modified values for every

observed variable and thus compute P(y, X=x^).

Figure 4.18.

Example of a poly-tree and its corresponding factor graph.

Figure 4.18. Full Alternative Text

Generalizations for Non-Tree

Graphs

The sum-product algorithm is guaranteed to optimally converge in the case of

trees using a single run of message passing in both directions of every edge.

This is because any two nodes in a tree have a unique path for the

transmission of messages. Furthermore, since every node in a tree has a

single parent, the joint probability involves only factors of at most two

variables. Hence, it is sufficient to consider potentials over edges and not

other generic substructures in the graph.

Both of the previous properties are violated in graphs that are not trees, thus

making it difficult to directly apply the sum-product algorithm for making

inferences. However, a number of variants of the sum-product algorithm have

been devised to perform inferences on a broader family of graphs than trees.

Many of these variants transform the original graph into an alternative treebased representation, and then apply the sum-product algorithm on the

transformed tree. In this section, we briefly discuss one such transformations

known as factor graphs.

Factor graphs are useful for making inferences over graphs that violate the

condition that every node has a single parent. Nonetheless, they still require

the absence of multiple paths between any two nodes, to guarantee

convergence. Such graphs are known as poly-trees. An example of a polytree is shown in Figure 4.18(a).

A poly-tree can be transformed into a tree-based representation with the help

of factor graphs. These graphs consist of two types of nodes, variables nodes

(that are represented using circles) and factor nodes (that are represented

using squares). The factor nodes represent conditional independence

relationships among the variables of the poly-tree. In particular, every

probability table can be represented as a factor node. The edges in a factor

graph are undirected in nature and relate a variable node to a factor node if

the variable is involved in the probability table corresponding to the factor

node. Figure 4.18(b) presents the factor graph representation of the poly-tree

shown in Figure 4.18(a).

Note that the factor graph of a poly-tree always forms a tree-like structure,

where there is a unique path of influence between any two nodes in the factor

graph. Hence, we can apply a modified form of sum-product algorithm to

transmit messages between variable nodes and factor nodes, which is

guaranteed to converge to optimal values.

Learning Model Parameters

In all our previous discussions on Bayesian networks, we had assumed that

the topology of the Bayesian network and the values in the probability tables

of every node were already known. In this section, we discuss approaches for

learning both the topology and the probability table values of a Bayesian

network from the training data.

Let us first consider the case where the topology of the network is known and

we are only required to compute the probability tables. If there are no

unobserved variables in the training data, then we can easily compute the

probability table for P(Xi|pa(Xi)), by counting the fraction of training

instances for every value of Xi and every combination of values in pa(Xi).

However, if there are unobserved variables in Xi or pa(Xi), then computing

the fraction of training instances for such variables is non-trivial and requires

the use of advances techniques such as the Expectation-Maximization

algorithm (described later in Chapter 8).

Learning the structure of the Bayesian network is a much more challenging

task than learning the probability tables. Although there are some scoring

approaches that attempt to find a graph structure that maximizes the training

likelihood, they are often computationally infeasible when the graph is large.

Hence, a common approach for constructing Bayesian networks is to use the

subjective knowledge of domain experts.

4.5.3 Characteristics of Bayesian

Networks

1. Bayesian networks provide a powerful approach for representing

probabilistic relationships between attributes and class labels with the

help of graphical models. They are able to capture complex forms of

dependencies among variables. Apart from encoding prior beliefs, they

are also able to model the presence of latent (unobserved) factors as

hidden variables in the graph. Bayesian networks are thus quite

expressive and provide predictive as well as descriptive insights about

the behavior of attributes and class labels.

2. Bayesian networks can easily handle the presence of correlated or

redundant attributes, as opposed to the naïve Bayes classifier. This is

because Bayesian networks do not use the naïve Bayes assumption

about conditional independence, but instead are able to express richer

forms of conditional independence.

3. Similar to the naïve Bayes classifier, Bayesian networks are also quite

robust to the presence of noise in the training data. Further, they can

handle missing values during training as well as testing. If a test instance

contains an attribute Xi with a missing value, then a Bayesian network

can perform inference by treating Xi as an unobserved node and

marginalizing out its effect on the target class. Hence, Bayesian

networks are well-suited for handling incompleteness in the data, and

can work with partial information. However, unless the pattern with

which missing values occurs is completely random, then their presence

will likely introduce some degree of error and/or bias into the analysis.

4. Bayesian networks are robust to irrelevant attributes that contain no

discriminatory information about the class labels. Such attributes show

no impact on the conditional probability of the target class, and are thus

rightfully ignored.

5. Learning the structure of a Bayesian network is a cumbersome task that

often requires assistance from expert knowledge. However, once the

structure has been decided, learning the parameters of the network can

be quite straightforward, especially if all the variables in the network are

observed.

6. Due to its additional ability of representing complex forms of

relationships, Bayesian networks are more susceptible to overfitting as

compared to the naïve Bayes classifier. Furthermore, Bayesian networks

typically require more training instances for effectively learning the

probability tables than the naïve Bayes classifier.

7. Although the sum-product algorithm provides computationally efficient

techniques for performing inference over tree-like graphs, the

complexity of the approach increase significantly when dealing with

generic graphs of large sizes. In situations where exact inference is

computationally infeasible, it is quite common to use approximate

inference techniques.

4.6 Logistic Regression

The naïve Bayes and the Bayesian network classifiers described in the

previous sections provide different ways of estimating the conditional

probability of an instance x given class *y*, P(x|y). Such models are known as

probabilistic generative models. Note that the conditional probability P(x|y)

essentially describes the behavior of instances in the attribute space that are

*generated *from class *y*. However, for the purpose of making predictions, we

are finally interested in computing the posterior probability P(y|x). For

example, computing the following ratio of posterior probabilities is sufficient

for inferring class labels in a binary classification problem:

P(y=1|x)P(y=0|x)

This ratio is known as the odds. If this ratio is greater than 1, then x is

classified as y=1. Otherwise, it is assigned to class y=0. Hence, one may

simply learn a model of the odds based on the attribute values of training

instances, without having to compute P(x|y) as an intermediate quantity in the

Bayes theorem.

Classification models that directly assign class labels without computing

class-conditional probabilities are called discriminative models. In this

section, we present a probabilistic discriminative model known as logistic

regression, which directly estimates the odds of a data instance x using its

attribute values. The basic idea of logistic regression is to use a linear

predictor, z=wTx+b, for representing the odds of x as follows:

P(y=1|x)P(y=0|x)=ez=ewTx+b, (4.37)

where w and *b *are the parameters of the model and aT denotes the transpose

of a vector a. Note that if wTx+b>0, then x belongs to class 1 since its odds is

greater than 1. Otherwise, x belongs to class 0.

Figure 4.19.

Plot of sigmoid (logistic) function, σ(z).

Since P(y=0|x)+P(y=1|x)=1, we can re-write Equation 4.37 as

P(y=1|x)1−P(y=1|x)=ez.

This can be further simplified to express P(y=1|x) as a function of *z*.

P(y=1|x)=11+e−z=σ(z), (4.38)

where the function σ(⋅) is known as the logistic or sigmoid function. Figure

4.19 shows the behavior of the sigmoid function as we vary *z*. We can see

that σ(z)≥0.5 only when z≥0. We can also derive P(y=0|x) using σ(z) as

follows:

P(y=0|x)=1−σ(z)=11+e−z (4.39)

Hence, if we have learned a suitable value of parameters w and *b*, we can use

Equations 4.38 and 4.39 to estimate the posterior probabilities of any data

instance x and determine its class label.

4.6.1 Logistic Regression as a

Generalized Linear Model

Since the posterior probabilities are real-valued, their estimation using the

previous equations can be viewed as solving a regression problem. In fact,

logistic regression belongs to a broader family of statistical regression

models, known as generalized linear models (GLM). In these models, the

target variable *y *is considered to be generated from a probability distribution

P(y|x), whose mean μ can be estimated using a link function g(⋅) as follows:

g(μ)=z=wT x + b. (4.40)

For binary classification using logistic regression, *y *follows a Bernoulli

distribution (*y *can either be 0 or 1) and μ is equal to P(y=1|x). The link

function g(⋅) of logistic regression, called the logit function, can thus be

represented as

g(μ)=log(μ1−μ).

Depending on the choice of link function g(⋅) and the form of probability

distribution P(y|x), GLMs are able to represent a broad family of regression

models, such as linear regression and Poisson regression. They require

different approaches for estimating their model parameters, (w, b). In this

chapter, we will only discuss approaches for estimating the model parameters

of logistic regression, although methods for estimating parameters of other

types of GLMs are often similar (and sometimes even simpler). (See

Bibliographic Notes for more details on GLMs.)

Note that even though logistic regression has relationships with regression

models, it is a classification model since the computed posterior probabilities

are eventually used to determine the class label of a data instance.

4.6.2 Learning Model Parameters

The parameters of logistic regression, (w, b), are estimated during training

using a statistical approach known as the maximum likelihood estimation

(MLE) method. This method involves computing the likelihood of observing

the training data given (w, b), and then determining the model parameters

(w*, b*) that yield maximum likelihood.

Let D.train={(x1, y1), (x2, y2), … , (xn, yn)} denote a set of *n *training

instances, where yi is a binary variable (0 or 1). For a given training instance

xi, we can compute its posterior probabilities using Equations 4.38 and 4.39.

We can then express the likelihood of observing yi given xi, w, and *b *as

P(yi|xi, w, b)=P(y=1|xi)yi×P(y=0|xi)1−yi,=(σ(zi))yi×(1−σ(zi))1−yi,=

(σ(wTxi+b))yi×(1−σ(wTxi+b))1−yi, (4.41)

where σ(⋅) is the sigmoid function as described above, Equation 4.41

basically means that the likelihood P(yi|xi, w, b) is equal to P(y=1|xi) when

yi=1, and equal to P(y=0|xi) when yi=0. The likelihood of all training

instances, L(w, b), can then be computed by taking the product of individual

likelihoods (assuming independence among training instances) as follows:

L(w, b)=∏i=1nP(yi|xi, w, b)=∏i=1nP(y=1|xi)yi×P(y=0|xi)1−yi. (4.42)

The previous equation involves multiplying a large number of probability

values, each of which are smaller than or equal to 1. Since this naïve

computation can easily become numerically unstable when *n *is large, a more

practical approach is to consider the negative logarithm (to base *e*) of the

likelihood function, also known as the cross entropy function:

−logL(w, b)=−∑i=1nyilog(P(y=1|xi))+(1−yi)log(P(y=0|xi)).=

−∑i=1nyilog(σ(wTxi+b))+(1−yi)log(1−σ(wTxi+b)).

The cross entropy is a loss function that measures how unlikely it is for the

training data to be generated from the logistic regression model with

parameters (w, b). Intuitively, we would like to find model parameters

(w*, b*) that result in the lowest cross entropy, −logL(w*, b*).

(w*, b*)=argmin(w, b)E(w, b)=argmin(w, b)−logL(w, b) (4.43)

where E(w, b)=−logL(w, b) is the loss function. It is worth emphasizing that

*E*(w, b) is a convex function, i.e., any minima of *E*(w, b) will be a global

minima. Hence, we can use any of the standard convex optimization

techniques to solve Equation 4.43, which are mentioned in Appendix E. Here,

we briefly describe the Newton-Raphson method that is commonly used for

estimating the parameters of logistic regression. For ease of representation,

we will use a single vector to describe w˜=(wT b)T, which is of size one

greater than w. Similarly, we will consider the concatenated feature vector x˜=

(xT 1)T, such that the linear predictor z=wTx+b can be succinctly written as

z=w˜Tx˜. Also, the concatenation of all training labels, y1 to yn, will be

represented as *y*, the set consisting of σ(z1) to σ(zn) will be represented as σ,

and the concatenation of x˜1 to x˜n will be represented as X˜.

The Newton-Raphson is an iterative method for finding w˜* that uses the

following equation to update the model parameters at every iteration:

w˜(new)=w˜(old)−H−1∇E(w˜), (4.44)

where ∇E(w˜) and H are the first- and second-order derivatives of the loss

function E( w˜) with respect to w˜, respectively. The key intuition behind

Equation 4.44 is to move the model parameters in the direction of maximum

gradient, such that w˜ takes larger steps when ∇E(w˜) is large. When w˜

arrives at a minima after some number of iterations, then ∇E(w˜) would

become equal to 0 and thus result in convergence. Hence, we start with some

initial values of w˜ (either randomly assigned or set to 0) and use Equation

4.44 to iteratively update w˜ till there are no significant changes in its value

(beyond a certain threshold).

The first-order derivative of E(w˜) is given by

∇E(w˜)=−∑i=1nyix˜i(1−σ(w˜Tx˜i))−(1−yi)x˜iσ(w˜Tx˜i),=−∑i=1n(σ(w˜Tx˜i)

−yi)x˜i,=X˜(σ−y), (4.45)

where we have used the fact that dσ(z)/dz=σ(z)(1−σ(z)). Using ∇E(w˜), we

can compute the second-order derivative of E(w˜) as

H=∇∇E(w˜)=∑i=1nσ(w˜Tx˜i)(1−σ(w˜Tx˜i)x˜ix˜iT)=X˜TRX˜, (4.46)

where R is a diagonal matrix whose *i*th diagonal element Rii=σi(1−σi). We

can now use the first- and second-order derivatives of E(w˜) in Equation 4.44

to obtain the following update equation at the *k*th iteration:

w˜(k+1)=w˜(k)−(X˜TRkX˜)−1X˜T(σk−y) (4.47)

where the subscript *k *under Rk and σk refers to using w˜(k) to compute both

terms.

4.6.3 Characteristics of Logistic

Regression

1. Logistic Regression is a discriminative model for classification that

directly computes the poster probabilities without making any

assumption about the class conditional probabilities. Hence, it is quite

generic and can be applied in diverse applications. It can also be easily

extended to multiclass classification, where it is known as multinomial

logistic regression. However, its expressive power is limited to learning

only linear decision boundaries.

2. Because there are different weights (parameters) for every attribute, the

learned parameters of logistic regression can be analyzed to understand

the relationships between attributes and class labels.

3. Because logistic regression does not involve computing densities and

distances in the attribute space, it can work more robustly even in highdimensional settings than distance-based methods such as nearest

neighbor classifiers. However, the objective function of logistic

regression does not involve any term relating to the complexity of the

model. Hence, logistic regression does not provide a way to make a

trade-off between model complexity and training performance, as

compared to other classification models such as support vector

machines. Nevertheless, variants of logistic regression can easily be

developed to account for model complexity, by including appropriate

terms in the objective function along with the cross entropy function.

4. Logistic regression can handle irrelevant attributes by learning weight

parameters close to 0 for attributes that do not provide any gain in

performance during training. It can also handle interacting attributes

since the learning of model parameters is achieved in a joint fashion by

considering the effects of all attributes together. Furthermore, if there are

redundant attributes that are duplicates of each other, then logistic

regression can learn equal weights for every redundant attribute, without

degrading classification performance. However, the presence of a large

number of irrelevant or redundant attributes in high-dimensional settings

can make logistic regression susceptible to model overfitting.

5. Logistic regression cannot handle data instances with missing values,

since the posterior probabilities are only computed by taking a weighted

sum of all the attributes. If there are missing values in a training

instance, it can be discarded from the training set. However, if there are

missing values in a test instance, then logistic regression would fail to

predict its class label.

4.7 Artificial Neural Network

(ANN)

Artificial neural networks (ANN) are powerful classification models that are

able to learn highly complex and nonlinear decision boundaries purely from

the data. They have gained widespread acceptance in several applications

such as vision, speech, and language processing, where they have been

repeatedly shown to outperform other classification models (and in some

cases even human performance). Historically, the study of artificial neural

networks was inspired by attempts to emulate biological neural systems. The

human brain consists primarily of nerve cells called neurons, linked together

with other neurons via strands of fiber called axons. Whenever a neuron is

stimulated (e.g., in response to a stimuli), it transmits nerve activations via

axons to other neurons. The receptor neurons collect these nerve activations

using structures called dendrites, which are extensions from the cell body of

the neuron. The strength of the contact point between a dendrite and an axon,

known as a synapse, determines the connectivity between neurons.

Neuroscientists have discovered that the human brain learns by changing the

strength of the synaptic connection between neurons upon repeated

stimulation by the same impulse.

The human brain consists of approximately 100 billion neurons that are interconnected in complex ways, making it possible for us to learn new tasks and

perform regular activities. Note that a single neuron only performs a simple

modular function, which is to respond to the nerve activations coming from

sender neurons connected at its dendrite, and transmit its activation to

receptor neurons via axons. However, it is the composition of these simple

functions that together is able to express complex functions. This idea is at

the basis of constructing artificial neural networks.

Analogous to the structure of a human brain, an artificial neural network is

composed of a number of processing units, called nodes, that are connected

with each other via directed links. The nodes correspond to neurons that

perform the basic units of computation, while the directed links correspond to

connections between neurons, consisting of axons and dendrites. Further, the

weight of a directed link between two neurons represents the strength of the

synaptic connection between neurons. As in biological neural systems, the

primary objective of ANN is to adapt the weights of the links until they fit the

input-output relationships of the underlying data.

The basic motivation behind using an ANN model is to extract useful

features from the original attributes that are most relevant for classification.

Traditionally, feature extraction has been achieved by using dimensionality

reduction techniques such as PCA (introduced in Chapter 2), which show

limited success in extracting nonlinear features, or by using hand-crafted

features provided by domain experts. By using a complex combination of

inter-connected nodes, ANN models are able to extract much richer sets of

features, resulting in good classification performance. Moreover, ANN

models provide a natural way of representing features at multiple levels of

abstraction, where complex features are seen as compositions of simpler

features. In many classification problems, modeling such a hierarchy of

features turns out to be very useful. For example, in order to detect a human

face in an image, we can first identify low-level features such as sharp edges

with different gradients and orientations. These features can then be

combined to identify facial parts such as eyes, nose, ears, and lips. Finally, an

appropriate arrangement of facial parts can be used to correctly identify a

human face. ANN models provide a powerful architecture to represent a

hierarchical abstraction of features, from lower levels of abstraction (e.g.,

edges) to higher levels (e.g., facial parts).

Artificial neural networks have had a long history of developments spanning

over five decades of research. Although classical models of ANN suffered

from several challenges that hindered progress for a long time, they have reemerged with widespread popularity because of a number of recent

developments in the last decade, collectively known as deep learning. In this

section, we examine classical approaches for learning ANN models, starting

from the simplest model called perceptrons to more complex architectures

called multi-layer neural networks. In the next section, we discuss some of

the recent advancements in the area of ANN that have made it possible to

effectively learn modern ANN models with deep architectures.

4.7.1 Perceptron

A perceptron is a basic type of ANN model that involves two types of nodes:

input nodes, which are used to represent the input attributes, and an output

node, which is used to represent the model output. Figure 4.20 illustrates the

basic architecture of a perceptron that takes three input attributes, x1, x2, and

x3, and produces a binary output *y*. The input node corresponding to an

attribute xi is connected via a weighted link wi to the output node. The

weighted link is used to emulate the strength of a synaptic connection

between neurons.

Figure 4.20.

Basic architecture of a perceptron.

The output node is a mathematical device that computes a weighted sum of

its inputs, adds a bias factor *b *to the sum, and then examines the sign of the

result to produce the output y^ as follows:

3^y={1,if wTx+b>0.−1,otherwise. (4.48)

To simplify notations, w and *b *can be concatenated to form w˜=(wT b)T,

while x can be appended with 1 at the end to form x˜=(xT 1)T. The output of

the perceptron y^ can then be written:

y^=sign(w˜Tx˜),

where the sign function acts as an activation function by providing an output

value of +1 if the argument is positive and −1 if its argument is negative.

Learning the Perceptron

Given a training set, we are interested in learning parameters w˜ such that y^

closely resembles the true *y *of training instances. This is achieved by using

the perceptron learning algorithm given in Algorithm 4.3. The key

computation for this algorithm is the iterative weight update formula given in

Step 8 of the algorithm:

wj(k+1)=wj(k)+λ(yi−yi^(k))xij, (4.49)

where w(k) is the weight parameter associated with the *i*th input link after the

*k*th iteration, λ is a parameter known as the learning rate, and xij is the value

of the *j*th attribute of the training example xi. The justification for Equation

4.49 is rather intuitive. Note that (yi−y^i) captures the discrepancy between

yi and y^i, such that its value is 0 only when the true label and the predicted

output match. Assume xij is positive. If y^=0 and y=1, then wj is increased at

the next iteration so that w˜Txi can become positive. On the other hand, if

y^=1 and y=0, then wj is decreased so that w˜Txi can become negative.

Hence, the weights are modified at every iteration to reduce the discrepancies

between y^ and *y *across all training instances. The learning rate λ, a

parameter whose value is between 0 and 1, can be used to control the amount

of adjustments made in each iteration. The algorithm halts when the average

number of discrepancies are smaller than a threshold γ.

Algorithm 4.3 Perceptron learning

algorithm.

1: Let D.train={(x˜i, yi)|i=1, 2, …, n} be the set of training instances.

2: Set *k *← 0.

3: Initialize the weight vector w˜(0) with random values.

4: repeat

5: for each training instance (x˜i, yi)∈D.train do

6: Compute the predicted output y^i(k) using w˜(k).

7: for each weight component *w**j *do

8: Update the weight, wj(k+1)=wj(k)+λ(yi−y^i(k))xij.

9: end for

10: Update *k *← *k *+ 1.

11: end for

12: until ∑i=1n|yi−y^i(k)|/n is less than a threshold γ

The perceptron is a simple classification model that is designed to learn linear

decision boundaries in the attribute space. Figure 4.21 shows the decision

boundary obtained by applying the perceptron learning algorithm to the data

set provided on the left of the figure. However, note that there can be multiple

decision boundaries that can separate the two classes, and the perceptron

arbitrarily learns one of these boundaries depending on the random initial

values of parameters. (The selection of the optimal decision boundary is a

problem that will be revisited in the context of support vector machines in

Section 4.9.) Further, the perceptron learning algorithm is only guaranteed to

converge when the classes are linearly separable. However, if the classes are

not linearly separable, the algorithm fails to converge. Figure 4.22 shows an

example of a nonlinearly separable data given by the XOR function. The

perceptron cannot find the right solution for this data because there is no

linear decision boundary that can perfectly separate the training instances.

Thus, the stopping condition at line 12 of Algorithm 4.3 would never be met

and hence, the perceptron learning algorithm would fail to converge. This is a

major limitation of perceptrons since real-world classification problems often

involve nonlinearly separable classes.

Figure 4.21.

Perceptron decision boundary for the data given on the left (+

represents a positively labeled instance while o represents a

negatively labeled instance.

Figure 4.21. Full Alternative Text

Figure 4.22.

XOR classification problem. No linear hyperplane can separate the

two classes.

Figure 4.22. Full Alternative Text

4.7.2 Multi-layer Neural Network

A multi-layer neural network generalizes the basic concept of a perceptron to

more complex architectures of nodes that are capable of learning nonlinear

decision boundaries. A generic architecture of a multi-layer neural network is

shown in Figure 4.23 where the nodes are arranged in groups called layers.

These layers are commonly organized in the form of a chain such that every

layer operates on the outputs of its preceding layer. In this way, the layers

represent different levels of *abstraction *that are applied on the input features

in a sequential manner. The composition of these abstractions generates the

final output at the last layer, which is used for making predictions. In the

following, we briefly describe the three types of layers used in multi-layer

neural networks.

Figure 4.23.

Example of a multi-layer artificial neural network (ANN).

The first layer of the network, called the input layer, is used for representing

inputs from attributes. Every numerical or binary attribute is typically

represented using a single node on this layer, while a categorical attribute is

either represented using a different node for each categorical value, or by

encoding the *k*-ary attribute using ⌈log2k ⌉ input nodes. These inputs are fed

into intermediary layers known as hidden layers, which are made up of

processing units known as hidden nodes. Every hidden node operates on

signals received from the input nodes or hidden nodes at the preceding layer,

and produces an activation value that is transmitted to the next layer. The

final layer is called the output layer and processes the activation values from

its preceding layer to produce predictions of output variables. For binary

classification, the output layer contains a single node representing the binary

class label. In this architecture, since the signals are propagated only in the

forward direction from the input layer to the output layer, they are also called

feedforward neural networks.

A major difference between multi-layer neural networks and perceptrons is

the inclusion of hidden layers, which dramatically improves their ability to

represent arbitrarily complex decision boundaries. For example, consider the

XOR problem described in the previous section. The instances can be

classified using two hyperplanes that partition the input space into their

respective classes, as shown in Figure 4.24(a). Because a perceptron can

create only one hyperplane, it cannot find the optimal solution. However, this

problem can be addressed by using a hidden layer consisting of two nodes, as

shown in Figure 4.24(b). Intuitively, we can think of each hidden node as a

perceptron that tries to construct one of the two hyperplanes, while the output

node simply combines the results of the perceptrons to yield the decision

boundary shown in Figure 4.24(a).

Figure 4.24.

A two-layer neural network for the XOR problem.

Figure 4.24. Full Alternative Text

The hidden nodes can be viewed as learning latent representations or *features*

that are useful for distinguishing between the classes. While the first hidden

layer directly operates on the input attributes and thus captures simpler

features, the subsequent hidden layers are able to combine them and construct

more complex features. From this perspective, multi-layer neural networks

learn a hierarchy of features at different levels of abstraction that are finally

combined at the output nodes to make predictions. Further, there are

combinatorially many ways we can combine the features learned at the

hidden layers of ANN, making them highly expressive. This property chiefly

distinguishes ANN from other classification models such as decision trees,

which can learn partitions in the attribute space but are unable to combine

them in exponential ways.

Figure 4.25.

Schematic illustration of the parameters of an ANN model with (L

−1) hidden layers.

Figure 4.25. Full Alternative Text

To understand the nature of computations happening at the hidden and output

nodes of ANN, consider the *i*th node at the *l*th layer of the network (l>0),

where the layers are numbered from 0 (input layer) to *L *(output layer), as

shown in Figure 4.25. The activation value generated at this node, ail, can be

represented as a function of the inputs received from nodes at the preceding

layer. Let wijl represent the weight of the connection from the *j*th node at

layer (l−1) to the *i*th node at layer *l*. Similarly, let us denote the bias term at

this node as bjl. The activation value ail can then be expressed as

ail=f(zil)=f(∑jwijlajl−1+bil),

where *z *is called the *linear predictor *and f(⋅) is the activation function that

converts *z *to *a*. Further, note that, by definition, aj0=xj at the input layer and

aL=y^ at the output node.

There are a number of alternate activation functions apart from the sign

function that can be used in multi-layer neural networks. Some examples

include linear, sigmoid (logistic), and hyperbolic tangent functions, as shown

in Figure 4.26. These functions are able to produce real-valued and nonlinear

activation values. Among these activation functions, the sigmoid σ(⋅) has

been widely used in many ANN models, although the use of other types of

activation functions in the context of deep learning will be discussed in

Section 4.8. We can thus represent ail as

Figure 4.26.

Types of activation functions used in multi-layer neural networks.

Figure 4.26. Full Alternative Text

ail=σ(zil)=11+e−zil. (4.50)

Learning Model Parameters

The weights and bias terms (w, b) of the ANN model are learned during

training so that the predictions on training instances match the true labels.

This is achieved by using a loss function

E(w, b)=∑k=1nLoss (yk, y^k) (4.51)

where yk is the true label of the *k*th training instance and y^k is equal to aL,

produced by using xk. A typical choice of the loss function is the squared

loss function:.

Loss (yk, y^k)=(yk, y^k)2. (4.52)

Note that *E*(w, b) is a function of the model parameters (w, b) because the

output activation value aL depends on the weights and bias terms. We are

interested in choosing (w, b) that minimizes the training loss *E*(w, b).

Unfortunately, because of the use of hidden nodes with nonlinear activation

functions, *E*(w, b) is not a convex function of w and b, which means that *E*(w,

b) can have local minima that are not globally optimal. However, we can still

apply standard optimization techniques such as the gradient descent method

to arrive at a locally optimal solution. In particular, the weight parameter wijl

and the bias term bil can be iteratively updated using the following equations:

wijl←wijl−λ∂E∂wijl, (4.53)

bil←bil−λ∂E∂bil, (4.54)

where λ is a hyper-parameter known as the learning rate. The intuition behind

this equation is to move the weights in a direction that reduces the training

loss. If we arrive at a minima using this procedure, the gradient of the

training loss will be close to 0, eliminating the second term and resulting in

the convergence of weights. The weights are commonly initialized with

values drawn randomly from a Gaussian or a uniform distribution.

A necessary tool for updating weights in Equation 4.53 is to compute the

partial derivative of *E *with respect to wijl. This computation is nontrivial

especially at hidden layers (l<L), since wijl does not directly affect y^=aL

(and hence the training loss), but has complex chains of influences via

activation values at subsequent layers. To address this problem, a technique

known as backpropagation was developed, which propagates the derivatives

backward from the output layer to the hidden layers. This technique can be

described as follows.

Recall that the training loss *E *is simply the sum of individual losses at

training instances. Hence the partial derivative of *E *can be decomposed as a

sum of partial derivatives of individual losses.

∂E∂wjl=∑k=1n∂ Loss (yk, y^k)∂wjl.

To simplify discussions, we will consider only the derivatives of the loss at

the *k*th training instance, which will be generically represented as Loss(y, aL).

By using the chain rule of differentiation, we can represent the partial

derivatives of the loss with respect to wijl as

∂ Loss∂wijl=∂ Loss∂ail×∂ail∂zil×∂zil∂wijl. (4.55)

The last term of the previous equation can be written as

∂zil∂wijl=∂(∑jwijlajl−1+bil)∂wijl=ajl−1.

Also, if we use the sigmoid activation function, then

∂ail∂zil=∂ σ(zil)∂zil=ail(1−ai1).

Equation 4.55 can thus be simplified as

∂ Loss∂wijl=δil×ail(1−ai1)×ajl−1,where δil=∂ Loss∂ail. (4.56)

A similar formula for the partial derivatives with respect to the bias terms bli

is given by

∂ Loss∂bil=δil×ail(1−ai1). (4.57)

Hence, to compute the partial derivatives, we only need to determine δil.

Using a squared loss function, we can easily write δL at the output node as

δL=∂ Loss∂aL=∂ (y−aL)2∂aL=2(aL−y). (4.58)

However, the approach for computing δjl at hidden nodes (l<L) is more

involved. Notice that ajl affects the activation values ail+1 of all nodes at the

next layer, which in turn influences the loss. Hence, again using the chain

rule of differentiation, δjl can be represented as

δjl=∂ Loss∂ajl=∑i(∂ Loss∂ail+1×∂ail+1∂ajl).=∑i(∂ Loss∂ail+1×∂ail+1∂zil+1×

(4.59)

The previous equation provides a concise representation of the δjl values at

layer *l *in terms of the δjl+1 values computed at layer l+1. Hence, proceeding

backward from the output layer *L *to the hidden layers, we can recursively

apply Equation 4.59 to compute δil at every hidden node. δil can then be used

in Equations 4.56 and 4.57 to compute the partial derivatives of the loss with

respect to wijl and bil, respectively. Algorithm 4.4 summarizes the complete

approach for learning the model parameters of ANN using backpropagation

and gradient descent method.

Algorithm 4.4 Learning ANN using

backpropagation and gradient

descent.

1: Let *D.train *= {(x*k*, *y**k*) | *k *= 1, 2, …, *n*} be the set of training instances.

2: Set counter *c *← 0.

3: Initialize the weight and bias terms (w(0), b(0)) with random values.

4: repeat

5: for each training instance (x*k*, *y**k*) ∈ *D.train *do

6: Compute the set of activations (ail)k by making *a *forward pass using

7: Compute the set (δil)k by backpropagation using Equations

8: Compute (∂ Loss/∂wijl, ∂ Loss/∂bil)k using Equations 4.56

9: end for

10: Compute ∂E/∂wijl←∑k=1n(∂ Loss/∂wijl)k.

11: Compute ∂E/∂bil←∑k=1n(∂ Loss/∂bil)k.

12: Update (w(*c *+ 1), b(*c *+ 1)) by gradient descent using Equations

13: Update *c *← *c *+ 1.

14: until (w(*c *+ 1), b(c + 1)) and (w(*c*), b(*c*)) converge to the same value

4.7.3 Characteristics of ANN

1. Multi-layer neural networks with at least one hidden layer are universal

approximators; i.e., they can be used to approximate any target

function. They are thus highly expressive and can be used to learn

complex decision boundaries in diverse applications. ANN can also be

used for multiclass classification and regression problems, by

appropriately modifying the output layer. However, the high model

complexity of classical ANN models makes it susceptible to overfitting,

which can be overcome to some extent by using deep learning

techniques discussed in Section 4.8.3.

2. ANN provides a natural way to represent a hierarchy of features at

multiple levels of abstraction. The outputs at the final hidden layer of the

ANN model thus represent features at the highest level of abstraction

that are most useful for classification. These features can also be used as

inputs in other supervised classification models, e.g., by replacing the

output node of the ANN by any generic classifier.

3. ANN represents complex high-level features as compositions of simpler

lower-level features that are easier to learn. This provides ANN the

ability to gradually increase the complexity of representations, by

adding more hidden layers to the architecture. Further, since simpler

features can be combined in combinatorial ways, the number of complex

features learned by ANN is much larger than traditional classification

models. This is one of the main reasons behind the high expressive

power of deep neural networks.

4. ANN can easily handle irrelevant attributes, by using zero weights for

attributes that do not help in improving the training loss. Also, redundant

attributes receive similar weights and do not degrade the quality of the

classifier. However, if the number of irrelevant or redundant attributes is

large, the learning of the ANN model may suffer from overfitting,

leading to poor generalization performance.

5. Since the learning of ANN model involves minimizing a non-convex

function, the solutions obtained by gradient descent are not guaranteed

to be globally optimal. For this reason, ANN has a tendency to get stuck

in local minima, a challenge that can be addressed by using deep

learning techniques discussed in Section 4.8.4.

6. Training an ANN is a time consuming process, especially when the

number of hidden nodes is large. Nevertheless, test examples can be

classified rapidly.

7. Just like logistic regression, ANN can learn in the presence of

interacting variables, since the model parameters are jointly learned over

all variables together. In addition, ANN cannot handle instances with

missing values in the training or testing phase.

4.8 Deep Learning

As described above, the use of hidden layers in ANN is based on the general

belief that complex high-level features can be constructed by combining

simpler lower-level features. Typically, the greater the number of hidden

layers, the deeper the hierarchy of features learned by the network. This

motivates the learning of ANN models with long chains of hidden layers,

known as deep neural networks. In contrast to “shallow” neural networks

that involve only a small number of hidden layers, deep neural networks are

able to represent features at multiple levels of abstraction and often require

far fewer nodes per layer to achieve generalization performance similar to

shallow networks.

Despite the huge potential in learning deep neural networks, it has remained

challenging to learn ANN models with a large number of hidden layers using

classical approaches. Apart from reasons related to limited computational

resources and hardware architectures, there have been a number of

algorithmic challenges in learning deep neural networks. First, learning a

deep neural network with low training error has been a daunting task because

of the saturation of sigmoid activation functions, resulting in slow

convergence of gradient descent. This problem becomes even more serious as

we move away from the output node to the hidden layers, because of the

compounded effects of saturation at multiple layers, known as the vanishing

gradient problem. Because of this reason, classical ANN models have

suffered from slow and ineffective learning, leading to poor training and test

performance. Second, the learning of deep neural networks is quite sensitive

to the initial values of model parameters, chiefly because of the non-convex

nature of the optimization function and the slow convergence of gradient

descent. Third, deep neural networks with a large number of hidden layers

have high model complexity, making them susceptible to overfitting. Hence,

even if a deep neural network has been trained to show low training error, it

can still suffer from poor generalization performance.

These challenges have deterred progress in building deep neural networks for

several decades and it is only recently that we have started to unlock their

immense potential with the help of a number of advances being made in the

area of deep learning. Although some of these advances have been around for

some time, they have only gained mainstream attention in the last decade,

with deep neural networks continually beating records in various

competitions and solving problems that were too difficult for other

classification approaches.

There are two factors that have played a major role in the emergence of deep

learning techniques. First, the availability of larger labeled data sets, e.g., the

ImageNet data set contains more than 10 million labeled images, has made it

possible to learn more complex ANN models than ever before, without

falling easily into the traps of model overfitting. Second, advances in

computational abilities and hardware infrastructures, such as the use of

graphical processing units (GPU) for distributed computing, have greatly

helped in experimenting with deep neural networks with larger architectures

that would not have been feasible with traditional resources.

In addition to the previous two factors, there have been a number of

algorithmic advancements to overcome the challenges faced by classical

methods in learning deep neural networks. Some examples include the use of

more responsive combinations of loss functions and activation functions,

better initialization of model parameters, novel regularization techniques,

more agile architecture designs, and better techniques for model learning and

hyper-parameter selection. In the following, we describe some of the deep

learning advances made to address the challenges in learning deep neural

networks. Further details on recent developments in deep learning can be

obtained from the Bibliographic Notes.

4.8.1 Using Synergistic Loss

Functions

One of the major realizations leading to deep learning has been the

importance of choosing appropriate combinations of activation and loss

functions. Classical ANN models commonly made use of the sigmoid

activation function at the output layer, because of its ability to produce real-

valued outputs between 0 and 1, which was combined with a squared loss

objective to perform gradient descent. It was soon noticed that this particular

combination of activation and loss function resulted in the saturation of

output activation values, which can be described as follows.

Saturation of Outputs

Although the sigmoid has been widely-used as an activation function, it

easily saturates at high and low values of inputs that are far away from 0.

Observe from Figure 4.27(a) that σ(z) shows variance in its values only when

*z *is close to 0. For this reason, ∂σ(z)/∂z is non-zero for only a small range of *z*

around 0, as shown in Figure 4.27(b). Since ∂σ(z)/∂z is one of the

components in the gradient of loss (see Equation 4.55), we get a diminishing

gradient value when the activation values are far from 0.

Figure 4.27.

Plots of sigmoid function and its derivative.

Figure 4.27. Full Alternative Text

To illustrate the effect of saturation on the learning of model parameters at

the output node, consider the partial derivative of loss with respect to the

weight wjL at the output node. Using the squared loss function, we can write

this as

∂ Loss∂wjL=2(aL−y)×σ(zL)(1−σ(zL))×ajL−1. (4.60)

In the previous equation, notice that when zL is highly negative, σ(zL) (and

hence the gradient) is close to 0. On the other hand, when zL is highly

positive, (1−σ(zL)) becomes close to 0, nullifying the value of the gradient.

Hence, irrespective of whether the prediction aL matches the true label *y *or

not, the gradient of the loss with respect to the weights is close to 0 whenever

zL is highly positive or negative. This causes an unnecessarily slow

convergence of the model parameters of the ANN model, often resulting in

poor learning.

Note that it is the combination of the squared loss function and the sigmoid

activation function at the output node that together results in diminishing

gradients (and thus poor learning) upon saturation of outputs. It is thus

important to choose a synergistic combination of loss function and activation

function that does not suffer from the saturation of outputs.

Cross entropy loss function

The cross entropy loss function, which was described in the context of

logistic regression in Section 4.6.2, can significantly avoid the problem of

saturating outputs when used in combination with the sigmoid activation

function. The cross entropy loss function of a real-valued prediction

y^∈(0, 1) on a data instance with binary label y∈{0, 1} can be defined as

Loss(y, y^)=−ylog(y^)−(1−y)log(1−y^), (4.61)

where log represents the natural logarithm (to base *e*) and 0 log(0)=0 for

convenience. The cross entropy function has foundations in information

theory and measures the amount of disagreement between *y *and y^. The

partial derivative of this loss function with respect to y^=aL can be given as

δL=∂ Loss∂aL=−yaL+(1−y)(1−aL).=(aL−y)aL(1−aL). (4.62)

Using this value of δL in Equation 4.56, we can obtain the partial derivative

of the loss with respect to the weight wjl at the output node as

∂ Loss∂wjL=(aL−y)aL(1−aL)×aL(1−aL)×ajL−1.=(aL−y)×ajL−1. (4.63)

Notice the simplicity of the previous formula using the cross entropy loss

function. The partial derivatives of the loss with respect to the weights at the

output node depend only on the difference between the prediction aL and the

true label *y*. In contrast to Equation 4.60, it does not involve terms such as

σ(zL)(1−σ(zL)) that can be impacted by saturation of zL. Hence, the

gradients are high whenever (aL−y) is large, promoting effective learning of

the model parameters at the output node. This has been a major breakthrough

in the learning of modern ANN models and it is now a common practice to

use the cross entropy loss function with sigmoid activations at the output

node.

4.8.2 Using Responsive Activation

Functions

Even though the cross entropy loss function helps in overcoming the problem

of saturating outputs, it still does not solve the problem of saturation at

hidden layers, arising due to the use of sigmoid activation functions at hidden

nodes. In fact, the effect of saturation on the learning of model parameters is

even more aggravated at hidden layers, a problem known as the vanishing

gradient problem. In the following, we describe the vanishing gradient

problem and the use of a more responsive activation function, called the

rectified linear output unit (ReLU), to overcome this problem.

Vanishing Gradient Problem

The impact of saturating activation values on the learning of model

parameters increases at deeper hidden layers that are farther away from the

output node. Even if the activation in the output layer does not saturate, the

repeated multiplications performed as we backpropagate the gradients from

the output layer to the hidden layers may lead to decreasing gradients in the

hidden layers. This is called the vanishing gradient problem, which has been

one of the major hindrances in learning deep neural networks.

To illustrate the vanishing gradient problem, consider an ANN model that

consists of a single node at every hidden layer of the network, as shown in

Figure 4.28. This simplified architecture involves a single chain of hidden

nodes where a single weighted link wl connects the node at layer l−1 to the

node at layer *l*. Using Equations 4.56 and 4.59, we can represent the partial

derivative of the loss with respect to wl as

∂ Loss∂wl=δl×al(1−al)×al−1,where δl=2(aL−y)×∏r=lL

−1(ar+1(1−ar+1)×wr+1). (4.64)

Notice that if any of the linear predictors zr+1 saturates at subsequent layers,

then the term ar+1(1−ar+1) becomes close to 0, thus diminishing the overall

gradient. The saturation of activations thus gets compounded and has

multiplicative effects on the gradients at hidden layers, making them highly

unstable and thus, unsuitable for use with gradient descent. Even though the

previous discussion only pertains to the simplified architecture involving a

single chain of hidden nodes, a similar argument can be made for any generic

ANN architecture involving multiple chains of hidden nodes. Note that the

vanishing gradient problem primarily arises because of the use of sigmoid

activation function at hidden nodes, which is known to easily saturate

especially after repeated multiplications.

Figure 4.28.

An example of an ANN model with only one node at every hidden

layer.

Figure 4.29.

Plot of the rectified linear unit (ReLU) activation function.

Rectified Linear Units (ReLU)

To overcome the vanishing gradient problem, it is important to use an

activation function *f*(*z*)at the hidden nodes that provides a stable and

significant value of the gradient whenever a hidden node is active, i.e., z>0.

This is achieved by using rectified linear units (ReLU) as activation functions

at hidden nodes, which can be defined as

a=f(z)={z,if z>0.0,otherwise. (4.65)

The idea of ReLU has been inspired from biological neurons, which are

either in an inactive state (f(z)=0) or show an activation value proportional to

the input. Figure 4.29 shows a plot of the ReLU function. We can see that it

is linear with respect to *z *when z>0. Hence, the gradient of the activation

value with respect to *z *can be written as

∂a∂z={1,if z>0.0,if z<0. (4.66)

Although *f*(*z*)is not differentiable at 0, it is common practice to use ∂a/∂z=0

when z=0. Since the gradient of the ReLU activation function is equal to 1

whenever z>0, it avoids the problem of saturation at hidden nodes, even after

repeated multiplications. Using ReLU, the partial derivatives of the loss with

respect to the weight and bias parameters can be given by

∂ Loss∂wijl=δil×I(zil)×ajl−1, (4.67)

∂ Loss∂bil=δil×I(zil),where δil=∑i=1n(δil+1×I(zil+1)×wijl+1),and I(z)=

{1,if z>0.0,otherwise. (4.68)

Notice that ReLU shows a linear behavior in the activation values whenever a

node is active, as compared to the nonlinear properties of the sigmoid

function. This linearity promotes better flows of gradients during

backpropagation, and thus simplifies the learning of ANN model parameters.

The ReLU is also highly responsive at large values of *z *away from 0, as

opposed to the sigmoid activation function, making it more suitable for

gradient descent. These differences give ReLU a major advantage over the

sigmoid function. Indeed, ReLU is used as the preferred choice of activation

function at hidden layers in most modern ANN models.

4.8.3 Regularization

A major challenge in learning deep neural networks is the high model

complexity of ANN models, which grows with the addition of hidden layers

in the network. This can become a serious concern, especially when the

training set is small, due to the phenomena of model overfitting. To

overcome this challenge, it is important to use techniques that can help in

reducing the complexity of the learned model, known as regularization

techniques. Classical approaches for learning ANN models did not have an

effective way to promote regularization of the learned model parameters.

Hence, they had often been sidelined by other classification methods, such as

support vector machines (SVM), which have in-built regularization

mechanisms. (SVMs will be discussed in more detail in Section 4.9).

One of the major advancements in deep learning has been the development of

novel regularization techniques for ANN models that are able to offer

significant improvements in generalization performance. In the following, we

discuss one of the regularization techniques for ANN, known as the dropout

method, that have gained a lot of attention in several applications.

Dropout

The main objective of dropout is to avoid the learning of spurious features at

hidden nodes, occurring due to model overfitting. It uses the basic intuition

that spurious features often “co-adapt” themselves such that they show good

training performance only when used in highly selective combinations. On

the other hand, relevant features can be used in a diversity of feature

combinations and hence are quite resilient to the removal or modification of

other features. The dropout method uses this intuition to break complex “coadaptations” in the learned features by randomly dropping input and hidden

nodes in the network during training.

Dropout belongs to a family of regularization techniques that uses the criteria

of resilience to random perturbations as a measure of the robustness (and

hence, simplicity) of a model. For example, one approach to regularization is

to inject noise in the input attributes of the training set and learn a model with

the noisy training instances. If a feature learned from the training data is

indeed generalizable, it should not be affected by the addition of noise.

Dropout can be viewed as a similar regularization approach that perturbs the

information content of the training set not only at the level of attributes but

also at multiple levels of abstractions, by dropping input and hidden nodes.

The dropout method draws inspiration from the biological process of gene

swapping in sexual reproduction, where half of the genes from both parents

are combined together to create the genes of the offspring. This favors the

selection of parent genes that are not only useful but can also inter-mingle

with diverse combinations of genes coming from the other parent. On the

other hand, co-adapted genes that function only in highly selective

combinations are soon eliminated in the process of evolution. This idea is

used in the dropout method for eliminating spurious co-adapted features. A

simplified description of the dropout method is provided in the rest of this

section.

Figure 4.30.

Examples of sub-networks generated in the dropout method using

γ=0.5.

Figure 4.30. Full Alternative Text

Let (wk, bk) represent the model parameters of the ANN model at the *k*th

iteration of the gradient descent method. At every iteration, we randomly

select a fraction γ of input and hidden nodes to be dropped from the network,

where γ∈(0, 1) is a hyper-parameter that is typically chosen to be 0.5. The

weighted links and bias terms involving the dropped nodes are then

eliminated, resulting in a “thinned” sub-network of smaller size. The model

parameters of the sub-network (wsk, bsk) are then updated by computing

activation values and performing backpropagation on this smaller subnetwork. These updated values are then added back in the original network to

obtain the updated model parameters, (wk+1, bk+1), to be used in the next

iteration.

Figure 4.30 shows some examples of sub-networks that can be generated at

different iterations of the dropout method, by randomly dropping input and

hidden nodes. Since every sub-network has a different architecture, it is

difficult to learn complex co-adaptations in the features that can result in

overfitting. Instead, the features at the hidden nodes are learned to be more

agile to random modifications in the network structure, thus improving their

generalization ability. The model parameters are updated using a different

random sub-network at every iteration, till the gradient descent method

converges.

Let (wkmax, bkmax) denote the model parameters at the last iteration kmax

of the gradient descent method. These parameters are finally scaled down by

a factor of (1−γ), to produce the weights and bias terms of the final ANN

model, as follows:

(w*, b*)=((1−γ)×wkmax, (1−γ)×bkmax)

We can now use the complete neural network with model parameters

(w*, b*) for testing. The dropout method has been shown to provide

significant improvements in the generalization performance of ANN models

in a number of applications. It is computationally cheap and can be applied in

combination with any of the other deep learning techniques. It also has a

number of similarities with a widely-used ensemble learning method known

as bagging, which learns multiple models using random subsets of the

training set, and then uses the *average output *of all the models to make

predictions. (Bagging will be presented in more detail later in Section 4.10.4).

In a similar vein, it can be shown that the predictions of the final network

learned using dropout approximates the average output of all possible 2n subnetworks that can be formed using *n *nodes. This is one of the reasons behind

the superior regularization abilities of dropout.

4.8.4 Initialization of Model

Parameters

Because of the non-convex nature of the loss function used by ANN models,

it is possible to get stuck in locally optimal but globally inferior solutions.

Hence, the initial choice of model parameter values plays a significant role in

the learning of ANN by gradient descent. The impact of poor initialization is

even more aggravated when the model is complex, the network architecture is

deep, or the classification task is difficult. In such cases, it is often advisable

to first learn a simpler model for the problem, e.g., using a single hidden

layer, and then incrementally increase the complexity of the model, e.g., by

adding more hidden layers. An alternate approach is to train the model for a

simpler task and then use the learned model parameters as initial parameter

choices in the learning of the original task. The process of initializing ANN

model parameters before the actual training process is known as pretraining.

Pretraining helps in initializing the model to a suitable region in the

parameter space that would otherwise be inaccessible by random

initialization. Pretraining also reduces the variance in the model parameters

by fixing the starting point of gradient descent, thus reducing the chances of

overfitting due to multiple comparisons. The models learned by pretraining

are thus more consistent and provide better generalization performance.

Supervised Pretraining

A common approach for pretraining is to incrementally train the ANN model

in a layer-wise manner, by adding one hidden layer at a time. This approach,

known as supervised pretraining, ensures that the parameters learned at

every layer are obtained by solving a simpler problem, rather than learning all

model parameters together. These parameter values thus provide a good

choice for initializing the ANN model. The approach for supervised

pretraining can be briefly described as follows.

We start the supervised pretraining process by considering a reduced ANN

model with only a single hidden layer. By applying gradient descent on this

simple model, we are able to learn the model parameters of the first hidden

layer. At the next run, we add another hidden layer to the model and apply

gradient descent to learn the parameters of the newly added hidden layer,

while keeping the parameters of the first layer fixed. This procedure is

recursively applied such that while learning the parameters of the *l*th hidden

layer, we consider a reduced model with only *l *hidden layers, whose first (l

−1) hidden layers are not updated on the *l*th run but are instead fixed using

pretrained values from previous runs. In this way, we are able to learn the

model parameters of all (L−1) hidden layers. These pretrained values are

used to initialize the hidden layers of the final ANN model, which is finetuned by applying a final round of gradient descent over all the layers.

Unsupervised Pretraining

Supervised pretraining provides a powerful way to initialize model

parameters, by gradually growing the model complexity from shallower to

deeper networks. However, supervised pretraining requires a sufficient

number of labeled training instances for effective initialization of the ANN

model. An alternate pretraining approach is unsupervised pretraining,

which initializes model parameters by using unlabeled instances that are often

abundantly available. The basic idea of unsupervised pretraining is to

initialize the ANN model in such a way that the learned features capture the

latent structure in the unlabeled data.

Figure 4.31.

The basic architecture of a single-layer autoencoder.

Unsupervised pretraining relies on the assumption that learning the

distribution of the input data can indirectly help in learning the classification

model. It is most helpful when the number of labeled examples is small and

the features for the supervised problem bear resemblance to the factors

generating the input data. Unsupervised pretraining can be viewed as a

different form of regularization, where the focus is not explicitly toward

finding simpler features but instead toward finding features that can best

explain the input data. Historically, unsupervised pretraining has played an

important role in reviving the area of deep learning, by making it possible to

train any generic deep neural network without requiring specialized

architectures.

Use of Autoencoders

One simple and commonly used approach for unsupervised pretraining is to

use an unsupervised ANN model known as an autoencoder. The basic

architecture of an autoencoder is shown in Figure 4.31. An autoencoder

attempts to learn a *reconstruction *of the input data by mapping the attributes

x to latent features c, and then re-projecting c back to the original attribute

space to create the reconstruction x^. The latent features are represented

using a hidden layer of nodes, while the input and output layers represent the

attributes and contain the same number of nodes. During training, the goal is

to learn an autoencoder model that provides the lowest reconstruction error,

RE(x, x^), on all input data instances. A typical choice of the reconstruction

error is the squared loss function:

RE(x, x^)=ǁx−x^ ǁ2.

The model parameters of the autoencoder can be learned by using a similar

gradient descent method as the one used for learning supervised ANN models

for classification. The key difference is the use of the reconstruction error on

all training instances as the training loss. Autoencoders that have multiple

layers of hidden layers are known as stacked autoencoders.

Autoencoders are able to capture complex representations of the input data by

the use of hidden nodes. However, if the number of hidden nodes is large, it

is possible for an autoencoder to learn the identity relationship, where the

input x is just copied and returned as the output x^, resulting in a trivial

solution. For example, if we use as many hidden nodes as the number of

attributes, then it is possible for every hidden node to copy an attribute and

simply pass it along to an output node, without extracting any useful

information. To avoid this problem, it is common practice to keep the number

of hidden nodes smaller than the number of input attributes. This forces the

autoencoder to learn a compact and useful *encoding *of the input data, similar

to a dimensionality reduction technique. An alternate approach is to corrupt

the input instances by adding random noise, and then learn the autoencoder to

reconstruct the original instance from the noisy input. This approach is

known as the denoising autoencoder, which offers strong regularization

capabilities and is often used to learn complex features even in the presence

of a large number of hidden nodes.

To use an autoencoder for unsupervised pretraining, we can follow a similar

layer-wise approach like supervised pretraining. In particular, to pretrain the

model parameters of the *l*th hidden layer, we can construct a reduced ANN

model with only *l *hidden layers and an output layer containing the same

number of nodes as the attributes and is used for reconstruction. The

parameters of the *l*th hidden layer of this network are then learned using a

gradient descent method to minimize the reconstruction error. The use of

unlabeled data can be viewed as providing *hints *to the learning of parameters

at every layer that aid in generalization. The final model parameters of the

ANN model are then learned by applying gradient descent over all the layers,

using the initial values of parameters obtained from pretraining.

Hybrid Pretraining

Unsupervised pretraining can also be combined with supervised pretraining

by using two output layers at every run of pretraining, one for reconstruction

and the other for supervised classification. The parameters of the *l*th hidden

layer are then learned by jointly minimizing the losses on both output layers,

usually weighted by a trade-off hyper-parameter α. Such a combined

approach often shows better generalization performance than either of the

approaches, since it provides a way to balance between the competing

objectives of representing the input data and improving classification

performance.

4.8.5 Characteristics of Deep

Learning

Apart from the basic characteristics of ANN discussed in Section 4.7.3, the

use of deep learning techniques provides the following additional

characteristics:

1. An ANN model trained for some task can be easily re-used for a

different task that involves the same attributes, by using pretraining

strategies. For example, we can use the learned parameters of the

original task as initial parameter choices for the target task. In this way,

ANN promotes *re-usability *of learning, which can be quite useful when

the target application has a smaller number of labeled training instances.

2. Deep learning techniques for regularization, such as the dropout method,

help in reducing the model complexity of ANN and thus promoting

good generalization performance. The use of regularization techniques is

especially useful in high-dimensional settings, where the number of

training labels is small but the classification problem is inherently

difficult.

3. The use of an autoencoder for pretraining can help eliminate irrelevant

attributes that are not related to other attributes. Further, it can help

reduce the impact of redundant attributes by representing them as copies

of the same attribute.

4. Although the learning of an ANN model can succumb to finding inferior

and locally optimal solutions, there are a number of deep learning

techniques that have been proposed to ensure adequate learning of an

ANN. Apart from the methods discussed in this section, some other

techniques involve novel architecture designs such as skip connections

between the output layer and lower layers, which aids the easy flow of

gradients during backpropagation.

5. A number of specialized ANN architectures have been designed to

handle a variety of input data sets. Some examples include

convolutional neural networks (CNN) for two-dimensional gridded

objects such as images, and recurrent neural network (RNN) for

sequences. While CNNs have been extensively used in the area of

computer vision, RNNs have found applications in processing speech

and language.

4.9 Support Vector Machine (SVM)

A support vector machine (SVM) is a discriminative classification model that

learns linear or nonlinear decision boundaries in the attribute space to

separate the classes. Apart from maximizing the separability of the two

classes, SVM offers strong regularization capabilities, i.e., it is able to control

the complexity of the model in order to ensure good generalization

performance. Due to its unique ability to innately regularize its learning,

SVM is able to learn highly expressive models without suffering from

overfitting. It has thus received considerable attention in the machine learning

community and is commonly used in several practical applications, ranging

from handwritten digit recognition to text categorization. SVM has strong

roots in statistical learning theory and is based on the principle of structural

risk minimization. Another unique aspect of SVM is that it represents the

decision boundary using only a subset of the training examples that are most

difficult to classify, known as the support vectors. Hence, it is a

discriminative model that is impacted only by training instances near the

boundary of the two classes, in contrast to learning the generative distribution

of every class.

To illustrate the basic idea behind SVM, we first introduce the concept of the

margin of a separating hyperplane and the rationale for choosing such a

hyperplane with maximum margin. We then describe how a linear SVM can

be trained to explicitly look for this type of hyperplane. We conclude by

showing how the SVM methodology can be extended to learn nonlinear

decision boundaries by using kernel functions.

4.9.1 Margin of a Separating

Hyperplane

The generic equation of a separating hyperplane can be written as

wTx+b=0,

where x represents the attributes and (w, b) represent the parameters of the

hyperplane. A data instance xi can belong to either side of the hyperplane

depending on the sign of (wTxi+b). For the purpose of binary classification,

we are interested in finding a hyperplane that places instances of both classes

on opposite sides of the hyperplane, thus resulting in a *separation *of the two

classes. If there exists a hyperplane that can perfectly separate the classes in

the data set, we say that the data set is linearly separable. Figure 4.32 shows

an example of linearly separable data involving two classes, squares and

circles. Note that there can be infinitely many hyperplanes that can separate

the classes, two of which are shown in Figure 4.32 as lines B1 and B2. Even

though every such hyperplane will have zero training error, they can provide

different results on previously unseen instances. Which separating hyperplane

should we thus finally choose to obtain the best generalization performance?

Ideally, we would like to choose a *simple *hyperplane that is robust to small

perturbations. This can be achieved by using the concept of the margin of a

separating hyperplane, which can be briefly described as follows.

Figure 4.32.

Margin of a hyperplane in a two-dimensional data set.

Figure 4.32. Full Alternative Text

For every separating hyperplane Bi, let us associate a pair of parallel

hyperplanes, bi1 and bi2, such that they touch the closest instances of both

classes, respectively. For example, if we move B1 parallel to its direction, we

can touch the first square using b11 and the first circle using b12. bi1 and bi2

are known as the margin hyperplanes of Bi and the distance between them

is known as the margin of the separating hyperplane Bi. From the diagram

shown in Figure 4.32, notice that the margin for B1 is considerably larger

than that for B2. In this example, b1 turns out to be the separating hyperplane

with the maximum margin, known as the maximum margin hyperplane.

Rationale for Maximum Margin

Hyperplanes with large margins tend to have better generalization

performance than those with small margins. Intuitively, if the margin is

small, then any slight perturbation in the hyperplane or the training instances

located at the boundary can have quite an impact on the classification

performance. Small margin hyperplanes are thus more susceptible to

overfitting, as they are barely able to separate the classes with a very narrow

room to allow perturbations. On the other hand, a hyperplane that is farther

away from training instances of both classes has sufficient leeway to be

robust to minor modifications in the data, and thus shows superior

generalization performance.

The idea of choosing the maximum margin separating hyperplane also has

strong foundations in statistical learning theory. It can be shown that the

margin of such a hyperplane is inversely related to the *VC*-dimension of the

classifier, which is a commonly used measure of the complexity of a model.

As discussed in Section 3.4 of the last chapter, a simpler model should be

preferred over a more complex model if they both show similar training

performance. Hence, maximizing the margin results in the selection of a

separating hyperplane with the lowest model complexity, which is expected

to show better generalization performance.

4.9.2 Linear SVM

A linear SVM is a classifier that searches for a separating hyperplane with the

largest margin, which is why it is often known as a maximal margin

classifier. The basic idea of SVM can be described as follows.

Consider a binary classification problem consisting of *n *training instances,

where every training instance xi is associated with a binary label yi∈{−1, 1}.

Let wTx+b=0 be the equation of a separating hyperplane that separates the

two classes by placing them on opposite sides. This means that

wTxi+b>0if yi=1,wTxi+b<0if yi=−1.

The distance of any point x from the hyperplane is then given by

D(x)=|wTx+b |ǁ w ǁ

where |⋅| denotes the absolute value and ǁ ⋅ ǁ denotes the length of a vector.

Let the distance of the closest point from the hyperplane with y=1 be k+>0.

Similarly, let k−>0 denote the distance of the closest point from class −1.

This can be represented using the following constraints:

wTxi+bǁ w ǁ≥k+if yi=1,wTxi+bǁ w ǁ≤−k−if yi=−1, (4.69)

The previous equations can be succinctly represented by using the product of

yi and (wTxi+b) as

yi(wTxi+b)≥Mǁwǁ (4.70)

where *M *is a parameter related to the margin of the hyperplane, i.e., if k+=k

−=M, then margin =k+−k−=2M. In order to find the maximum margin

hyperplane that adheres to the previous constraints, we can consider the

following optimization problem:

maxw, bMsubject toyi(wTxi+b)≥Mǁ w ǁ. (4.71)

To find the solution to the previous problem, note that if w and *b *satisfy the

constraints of the previous problem, then any scaled version of w and *b *would

satisfy them too. Hence, we can conveniently choose ǁwǁ=1/M to simplify the

right-hand side of the inequalities. Furthermore, maximizing *M *amounts to

minimizing ǁwǁ2. Hence, the optimization problem of SVM is commonly

represented in the following form:

minw, bǁ w ǁ22subject toyi(wTxi+b)≥1. (4.72)

Learning Model Parameters

Equation 4.72 represents a constrained optimization problem with linear

inequalities. Since the objective function is convex and quadratic with respect

to w, it is known as a quadratic programming problem (QPP), which can be

solved using standard optimization techniques, as described in Appendix E.

In the following, we present a brief sketch of the main ideas for learning the

model parameters of SVM.

First, we rewrite the objective function in a form that takes into account the

constraints imposed on its solutions. The new objective function is known as

the Lagrangian primal problem, which can be represented as follows,

LP=12ǁ w ǁ2−∑i=1nλi(yi(wTxi+b)−1), (4.73)

where the parameters λi≥0 correspond to the constraints and are called the

Lagrange multipliers. Next, to minimize the Lagrangian, we take the

derivative of LP with respect to w and *b *and set them equal to zero:

∂LP∂w=0⇒w=∑i=1nλiyixi, (4.74)

∂LP∂b=0⇒∑i=1nλiyi=0. (4.75)

Note that using Equation 4.74, we can represent w completely in terms of the

Lagrange multipliers. There is another relationship between (w, *b*) and λi that

is derived from the Karush-Kuhn-Tucker (KKT) conditions, a commonly

used technique for solving QPP. This relationship can be described as

λi[yi(wTxi+b)−1]=0. (4.76)

Equation 4.76 is known as the complementary slackness condition, which

sheds light on a valuable property of SVM. It states that the Lagrange

multiplier λi is strictly greater than 0 only when xi satisfies the equation

yi(w⋅xi+b)=1, which means that xi lies exactly on a margin hyperplane.

However, if xi is farther away from the margin hyperplanes such that

yi(w⋅xi+b)>1, then λi is necessarily 0. Hence, λi>0 for only a small number

of instances that are closest to the separating hyperplane, which are known as

support vectors. Figure 4.33 shows the support vectors of a hyperplane as

filled circles and squares. Further, if we look at Equation 4.74, we will

observe that training instances with λi=0 do not contribute to the weight

parameter w. This suggests that w can be concisely represented only in terms

of the support vectors in the training data, which are quite fewer than the

overall number of training instances. This ability to represent the decision

function only in terms of the support vectors is what gives this classifier the

name support vector machines.

Figure 4.33.

Support vectors of a hyperplane shown as filled circles and squares.

Figure 4.33. Full Alternative Text

Using equations 4.74, 4.75, and 4.76 in Equation 4.73, we obtain the

following optimization problem in terms of the Lagrange multipliers λi:

maxλi∑i=1nλi−12∑i=1n∑j=1nλiλjyiyjxiTxjsubject to∑i=1nλiyi=0,λi≥0.

(4.77)

The previous optimization problem is called the dual optimization problem.

Maximizing the dual problem with respect to λi is equivalent to minimizing

the primal problem with respect to w and *b*.

The key differences between the dual and primal problems are as follows:

1. Solving the dual problem helps us identify the support vectors in the

data that have non-zero values of λi. Further, the solution of the dual

problem is influenced only by the support vectors that are closest to the

decision boundary of SVM. This helps in summarizing the learning of

SVM solely in terms of its support vectors, which are easier to manage

computationally. Further, it represents a unique ability of SVM to be

dependent only on the instances closest to the boundary, which are

harder to classify, rather than the distribution of instances farther away

from the boundary.

2. The objective of the dual problem involves only terms of the form xiTxj,

which are basically inner products in the attribute space. As we will see

later in Section 4.9.4, this property will prove to be quite useful in

learning nonlinear decision boundaries using SVM.

Because of these differences, it is useful to solve the dual optimization

problem using any of the standard solvers for QPP. Having found an optimal

solution for λi, we can use Equation 4.74 to solve for w. We can then use

Equation 4.76 on the support vectors to solve for *b *as follows:

b=1nS∑i∈S1−yiwTxiyi (4.78)

where *S *represents the set of support vectors (S={i|λi>0}) and nS is the

number of support vectors. The maximum margin hyperplane can then be

expressed as

f(x)=(∑i=1nλiyixiTx)+b=0. (4.79)

Using this separating hyperplane, a test instance x can be assigned a class

label using the sign of *f*(x).

Example 4.7.

Consider the two-dimensional data set shown in Figure 4.34, which contains

eight training instances. Using quadratic programming, we can solve the

optimization problem stated in Equation 4.77 to obtain the Lagrange

multiplier λi for each training instance. The Lagrange multipliers are depicted

in the last column of the table. Notice that only the first two instances have

non-zero Lagrange multipliers. These instances correspond to the support

vectors for this data set.

Let w=(w1, w2) and *b *denote the parameters of the decision boundary. Using

Equation 4.74, we can solve for w1 and w2 in the following way:

w1=∑iλiyixi1=65.5261×1×0.3858+65.5261×−1×0.4871=

−6.64.w2=∑iλiyixi2=65.5261×1×0.4687+65.5261×−1×0.611=−9.32.

Figure 4.34.

Example of a linearly separable data set.

Figure 4.34. Full Alternative Text

The bias term *b *can be computed using Equation 4.76 for each support vector:

b(1)=1−w⋅x1=1−(−6.64)(0.3858)−(−9.32)(0.4687)=7.9300.b(2)=1−w⋅x2=

−1−(−6.64)(0.4871)−(−9.32)(0.611)=7.9289.

Averaging these values, we obtain b=7.93. The decision boundary

corresponding to these parameters is shown in Figure 4.34.

4.9.3 Soft-margin SVM

Figure 4.35 shows a data set that is similar to Figure 4.32, except it has two

new examples, *P *and *Q*. Although the decision boundary B1 misclassifies the

new examples, while B2 classifies them correctly, this does not mean that B2

is a better decision boundary than B1 because the new examples may

correspond to noise in the training data. B1 should still be preferred over B2

because it has a wider margin, and thus, is less susceptible to overfitting.

However, the SVM formulation presented in the previous section only

constructs decision boundaries that are mistake-free.

Figure 4.35.

Decision boundary of SVM for the non-separable case.

Figure 4.35. Full Alternative Text

This section examines how the formulation of SVM can be modified to learn

a separating hyperplane that is tolerable to small number of training errors

using a method known as the soft-margin approach. More importantly, the

method presented in this section allows SVM to learn linear hyperplanes even

in situations where the classes are not linearly separable. To do this, the

learning algorithm in SVM must consider the trade-off between the width of

the margin and the number of training errors committed by the linear

hyperplane.

To introduce the concept of training errors in the SVM formulation, let us

relax the inequality constraints to accommodate for some violations on a

small number of training instances. This can be done by introducing a slack

variable ξ≥0 for every training instance xi as follows:

yi(wTxi+b)≥1−ξi (4.80)

The variable ξi allows for some slack in the inequalities of the SVM such that

every instance xi does not need to strictly satisfy yi(wTxi+b)≥1. Further, ξi is

non-zero only if the margin hyperplanes are not able to place xi on the same

side as the rest of the instances belonging to yi. To illustrate this, Figure 4.36

shows a circle *P *that falls on the opposite side of the separating hyperplane as

the rest of the circles, and thus satisfies wTx+b=−1+ξ. The distance between

*P *and the margin hyperplane wTx+b=−1 is equal to ξ/ǁ w ǁ. Hence, ξi

provides a measure of the error of SVM in representing xi using *soft*

inequality constraints.

Figure 4.36.

Slack variables used in soft-margin SVM.

Figure 4.36. Full Alternative Text

In the presence of slack variables, it is important to learn a separating

hyperplane that jointly maximizes the margin (ensuring good generalization

performance) and minimizes the values of slack variables (ensuring low

training error). This can be achieved by modifying the optimization problem

of SVM as follows:

minw, b, ξiǁ w ǁ22+c∑i=1nξisubject toyi(wTxi+b)≥1−ξi,ξi≥0. (4.81)

where *C *is a hyper-parameter that makes a trade-off between maximizing the

margin and minimizing the training error. A large value of *C *pays more

emphasis on minimizing the training error than maximizing the margin.

Notice the similarity of the previous equation with the generic formula of

generalization error rate introduced in Section 3.4 of the previous chapter.

Indeed, SVM provides a natural way to balance between model complexity

and training error in order to maximize generalization performance.

To solve Equation 4.81 we apply the Lagrange multiplier method and convert

the primal problem to its corresponding dual problem, similar to the approach

described in the previous section. The Lagrangian primal problem of

Equation 4.81 can be written as follows:

LP=12ǁ w ǁ2+C∑i=1nξi−∑i=1nλi(yi(wTxi+b)−1+ξi)−∑i=1nμi(ξi), (4.82)

where λi≥0 and μi≥0 are the Lagrange multipliers corresponding to the

inequality constraints of Equation 4.81. Setting the derivative of LP with

respect to w, *b*, and ξi equal to 0, we obtain the following equations:

∂LP∂w=0⇒w=∑i=1nλiyixi. (4.83)

∂L∂b=0⇒∑i=1nλiyi=0. (4.84)

∂L∂ξi=0⇒λi+μi=C. (4.85)

We can also obtain the complementary slackness conditions by using the

following KKT conditions:

λi(yi(wTxi+b)−1+ξi)=0, (4.86)

μiξi=0. (4.87)

Equation 4.86 suggests that λi is zero for all training instances except those

that reside on the margin hyperplanes wTxi+b=±1, or have ξi>0. These

instances with λi>0 are known as support vectors. On the other hand, μi given

in Equation 4.87 is zero for any training instance that is misclassified, i.e.,

ξi>0. Further, λi and μi are related with each other by Equation 4.85. This

results in the following three configurations of (λi, μi):

1. If λi=0 and μi=C, then xi does not reside on the margin hyperplanes and

is correctly classified on the same side as other instances belonging to

yi.

2. If λi=C and μi=0, then xi is misclassified and has a non-zero slack

variable ξi.

3. If 0<λi<C and 0<μi<C, then xi resides on one of the margin

hyperplanes.

Substituting Equations 4.83 to 4.87 into Equation 4.82, we obtain the

following dual optimization problem:

maxλi∑i=1nλi−12∑i=1n∑j=1nλiλjyiyjxiTxjsubject to∑i=1nλiyi=0,0≤λi≤C.

(4.88)

Notice that the previous problem looks almost identical to the dual problem

of SVM for the linearly separable case (Equation 4.77), except that λi is

required to not only be greater than 0 but also smaller than a constant value

*C*. Clearly, when *C *reaches infinity, the previous optimization problem

becomes equivalent to Equation 4.77, where the learned hyperplane perfectly

separates the classes (with no training errors). However, by capping the

values of λi to *C*, the learned hyperplane is able to tolerate a few training

errors that have ξi>0.

Figure 4.37.

Hinge loss as a function of yy^.

As before, Equation 4.88 can be solved by using any of the standard solvers

for QPP, and the optimal value of w can be obtained by using Equation 4.83.

To solve for *b*, we can use Equation 4.86 on the support vectors that reside on

the margin hyperplanes as follows:

b=1nS∑i∈S1−yiwTxiyi (4.89)

where *S *represents the set of support vectors residing on the margin

hyperplanes (S={i|0<λi<C}) and nS is the number of elements in *S*.

SVM as a Regularizer of Hinge Loss

SVM belongs to a broad class of regularization techniques that use a loss

function to represent the training errors and a norm of the model parameters

to represent the model complexity. To realize this, notice that the slack

variable ξ, used for measuring the training errors in SVM, is equivalent to the

hinge loss function, which can be defined as follows:

Loss (y, y^) =max(0, 1−yy^),

where y∈{+1, −1}. In the case of SVM, y^ corresponds to wTx+b. Figure

4.37 shows a plot of the hinge loss function as we vary yy^. We can see that

the hinge loss is equal to 0 as long as *y *and y^ have the same sign and |y^|≥1.

However, the hinge loss grows linearly with |y^| whenever *y *and y^ are of the

opposite sign or |y^|<1. This is similar to the notion of the slack variable ξ,

which is used to measure the distance of a point from its margin hyperplane.

Hence, the optimization problem of SVM can be represented in the following

equivalent form:

minw, bǁ w ǁ22+C∑i=1nLoss (yi, wTxi+b) (4.90)

Note that using the hinge loss ensures that the optimization problem is

convex and can be solved using standard optimization techniques. However,

if we use a different loss function, such as the squared loss function that was

introduced in Section 4.7 on ANN, it will result in a different optimization

problem that may or may not remain convex. Nevertheless, different loss

functions can be explored to capture varying notions of training error,

depending on the characteristics of the problem.

Another interesting property of SVM that relates it to a broader class of

regularization techniques is the concept of a margin. Although minimizing ǁ

w ǁ2 has the geometric interpretation of maximizing the margin of a

separating hyperplane, it is essentially the squared L2 norm of the model

parameters, ǁ w ǁ22. In general, the Lq norm of w, ǁ w ǁq, is equal to the

Minkowski distance of order *q *from w to the origin, i.e.,

ǁ w ǁq=(∑ipwiq)1/q

Minimizing the Lq norm of w to achieve lower model complexity is a generic

regularization concept that has several interpretations. For example,

minimizing the L2 norm amounts to finding a solution on a hypersphere of

smallest radius that shows suitable training performance. To visualize this in

two-dimensions, Figure 4.38(a) shows the plot of a circle with constant radius

*r*, where every point has the same L2 norm. On the other hand, using the L1

norm ensures that the solution lies on the surface of a hypercube with

smallest size, with vertices along the axes. This is illustrated in Figure 4.38(b)

as a square with vertices on the axes at a distance of *r *from the origin. The L1

norm is commonly used as a regularizer to obtain *sparse *model parameters

with only a small number of non-zero parameter values, such as the use of

Lasso in regression problems (see Bibliographic Notes).

Figure 4.38.

Plots showing the behavior of two-dimensional solutions with

constant L2 and L1 norms.

Figure 4.38. Full Alternative Text

In general, depending on the characteristics of the problem, different

combinations of Lq norms and training loss functions can be used for

learning the model parameters, each requiring a different optimization solver.

This forms the backbone of a wide range of modeling techniques that attempt

to improve the generalization performance by jointly minimizing training

error and model complexity. However, in this section, we focus only on the

squared L2 norm and the hinge loss function, resulting in the classical

formulation of SVM.

4.9.4 Nonlinear SVM

The SVM formulations described in the previous sections construct a linear

decision boundary to separate the training examples into their respective

classes. This section presents a methodology for applying SVM to data sets

that have nonlinear decision boundaries. The basic idea is to transform the

data from its original attribute space in x into a new space φ(x) so that a linear

hyperplane can be used to separate the instances in the transformed space,

using the SVM approach. The learned hyperplane can then be projected back

to the original attribute space, resulting in a nonlinear decision boundary.

Figure 4.39.

Classifying data with a nonlinear decision boundary.

Figure 4.39. Full Alternative Text

Attribute Transformation

To illustrate how attribute transformation can lead to a linear decision

boundary, Figure 4.39(a) shows an example of a two-dimensional data set

consisting of squares (classified as y=1) and circles (classified as y=−1). The

data set is generated in such a way that all the circles are clustered near the

center of the diagram and all the squares are distributed farther away from the

center. Instances of the data set can be classified using the following

equation:

y={1if (x1−0.5)2+(x2−0.5)2>0.2,−1otherwise. (4.91)

The decision boundary for the data can therefore be written as follows:

(x1−0.5)2+(x2−0.5)2>0.2,

which can be further simplified into the following quadratic equation:

x12−x1+x22−x2=−0.46.

A nonlinear transformation φ is needed to map the data from its original

attribute space into a new space such that a linear hyperplane can separate the

classes. This can be achieved by using the following simple transformation:

φ:(x1, x2)→(x12−x1, x22−x2). (4.92)

Figure 4.39(b) shows the points in the transformed space, where we can see

that all the circles are located in the lower left-hand side of the diagram. A

linear hyperplane with parameters w and *b *can therefore be constructed in the

transformed space, to separate the instances into their respective classes.

One may think that because the nonlinear transformation possibly increases

the dimensionality of the input space, this approach can suffer from the curse

of dimensionality that is often associated with high-dimensional data.

However, as we will see in the following section, nonlinear SVM is able to

avoid this problem by using kernel functions.

Learning a Nonlinear SVM Model

Using a suitable function, φ(⋅), we can transform any data instance x to φ(x).

(The details on how to choose φ(⋅) will become clear later.) The linear

hyperplane in the transformed space can be expressed as wTφ(x)+b=0. To

learn the optimal separating hyperplane, we can substitute φ(x) for x in the

formulation of SVM to obtain the following optimization problem:

minw, b, ξiǁ w ǁ22+C∑i=1nξisubject toyi(wTφ(xi)+b)≥1−ξi,ξi≥0. (4.93)

Using Lagrange multipliers λi, this can be converted into a dual optimization

problem: max

maxλi∑i=1nλi−12∑i=1n∑j=1nλiλjyiyj〈φ(xi), φ(xj)

〉subject to∑i=1nλjyi=0,0≤λi≤C, (4.94)

where 〈 a, b 〉 denotes the inner product between vectors a and b. Also, the

equation of the hyperplane in the transformed space can be represented using

λi as follows:

∑i=1nλiyi〈φ(xi), φ(x) 〉+b=0. (4.95)

Further, *b *is given by

b=1nS(∑i∈S1yi−∑i∈S∑j=1nλjyiyj〈φ(xi), φ(xj) 〉yi), (4.96)

where S={i|0>λi<C} is the set of support vectors residing on the margin

hyperplanes and nS is the number of elements in *S*.

Note that in order to solve the dual optimization problem in Equation 4.94, or

to use the learned model parameters to make predictions using Equations

4.95 and 4.96, we need only inner products of φ(x). Hence, even though φ(x)

may be nonlinear and high-dimensional, it suffices to use a function of the

inner products of φ(x) in the transformed space. This can be achieved by

using a kernel trick, which can be described as follows.

The inner product between two vectors is often regarded as a measure of

similarity between the vectors. For example, the cosine similarity described

in Section 2.4.5 on page 79 can be defined as the dot product between two

vectors that are normalized to unit length. Analogously, the inner product

φ(xi), φ(xj) can also be regarded as a measure of similarity between two

instances, xi and xj, in the transformed space. The kernel trick is a method

for computing this similarity as a function of the original attributes.

Specifically, the kernel function *K*(u, v) between two instances *u *and *v *can be

defined as follows:

K(u, v)=〈φ(u), φ(v) 〉=f(u, v) (4.97)

where f(⋅) is a function that follows certain conditions as stated by the

Mercer’s Theorem. Although the details of this theorem are outside the scope

of the book, we provide a list of some of the commonly used kernel

functions:

Polynomial kernelK(u, v)=(uTv+1)p (4.98)

Radial Basis Function kernelK(u, v)=e−ǁu−v ǁ2/(2σ2) (4.99)

Sigmoid kernelK(u, v)=tanh(kuTv−δ) (4.100)

By using a kernel function, we can directly work with inner products in the

transformed space without dealing with the exact forms of the nonlinear

transformation function φ. Specifically, this allows us to use highdimensional transformations (sometimes even involving infinitely many

dimensions), while performing calculations only in the original attribute

space. Computing the inner products using kernel functions is also

considerably cheaper than using the transformed attribute set φ(x). Hence, the

use of kernel functions provides a significant advantage in representing

nonlinear decision boundaries, without suffering from the curse of

dimensionality. This has been one of the major reasons behind the

widespread usage of SVM in highly complex and nonlinear problems.

Figure 4.40.

Decision boundary produced by a nonlinear SVM with polynomial

kernel.

Figure 4.40 shows the nonlinear decision boundary obtained by SVM using

the polynomial kernel function given in Equation 4.98. We can see that the

learned decision boundary is quite close to the true decision boundary shown

in Figure 4.39(a). Although the choice of kernel function depends on the

characteristics of the input data, a commonly used kernel function is the

radial basis function (RBF) kernel, which involves a single hyper-parameter

σ, known as the standard deviation of the RBF kernel.

4.9.5 Characteristics of SVM

1. The SVM learning problem can be formulated as a convex optimization

problem, in which efficient algorithms are available to find the global

minimum of the objective function. Other classification methods, such

as rule-based classifiers and artificial neural networks, employ a greedy

strategy to search the hypothesis space. Such methods tend to find only

locally optimum solutions.

2. SVM provides an effective way of regularizing the model parameters by

maximizing the margin of the decision boundary. Furthermore, it is able

to create a balance between model complexity and training errors by

using a hyper-parameter *C*. This trade-off is generic to a broader class of

model learning techniques that capture the model complexity and the

training loss using different formulations.

3. Linear SVM can handle irrelevant attributes by learning zero weights

corresponding to such attributes. It can also handle redundant attributes

by learning similar weights for the duplicate attributes. Furthermore, the

ability of SVM to regularize its learning makes it more robust to the

presence of a large number of irrelevant and redundant attributes than

other classifiers, even in high-dimensional settings. For this reason,

nonlinear SVMs are less impacted by irrelevant and redundant attributes

than other highly expressive classifiers that can learn nonlinear decision

boundaries such as decision trees.

To compare the effect of irrelevant attributes on the performance of

nonlinear SVMs and decision trees, consider the two-dimensional data

set shown in Figure 4.41(a) containing 500+ and 500o instances, where

the two classes can be easily separated using a nonlinear decision

boundary. We incrementally add irrelevant attributes to this data set and

compare the performance of two classifiers: decision tree and nonlinear

SVM (using radial basis function kernel), using 70% of the data for

training and the rest for testing. Figure 4.41(b) shows the test error rates

of the two classifiers as we increase the number of irrelevant attributes.

We can see that the test error rate of decision trees swiftly reaches 0.5

(same as random guessing) in the presence of even a small number of

irrelevant attributes. This can be attributed to the problem of multiple

comparisons while choosing splitting attributes at internal nodes as

discussed in Example 3.7 of the previous chapter. On the other hand,

nonlinear SVM shows a more robust and steady performance even after

adding a moderately large number of irrelevant attributes. Its test error

rate gradually declines and eventually reaches close to 0.5 after adding

125 irrelevant attributes, at which point it becomes difficult to discern

the discriminative information in the original two attributes from the

noise in the remaining attributes for learning nonlinear decision

boundaries.

Figure 4.41.

Comparing the effect of adding irrelevant attributes on the

performance of nonlinear SVMs and decision trees.

Figure 4.41. Full Alternative Text

4. SVM can be applied to categorical data by introducing dummy variables

for each categorical attribute value present in the data. For example, if

Marital Status has three values {Single,Married,Divorced}, we can

introduce a binary variable for each of the attribute values.

5. The SVM formulation presented in this chapter is for binary class

problems. However, multiclass extensions of SVM have also been

proposed.

6. Although the training time of an SVM model can be large, the learned

parameters can be succinctly represented with the help of a small

number of support vectors, making the classification of test instances

quite fast.

4.10 Ensemble Methods

This section presents techniques for improving classification accuracy by

aggregating the predictions of multiple classifiers. These techniques are

known as ensemble or classifier combination methods. An ensemble

method constructs a set of base classifiers from training data and performs

classification by taking a vote on the predictions made by each base

classifier. This section explains why ensemble methods tend to perform better

than any single classifier and presents techniques for constructing the

classifier ensemble.

4.10.1 Rationale for Ensemble

Method

The following example illustrates how an ensemble method can improve a

classifier’s performance.

Example 4.8.

Consider an ensemble of 25 binary classifiers, each of which has an error rate

of ∈=0.35. The ensemble classifier predicts the class label of a test example

by taking a majority vote on the predictions made by the base classifiers. If

the base classifiers are identical, then all the base classifiers will commit the

same mistakes. Thus, the error rate of the ensemble remains 0.35. On the

other hand, if the base classifiers are independent—i.e., their errors are

uncorrelated—then the ensemble makes a wrong prediction only if more than

half of the base classifiers predict incorrectly. In this case, the error rate of the

ensemble classifier is

eensemble=∑i=1325(25i)∈i(1−∈)25−i=0.06, (4.101)

which is considerably lower than the error rate of the base classifiers.

Figure 4.42 shows the error rate of an ensemble of 25 binary classifiers

(eensemble) for different base classifier error rates (∈). The diagonal line

represents the case in which the base classifiers are identical, while the solid

line represents the case in which the base classifiers are independent. Observe

that the ensemble classifier performs worse than the base classifiers when ∈

is larger than 0.5.

The preceding example illustrates two necessary conditions for an ensemble

classifier to perform better than a single classifier: (1) the base classifiers

should be independent of each other, and (2) the base classifiers should do

better than a classifier that performs random guessing. In practice, it is

difficult to ensure total independence among the base classifiers.

Nevertheless, improvements in classification accuracies have been observed

in ensemble methods in which the base classifiers are somewhat correlated.

4.10.2 Methods for Constructing an

Ensemble Classifier

A logical view of the ensemble method is presented in Figure 4.43. The basic

idea is to construct multiple classifiers from the original data and then

aggregate their predictions when classifying unknown examples. The

ensemble of classifiers can be constructed in many ways:

1. By manipulating the training set. In this approach, multiple training sets

are created by resampling the original data according to some sampling

distribution and constructing a classifier from each training set. The

sampling distribution determines how likely it is that an example will be

selected for training, and it may vary from one trial to another. Bagging

and boosting are two examples of ensemble methods that manipulate

their training sets. These methods are described in further detail in

Sections 4.10.4 and 4.10.5.

Figure 4.42.

Comparison between errors of base classifiers and errors of

the ensemble classifier.

Figure 4.43.

A logical view of the ensemble learning method.

Figure 4.43. Full Alternative Text

2. By manipulating the input features. In this approach, a subset of input

features is chosen to form each training set. The subset can be either

chosen randomly or based on the recommendation of domain experts.

Some studies have shown that this approach works very well with data

sets that contain highly redundant features. Random forest, which is

described in Section 4.10.6, is an ensemble method that manipulates its

input features and uses decision trees as its base classifiers.

3. By manipulating the class labels. This method can be used when the

number of classes is sufficiently large. The training data is transformed

into a binary class problem by randomly partitioning the class labels into

two disjoint subsets, A0 and A1. Training examples whose class label

belongs to the subset A0 are assigned to class 0, while those that belong

to the subset A1 are assigned to class 1. The relabeled examples are then

used to train a base classifier. By repeating this process multiple times,

an ensemble of base classifiers is obtained. When a test example is

presented, each base classifier Ci is used to predict its class label. If the

test example is predicted as class 0, then all the classes that belong to A0

will receive a vote. Conversely, if it is predicted to be class 1, then all

the classes that belong to A1 will receive a vote. The votes are tallied

and the class that receives the highest vote is assigned to the test

example. An example of this approach is the error-correcting output

coding method described on page 331.

4. By manipulating the learning algorithm. Many learning algorithms can

be manipulated in such a way that applying the algorithm several times

on the same training data will result in the construction of different

classifiers. For example, an artificial neural network can change its

network topology or the initial weights of the links between neurons.

Similarly, an ensemble of decision trees can be constructed by injecting

randomness into the tree-growing procedure. For example, instead of

choosing the best splitting attribute at each node, we can randomly

choose one of the top *k *attributes for splitting.

The first three approaches are generic methods that are applicable to any

classifier, whereas the fourth approach depends on the type of classifier used.

The base classifiers for most of these approaches can be generated

sequentially (one after another) or in parallel (all at once). Once an ensemble

of classifiers has been learned, a test example x is classified by combining the

predictions made by the base classifiers Ci(x):

C*(x)=f(C1(x), C2(x), …, Ck(x)).

where *f *is the function that combines the ensemble responses. One simple

approach for obtaining C*(x) is to take a majority vote of the individual

predictions. An alternate approach is to take a *weighted *majority vote, where

the weight of a base classifier denotes its accuracy or relevance.

Ensemble methods show the most improvement when used with unstable

classifiers, i.e., base classifiers that are sensitive to minor perturbations in the

training set, because of high model complexity. Although unstable classifiers

may have a low *bias *in finding the optimal decision boundary, their

predictions have a high *variance *for minor changes in the training set or

model selection. This trade-off between bias and variance is discussed in

detail in the next section. By aggregating the responses of multiple unstable

classifiers, ensemble learning attempts to minimize their variance without

worsening their bias.

4.10.3 Bias-Variance Decomposition

Bias-variance decomposition is a formal method for analyzing the

generalization error of a predictive model. Although the analysis is slightly

different for classification than regression, we first discuss the basic intuition

of this decomposition by using an analogue of a regression problem.

Consider the illustrative task of reaching a target *y *by firing projectiles from a

starting position x, as shown in Figure 4.44. The target corresponds to the

desired output at a test instance, while the starting position corresponds to its

observed attributes. In this analogy, the projectile represents the model used

for predicting the target using the observed attributes. Let y^ denote the point

where the projectile hits the ground, which is analogous of the prediction of

the model.

Figure 4.44.

Bias-variance decomposition.

Figure 4.44. Full Alternative Text

Ideally, we would like our predictions to be as close to the true target as

possible. However, note that different trajectories of projectiles are possible

based on differences in the training data or in the approach used for model

selection. Hence, we can observe a variance in the predictions y^ over

different runs of projectile. Further, the target in our example is not fixed but

has some freedom to move around, resulting in a noise component in the true

target. This can be understood as the non-deterministic nature of the output

variable, where the same set of attributes can have different output values.

Let y^avg represent the *average *prediction of the projectile over multiple

runs, and yavg denote the average target value. The difference between y^avg

and yavg is known as the bias of the model.

In the context of classification, it can be shown that the generalization error

of a classification model *m *can be decomposed into terms involving the bias,

variance, and noise components of the model in the following way:

gen.error(m)=c1×noise+bias(m)+c2×variance(m)

where c1 and c2 are constants that depend on the characteristics of training

and test sets. Note that while the noise term is intrinsic to the target class, the

bias and variance terms depend on the choice of the classification model. The

bias of a model represents how close the average prediction of the model is to

the average target. Models that are able to learn complex decision boundaries,

e.g., models produced by *k*-nearest neighbor and multi-layer ANN, generally

show low bias. The variance of a model captures the stability of its

predictions in response to minor perturbations in the training set or the model

selection approach.

We can say that a model shows better generalization performance if it has a

lower bias and lower variance. However, if the complexity of a model is high

but the training size is small, we generally expect to see a lower bias but

*higher variance*, resulting in the phenomena of overfitting. This phenomena

is pictorially represented in Figure 4.45(a). On the other hand, an overly

simplistic model that suffers from underfitting may show a lower variance

but would suffer from a *high bias*, as shown in Figure 4.45(b). Hence, the

trade-off between bias and variance provides a useful way for interpreting the

effects of underfitting and overfitting on the generalization performance of a

model.

Figure 4.45.

Plots showing the behavior of two-dimensional solutions with

constant L2 and L1 norms.

Figure 4.45. Full Alternative Text

The bias-variance trade-off can be used to explain why ensemble learning

improves the generalization performance of unstable classifiers. If a base

classifier show low bias but high variance, it can become susceptible to

overfitting, as even a small change in the training set will result in different

predictions. However, by combining the responses of multiple base

classifiers, we can expect to reduce the overall variance. Hence, ensemble

learning methods show better performance primarily by lowering the

variance in the predictions, although they can even help in reducing the bias.

One of the simplest approaches for combining predictions and reducing their

variance is to compute their average. This forms the basis of the bagging

method, described in the following subsection.

4.10.4 Bagging

Bagging, which is also known as bootstrap aggregating, is a technique that

repeatedly samples (with replacement) from a data set according to a uniform

probability distribution. Each bootstrap sample has the same size as the

original data. Because the sampling is done with replacement, some instances

may appear several times in the same training set, while others may be

omitted from the training set. On average, a bootstrap sample Di contains

approximately 63% of the original training data because each sample has a

probability 1−(1−1/N)N of being selected in each Di. If *N *is sufficiently

large, this probability converges to 1−1/e≃0.632. The basic procedure for

bagging is summarized in Algorithm 4.5. After training the *k *classifiers, a test

instance is assigned to the class that receives the highest number of votes.

To illustrate how bagging works, consider the data set shown in Table 4.4.

Let *x *denote a one-dimensional attribute and *y *denote the class label. Suppose

we use only one-level binary decision trees, with a test condition x≤k, where

*k *is a split point chosen to minimize the entropy of the leaf nodes. Such a tree

is also known as a decision stump.

Table 4.4. Example of data set

used to construct an ensemble

of bagging classifiers.

*x *0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

*y *1 1 1 −1 −1 −1 −1 1 1 1

Without bagging, the best decision stump we can produce splits the instances

at either x≤0.35 or x≤0.75. Either way, the accuracy of the tree is at most

70%. Suppose we apply the bagging procedure on the data set using 10

bootstrap samples. The examples chosen for training in each bagging round

are shown in Figure 4.46. On the right-hand side of each table, we also

describe the decision stump being used in each round.

We classify the entire data set given in Table 4.4 by taking a majority vote

among the predictions made by each base classifier. The results of the

predictions are shown in Figure 4.47. Since the class labels are either −1 or

+1, taking the majority vote is equivalent to summing up the predicted values

of *y *and examining the sign of the resulting sum (refer to the second to last

row in Figure 4.47). Notice that the ensemble classifier perfectly classifies all

10 examples in the original data.

Algorithm 4.5 Bagging algorithm.

1: Let *k *be the number of bootstrap samples.

2: for *i *= 1 to *k *do

3: Create a bootstrap sample of size *N*, *D*i.

4: Train a base classifier *C*

i on the bootstrap sample *D**i*.

5: end for

6: C*(x)=argmaxy∑iδ(Ci(x)=y).

{δ(⋅)=1 if its argument is true and 0 otherwise.}

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Figure 4.46.

Example of bagging.

Figure 4.46. Full Alternative Text

The preceding example illustrates another advantage of using ensemble

methods in terms of enhancing the representation of the target function. Even

though each base classifier is a decision stump, combining the classifiers can

lead to a decision boundary that mimics a decision tree of depth 2.

Bagging improves generalization error by reducing the variance of the base

classifiers. The performance of bagging depends on the stability of the base

classifier. If a base classifier is unstable, bagging helps to reduce the errors

associated with random fluctuations in the training data. If a base classifier is

stable, i.e., robust to minor perturbations in the training set, then the error of

the ensemble is primarily caused by bias in the base classifier. In this

situation, bagging may not be able to improve the performance of the base

classifiers significantly. It may even degrade the classifier’s performance

because the effective size of each training set is about 37% smaller than the

original data.

Figure 4.47.

Example of combining classifiers constructed using the bagging

approach.

4.10.5 Boosting

Boosting is an iterative procedure used to adaptively change the distribution

of training examples for learning base classifiers so that they increasingly

focus on examples that are hard to classify. Unlike bagging, boosting assigns

a weight to each training example and may adaptively change the weight at

the end of each boosting round. The weights assigned to the training

examples can be used in the following ways:

1. They can be used to inform the sampling distribution used to draw a set

of bootstrap samples from the original data.

2. They can be used to learn a model that is biased toward examples with

higher weight.

This section describes an algorithm that uses weights of examples to

determine the sampling distribution of its training set. Initially, the examples

are assigned equal weights, 1/*N*, so that they are equally likely to be chosen

for training. A sample is drawn according to the sampling distribution of the

training examples to obtain a new training set. Next, a classifier is built from

the training set and used to classify all the examples in the original data. The

weights of the training examples are updated at the end of each boosting

round. Examples that are classified incorrectly will have their weights

increased, while those that are classified correctly will have their weights

decreased. This forces the classifier to focus on examples that are difficult to

classify in subsequent iterations.

The following table shows the examples chosen during each boosting round,

when applied to the data shown in Table 4.4.

Boosting (Round 1): 7 3 2 8 7 9 4 10 6 3

Boosting (Round 2): 5 4 9 4 2 5 1 7 4 2

Boosting (Round 3): 4 4 8 10 4 5 4 6 3 4

Initially, all the examples are assigned the same weights. However, some

examples may be chosen more than once, e.g., examples 3 and 7, because the

sampling is done with replacement. A classifier built from the data is then

used to classify all the examples. Suppose example 4 is difficult to classify.

The weight for this example will be increased in future iterations as it gets

misclassified repeatedly. Meanwhile, examples that were not chosen in the

previous round, e.g., examples 1 and 5, also have a better chance of being

selected in the next round since their predictions in the previous round were

likely to be wrong. As the boosting rounds proceed, examples that are the

hardest to classify tend to become even more prevalent. The final ensemble is

obtained by aggregating the base classifiers obtained from each boosting

round.

Over the years, several implementations of the boosting algorithm have been

developed. These algorithms differ in terms of (1) how the weights of the

training examples are updated at the end of each boosting round, and (2) how

the predictions made by each classifier are combined. An implementation

called AdaBoost is explored in the next section.

AdaBoost

Let {(xj, yj)|j=1, 2, …, N} denote a set of *N *training examples. In the

AdaBoost algorithm, the importance of a base classifier Ci depends on its

Figure 4.48.

Plot of α as a function of training error ∈.

error rate, which is defined as

∈i=1N[∑j=1Nwj I(Ci(xj)≠yj) ], (4.102)

where I(p)=1 if the predicate *p *is true, and 0 otherwise. The importance of a

classifier Ci is given by the following parameter,

αi=12ln (1−∈i∈i).

Note that αi has a large positive value if the error rate is close to 0 and a large

negative value if the error rate is close to 1, as shown in Figure 4.48.

The αi parameter is also used to update the weight of the training examples.

To illustrate, let wi(j) denote the weight assigned to example (xi, yi) during

the *j*th boosting round. The weight update mechanism for AdaBoost is given

by the equation:

wi(j+1)=wi(j)Zj×{e−αjif Cj(xi)=yi,eαjif Cj(xi)≠yi (4.103)

where Zj is the normalization factor used to ensure that ∑iwi(j+1)=1. The

weight update formula given in Equation 4.103 increases the weights of

incorrectly classified examples and decreases the weights of those classified

correctly.

Instead of using a majority voting scheme, the prediction made by each

classifier Cj is weighted according to αj. This approach allows AdaBoost to

penalize models that have poor accuracy, e.g., those generated at the earlier

boosting rounds. In addition, if any intermediate rounds produce an error rate

higher than 50%, the weights are reverted back to their original uniform

values, wi=1/N, and the resampling procedure is repeated. The AdaBoost

algorithm is summarized in Algorithm 4.6.

Algorithm 4.6 AdaBoost algorithm.

1: w = {*w*j = 1/*N *| *j *= 1, 2,…,*N*}. {Initialize the weights for all

2: Let *k *be the number of boosting rounds.

3: for *i *= 1 to *k *do

4: Create training set *D**i *by sampling (with replacement) from

5: Train a base classifier *C*

*i *on *D**i*.

6: Apply *C**i *to all examples in the original training set, *D*.

7: ∈i=1N[ ∑jwj δ(Ci(xj)≠yj) ] {Calculate the weighted error.}

8: if ∈i > 0.5 then

9: w = {*w**j *= 1/*N *| *j *= 1, 2,…,*N*}. {Reset the weights for all

10: Go back to Step 4.

11: end if

12: αi=12ln1−∈i∈i.

13: Update the weight of each example according to Equation 4.103

14: end for

15: C*(x)=argmaxy∑j=1Tαjδ(Cj(x)=y))..

Let us examine how the boosting approach works on the data set shown in

Table 4.4. Initially, all the examples have identical weights. After three

boosting rounds, the examples chosen for training are shown in Figure

4.49(a). The weights for each example are updated at the end of each

boosting round using Equation 4.103, as shown in Figure 4.50(b).

Without boosting, the accuracy of the decision stump is, at best, 70%. With

AdaBoost, the results of the predictions are given in Figure 4.50(b). The final

prediction of the ensemble classifier is obtained by taking a weighted average

of the predictions made by each base classifier, which is shown in the last

row of Figure 4.50(b). Notice that AdaBoost perfectly classifies all the

examples in the training data.

Figure 4.49.

Example of boosting.

Figure 4.49. Full Alternative Text

An important analytical result of boosting shows that the training error of the

ensemble is bounded by the following expression:

eensemble≤∏i[∈i(1−∈i) ], (4.104)

where ∈i is the error rate of each base classifier *i*. If the error rate of the base

classifier is less than 50%, we can write ∈i=0.5 −γi, where γi measures how

much better the classifier is than random guessing. The bound on the training

error of the ensemble becomes

eensemble≤∏i1−4γi2≤exp(−2∑iγi2). (4.105)

Hence, the training error of the ensemble decreases exponentially, which

leads to the fast convergence of the algorithm. By focusing on examples that

are difficult to classify by base classifiers, it is able to reduce the bias of the

final predictions along with the variance. AdaBoost has been shown to

provide significant improvements in performance over base classifiers on a

range of data sets. Nevertheless, because of its tendency to focus on training

examples that are wrongly classified, the boosting technique can be

susceptible to overfitting, resulting in poor generalization performance in

some scenarios.

Figure 4.50.

Example of combining classifiers constructed using the

AdaBoost approach.

4.10.6 Random Forests

Random forests attempt to improve the generalization performance by

constructing an ensemble of *decorrelated *decision trees. Random forests

build on the idea of bagging to use a different bootstrap sample of the

training data for learning decision trees. However, a key distinguishing

feature of random forests from bagging is that at every internal node of a tree,

the best splitting criterion is chosen among a small set of randomly selected

attributes. In this way, random forests construct ensembles of decision trees

by not only manipulating training instances (by using bootstrap samples

similar to bagging), but also the input attributes (by using different subsets of

attributes at every internal node).

Given a training set *D *consisting of *n *instances and *d *attributes, the basic

procedure of training a random forest classifier can be summarized using the

following steps:

1. Construct a bootstrap sample Di of the training set by randomly

sampling *n *instances (with replacement) from *D*.

2. Use Di to learn a decision tree Ti as follows. At every internal node of

Ti, randomly sample a set of *p *attributes and choose an attribute from

this subset that shows the maximum reduction in an impurity measure

for splitting. Repeat this procedure till every leaf is pure, i.e., containing

instances from the same class.

Once an ensemble of decision trees have been constructed, their average

prediction (majority vote) on a test instance is used as the final prediction of

the random forest. Note that the decision trees involved in a random forest

are *unpruned *trees, as they are allowed to grow to their largest possible size

till every leaf is pure. Hence, the base classifiers of random forest represent

unstable classifiers that have low bias but high variance, because of their

large size.

Another property of the base classifiers learned in random forests is the *lack*

*of correlation *among their model parameters and test predictions. This can be

attributed to the use of an independently sampled data set Di for learning

every decision tree Ti, similar to the bagging approach. However, random

forests have the additional advantage of choosing a splitting criterion at every

internal node using a different (and randomly selected) subset of attributes.

This property significantly helps in breaking the correlation structure, if any,

among the decision trees Ti.

To realize this, consider a training set involving a large number of attributes,

where only a small subset of attributes are strong predictors of the target

class, whereas other attributes are weak indicators. Given such a training set,

even if we consider different bootstrap samples Di for learning Ti, we would

mostly be choosing the same attributes for splitting at internal nodes, because

the weak attributes would be largely overlooked when compared with the

strong predictors. This can result in a considerable correlation among the

trees. However, if we restrict the choice of attributes at every internal node to

a random subset of attributes, we can ensure the selection of both strong and

weak predictors, thus promoting diversity among the trees. This principle is

utilized by random forests for creating decorrelated decision trees.

By aggregating the predictions of an ensemble of strong and decorrelated

decision trees, random forests are able to reduce the variance of the trees

without negatively impacting their low bias. This makes random forests quite

robust to overfitting. Additionally, because of their ability to consider only a

small subset of attributes at every internal node, random forests are

computationally fast and robust even in high-dimensional settings.

The number of attributes to be selected at every node, *p*, is a hyper-parameter

of the random forest classifier. A small value of *p *can reduce the correlation

among the classifiers but may also reduce their strength. A large value can

improve their strength but may result in correlated trees similar to bagging.

Although common suggestions for *p *in the literature include d and log2d+1, a

suitable value of *p *for a given training set can always be selected by tuning it

over a validation set, as described in the previous chapter. However, there is

an alternative way for selecting hyper-parameters in random forests, which

does not require using a separate validation set. It involves computing a

reliable estimate of the generalization error rate directly during training,

known as the out-of-bag (oob) error estimate. The oob estimate can be

computed for any generic ensemble learning method that builds independent

base classifiers using bootstrap samples of the training set, e.g., bagging and

random forests. The approach for computing oob estimate can be described

as follows.

Consider an ensemble learning method that uses an independent base

classifier Ti built on a bootstrap sample of the training set Di. Since every

training instance x will be used for training approximately 63% of base

classifiers, we can call x as an out-of-bag sample for the remaining 27% of

base classifiers that did not use it for training. If we use these remaining 27%

classifiers to make predictions on x, we can obtain the oob error on x by

taking their majority vote and comparing it with its class label. Note that the

oob error estimates the error of 27% classifiers on an instance that was not

used for training those classifiers. Hence, the oob error can be considered as a

reliable estimate of generalization error. By taking the average of oob errors

of all training instances, we can compute the overall oob error estimate. This

can be used as an alternative to the validation error rate for selecting hyperparameters. Hence, random forests do not need to use a separate partition of

the training set for validation, as it can simultaneously train the base

classifiers and compute generalization error estimates on the same data set.

Random forests have been empirically found to provide significant

improvements in generalization performance that are often comparable, if not

superior, to the improvements provided by the AdaBoost algorithm. Random

forests are also more robust to overfitting and run much faster than the

AdaBoost algorithm.

4.10.7 Empirical Comparison

among Ensemble Methods

Table 4.5 shows the empirical results obtained when comparing the

performance of a decision tree classifier against bagging, boosting, and

random forest. The base classifiers used in each ensemble method consist of

50 decision trees. The classification accuracies reported in this table are

obtained from tenfold cross-validation. Notice that the ensemble classifiers

generally outperform a single decision tree classifier on many of the data sets.

Table 4.5. Comparing the

accuracy of a decision tree

classifier against three

ensemble methods.

Data Set

Number of

(Attributes,

Classes,

Instances)

Decision

Tree (%) Bagging(%) Boosting(%) RF(%)

Anneal (39, 6,

898) 92.09 94.43 95.43 95.43

Australia (15, 2,

690) 85.51 87.10 85.22 85.80

Auto (26, 7,

205) 81.95 85.37 85.37 84.39

Breast (11, 2,

699) 95.14 96.42 97.28 96.14

Cleve (14, 2,

303) 76.24 81.52 82.18 82.18

Credit (16, 2,

690) 85.8 86.23 86.09 85.8

Diabetes (9, 2, 768) 72.40 76.30 73.18 75.13

German (21, 2,

1000) 70.90 73.40 73.00 74.5

Glass (10, 7,

214) 67.29 76.17 77.57 78.04

Heart (14, 2,

270) 80.00 81.48 80.74 83.33

Hepatitis (20, 2,

155) 81.94 81.29 83.87 83.23

Horse (23, 2,

368) 85.33 85.87 81.25 85.33

Ionosphere (35, 2,

351) 89.17 92.02 93.73 93.45

Iris (5, 3, 150) 94.67 94.67 94.00 93.33

Labor (17, 2, 57) 78.95 84.21 89.47 84.21

Led7 (8, 10,

3200) 73.34 73.66 73.34 73.06

Lymphography (19, 4, 77.03 79.05 85.14 82.43

148)

Pima (9, 2, 768) 74.35 76.69 73.44 77.60

Sonar (61, 2,

208) 78.85 78.85 84.62 85.58

Tic-tac-toe (10, 2,

958) 83.72 93.84 98.54 95.82

Vehicle (19, 4,

846) 71.04 74.11 78.25 74.94

Waveform (22, 3,

5000) 76.44 83.30 83.90 84.04

Wine (14, 3,

178) 94.38 96.07 97.75 97.75

Zoo (17, 7,

101) 93.07 93.07 95.05 97.03

4.11 Class Imbalance Problem

In many data sets there are a disproportionate number of instances that

belong to different classes, a property known as skew or class imbalance.For

example, consider a health-care application where diagnostic reports are used

to decide whether a person has a rare disease. Because of the infrequent

nature of the disease, we can expect to observe a smaller number of subjects

who are positively diagnosed. Similarly, in credit card fraud detection,

fraudulent transactions are greatly outnumbered by legitimate transactions.

The degree of imbalance between the classes varies across different

applications and even across different data sets from the same application.

For example, the risk for a rare disease may vary across different populations

of subjects depending on their dietary and lifestyle choices. However, despite

their infrequent occurrences, a correct classification of the rare class often has

greater value than a correct classification of the majority class. For example,

it may be more dangerous to ignore a patient suffering from a disease than to

misdiagnose a healthy person.

More generally, class imbalance poses two challenges for classification. First,

it can be difficult to find sufficiently many labeled samples of a rare class.

Note that many of the classification methods discussed so far work well only

when the training set has a balanced representation of both classes. Although

some classifiers are more effective at handling imbalance in the training data

than others, e.g., rule-based classifiers and *k*-NN, they are all impacted if the

minority class is not well-represented in the training set. In general, a

classifier trained over an imbalanced data set shows a bias toward improving

its performance over the majority class, which is often not the desired

behavior. As a result, many existing classification models, when trained on

an imbalanced data set, may not effectively detect instances of the rare class.

Second, accuracy, which is the traditional measure for evaluating

classification performance, is not well-suited for evaluating models in the

presence of class imbalance in the test data. For example, if 1% of the credit

card transactions are fraudulent, then a trivial model that predicts every

transaction as legitimate will have an accuracy of 99% even though it fails to

detect any of the fraudulent activities. Thus, there is a need to use alternative

evaluation metrics that are sensitive to the skew and can capture different

criteria of performance than accuracy.

In this section, we first present some of the generic methods for building

classifiers when there is class imbalance in the training set. We then discuss

methods for evaluating classification performance and adapting classification

decisions in the presence of a skewed test set. In the remainder of this

section, we will consider binary classification problems for simplicity, where

the minority class is referred as the positive (+) class while the majority class

is referred as the negative (−) class.

4.11.1 Building Classifiers with

Class Imbalance

There are two primary considerations for building classifiers in the presence

of class imbalance in the training set. First, we need to ensure that the

learning algorithm is trained over a data set that has adequate representation

of both the majority as well as the minority classes. Some common

approaches for ensuring this includes the methodologies of oversampling and

undersampling the training set. Second, having learned a classification

model, we need a way to adapt its classification decisions (and thus create an

appropriately tuned classifier) to best match the requirements of the

imbalanced test set. This is typically done by converting the outputs of the

classification model to real-valued scores, and then selecting a suitable

threshold on the classification score to match the needs of a test set. Both

these considerations are discussed in detail in the following.

Oversampling and Undersampling

The first step in learning with imbalanced data is to transform the training set

to a balanced training set, where both classes have nearly equal

representation. The balanced training set can then be used with any of the

existing classification techniques (without making any modifications in the

learning algorithm) to learn a model that gives equal emphasis to both

classes. In the following, we present some of the common techniques for

transforming an imbalanced training set to a balanced one.

A basic approach for creating balanced training sets is to generate a *sample *of

training instances where the rare class has adequate representation. There are

two types of sampling methods that can be used to enhance the representation

of the minority class: (a) undersampling, where the frequency of the

majority class is reduced to match the frequency of the minority class, and (b)

oversampling, where artificial examples of the minority class are created to

make them equal in proportion to the number of negative instances.

To illustrate undersampling, consider a training set that contains 100 positive

examples and 1000 negative examples. To overcome the skew among the

classes, we can select a random sample of 100 examples from the negative

class and use them with the 100 positive examples to create a balanced

training set. A classifier built over the resultant balanced set will then be

unbiased toward both classes. However, one limitation of undersampling is

that some of the useful negative examples (e.g., those closer to the actual

decision boundary) may not be chosen for training, therefore, resulting in an

inferior classification model. Another limitation is that the smaller sample of

100 negative instances may have a higher variance than the larger set of

1000.

Oversampling attempts to create a balanced training set by artificially

generating new positive examples. A simple approach for oversampling is to

duplicate every positive instance n−/n+ times, where n+ and n− are the

numbers of positive and negative training instances, respectively. Figure 4.51

illustrates the effect of oversampling on the learning of a decision boundary

using a classifier such as a decision tree. Without oversampling, only the

positive examples at the bottom right-hand side of Figure 4.51(a) are

classified correctly. The positive example in the middle of the diagram is

misclassified because there are not enough examples to justify the creation of

a new decision boundary to separate the positive and negative instances.

Oversampling provides the additional examples needed to ensure that the

decision boundary surrounding the positive example is not pruned, as

illustrated in Figure 4.51(b). Note that duplicating a positive instance is

analogous to doubling its weight during the training stage. Hence, the effect

of oversampling can be alternatively achieved by assigning higher weights to

positive instances than negative instances. This method of weighting

instances can be used with a number of classifiers such as logistic regression,

ANN, and SVM.

Figure 4.51.

Illustrating the effect of oversampling of the rare class.

One limitation of the duplication method for oversampling is that the

replicated positive examples have an artificially lower variance when

compared with their true distribution in the overall data. This can bias the

classifier to the specific distribution of training instances, which may not be

representative of the overall distribution of test instances, leading to poor

generalizability. To overcome this limitation, an alternative approach for

oversampling is to generate *synthetic *positive instances in the neighborhood

of existing positive instances. In this approach, called the Synthetic Minority

Oversampling Technique (SMOTE), we first determine the *k*-nearest positive

neighbors of every positive instance x, and then generate a synthetic positive

instance at some intermediate point along the line segment joining x to one of

its randomly chosen *k*-nearest neighbor, xk. This process is repeated until the

desired number of positive instances is reached. However, one limitation of

this approach is that it can only generate new positive instances in the convex

hull of the existing positive class. Hence, it does not help improve the

representation of the positive class outside the boundary of existing positive

instances. Despite their complementary strengths and weaknesses,

undersampling and oversampling provide useful directions for generating

balanced training sets in the presence of class imbalance.

Assigning Scores to Test Instances

If a classifier returns an ordinal score *s*(x)for every test instance x such that a

higher score denotes a greater likelihood of x belonging to the positive class,

then for every possible value of score threshold, sT, we can create a new

binary classifier where a test instance x is classified positive only if s(x)>sT.

Thus, every choice of sT can potentially lead to a different classifier, and we

are interested in finding the classifier that is best suited for our needs.

Ideally, we would like the classification score to vary monotonically with the

actual posterior probability of the positive class, i.e., if s(x1) and s(x2) are the

scores of any two instances, x1 and x2, then

s(x1)≥s(x2)⇒P(y=1|x1)≥P(y=1|x2). However, this is difficult to guarantee in

practice as the properties of the classification score depends on several

factors such as the complexity of the classification algorithm and the

representative power of the training set. In general, we can only expect the

classification score of a reasonable algorithm to be weakly related to the

actual posterior probability of the positive class, even though the relationship

may not be strictly monotonic. Most classifiers can be easily modified to

produce such a real valued score. For example, the signed distance of an

instance from the positive margin hyperplane of SVM can be used as a

classification score. As another example, test instances belonging to a leaf in

a decision tree can be assigned a score based on the fraction of training

instances labeled as positive in the leaf. Also, probabilistic classifiers such as

naïve Bayes, Bayesian networks, and logistic regression naturally output

estimates of posterior probabilities, P(y=1|x). Next, we discuss some

evaluation measures for assessing the goodness of a classifier in the presence

of class imbalance.

Table 4.6. A confusion matrix

for a binary classification

problem in which the classes

are not equally important.

Predicted Class

+ −

Actual

class

+ f++ (TP) f+− (FN)

− f−+ (FP) f−− (TN)

4.11.2 Evaluating Performance with

Class Imbalance

The most basic approach for representing a classifier’s performance on a test

set is to use a confusion matrix, as shown in Table 4.6. This table is

essentially the same as Table 3.4, which was introduced in the context of

evaluating classification performance in Section 3.2. A confusion matrix

summarizes the number of instances predicted correctly or incorrectly by a

classifier using the following four counts:

True positive (TP) or f++, which corresponds to the number of positive

examples correctly predicted by the classifier.

False positive (FP) or f−+ (also known as Type I error), which

corresponds to the number of negative examples wrongly predicted as

positive by the classifier.

False negative (FN) or f+− (also known as Type II error), which

corresponds to the number of positive examples wrongly predicted as

negative by the classifier.

True negative (TN) or f−−, which corresponds to the number of negative

examples correctly predicted by the classifier.

The confusion matrix provides a concise representation of classification

performance on a given test data set. However, it is often difficult to interpret

and compare the performance of classifiers using the four-dimensional

representations (corresponding to the four counts) provided by their

confusion matrices. Hence, the counts in the confusion matrix are often

summarized using a number of evaluation measures. Accuracy is an

example of one such measure that combines these four counts into a single

value, which is used extensively when classes are balanced. However, the

accuracy measure is not suitable for handling data sets with imbalanced class

distributions as it tends to favor classifiers that correctly classify the majority

class. In the following, we describe other possible measures that capture

different criteria of performance when working with imbalanced classes.

A basic evaluation measure is the true positive rate (TPR), which is defined

as the fraction of positive test instances correctly predicted by the classifier:

TPR=TPTP+FN.

In the medical community, TPR is also known as sensitivity, while in the

information retrieval literature, it is also called recall (r). A classifier with a

high TPR has a high chance of correctly identifying the positive instances of

the data.

Analogously to TPR, the true negative rate (TNR) (also known as

specificity) is defined as the fraction of negative test instances correctly

predicted by the classifier, i.e.,

TNR=TNFP+TN.

A high TNR value signifies that the classifier correctly classifies any

randomly chosen negative instance in the test set. A commonly used

evaluation measure that is closely related to TNR is the false positive rate

(FPR), which is defined as 1−TNR.

FPR=FPFP+TN.

Similarly, we can define false negative rate (FNR) as 1−TPR.

FNR=FNFN+TP.

Note that the evaluation measures defined above do not take into account the

skew among the classes, which can be formally defined as α=P/(P+N), where

*P *and *N *denote the number of actual positives and actual negatives,

respectively. As a result, changing the relative numbers of *P *and *N *will have

no effect on TPR, TNR, FPR, or FNR, since they depend only on the fraction

of correct classifications for every class, independently of the other class.

Furthermore, knowing the values of TPR and TNR (and consequently FNR

and FPR) does not by itself help us uniquely determine all four entries of the

confusion matrix. However, together with information about the skew factor,

α, and the total number of instances, *N*, we can compute the entire confusion

matrix using TPR and TNR, as shown in Table 4.7.

Table 4.7. Entries of the

confusion matrix in terms of

the TPR, TNR, skew, α, and

total number of instances, *N*.

Predicted + Predicted −

Actual + TPR×α×N (1−TPR)×α×N α×N

Actual − (1−TNR)×(1−α)×N TNR×(1−α)×N (1−α)×N

N

An evaluation measure that is sensitive to the skew is precision, which can

be defined as the fraction of correct predictions of the positive class over the

total number of positive predictions, i.e.,

Precision, p=TPTP+FP.

Precision is also referred as the positive predicted value (PPV). A classifier

that has a high precision is likely to have most of its positive predictions

correct. Precision is a useful measure for highly skewed test sets where the

positive predictions, even though small in numbers, are required to be mostly

correct. A measure that is closely related to precision is the false discovery

rate (FDR), which can be defined as 1−p.

FDR=FPTP+FP.

Although both FDR and FPR focus on FP, they are designed to capture

different evaluation objectives and thus can take quite contrasting values,

especially in the presence of class imbalance. To illustrate this, consider a

classifier with the following confusion matrix.

Predicted Class

+ −

Actual

Class

+ 100 0

− 100 900

Since half of the positive predictions made by the classifier are incorrect, it

has a FDR value of 100/(100+100)=0.5. However, its FPR is equal to

100/(100+900)=0.1, which is quite low. This example shows that in the

presence of high skew (i.e., very small value of α), even a small FPR can

result in high FDR. See Section 10.6 for further discussion of this issue.

Note that the evaluation measures defined above provide an incomplete

representation of performance, because they either only capture the effect of

false positives (e.g., FPR and precision) or the effect of false negatives (e.g.,

TPR or recall), but not both. Hence, if we optimize only one of these

evaluation measures, we may end up with a classifier that shows low FN but

high FP, or vice-versa. For example, a classifier that declares every instance

to be positive will have a perfect recall, but high FPR and very poor

precision. On the other hand, a classifier that is very conservative in

classifying an instance as positive (to reduce FP) may end up having high

precision but very poor recall. We thus need evaluation measures that

account for both types of misclassifications, FP and FN. Some examples of

such evaluation measures are summarized by the following definitions.

Positive Likelihood Ratio=TPRFPR.F1 measure=2rpr+p=2×TP2×TP+FP+FN.

(TP+FN).

While some of these evaluation measures are invariant to the skew (e.g., the

positive likelihood ratio), others (e.g., precision and the F1 measure) are

sensitive to skew. Further, different evaluation measures capture the effects

of different types of misclassification errors in various ways. For example,

the F1 measure represents a harmonic mean between recall and precision, i.e.,

F1=21r+1p.

Because the harmonic mean of two numbers tends to be closer to the smaller

of the two numbers, a high value of F1-measure ensures that both precision

and recall are reasonably high. Similarly, the *G *measure represents the

geometric mean between recall and precision. A comparison among

harmonic, geometric, and arithmetic means is given in the next example.

Example 4.9.

Consider two positive numbers a=1 and b=5. Their arithmetic mean is μa=

(a+b)/2=3 and their geometric mean is μg=ab=2.236. Their harmonic mean is

μh=(2×1×5)/6=1.667, which is closer to the smaller value between *a *and *b*

than the arithmetic and geometric means.

A generic extension of the F1 measure is the Fβ measure, which can be

defined as follows.

Fβ=(β2+1)rpr+β2p=(β2+1)×TP(β2+1)TP+β2FP+FN. (4.106)

Both precision and recall can be viewed as special cases of Fβ by setting β=0

and β=∞, respectively. Low values of β make Fβ closer to precision, and high

values make it closer to recall.

A more general measure that captures Fβ as well as accuracy is the weighted

accuracy measure, which is defined by the following equation:

Weighted accuracy=w1TP+w4TNw1TP+w2FP+w3FN+w4TN. (4.107)

The relationship between weighted accuracy and other performance measures

is summarized in the following table:

Measure w1 w2 w3 w4

Recall 1 1 0 0

Precision 1 0 1 0

Fβ β2+1 β2 1 0

Accuracy 1 1 1 1

4.11.3 Finding an Optimal Score

Threshold

Given a suitably chosen evaluation measure *E *and a distribution of

classification scores, s(x), on a validation set, we can obtain the optimal score

threshold s* on the validation set using the following approach:

1. Sort the scores in increasing order of their values.

2. For every unique value of score, *s*, consider the classification model that

assigns an instance x as positive only if s(x)>s. Let *E*(*s*) denote the

performance of this model on the validation set.

3. Find s* that maximizes the evaluation measure, *E*(*s*).

s*=argmaxs E(s).

Note that s* can be treated as a hyper-parameter of the classification

algorithm that is learned during model selection. Using s*, we can assign a

positive label to a future test instance x only if s(x)>s*. If the evaluation

measure *E *is skew invariant (e.g., Positive Likelihood Ratio), then we can

select s* without knowing the skew of the test set, and the resultant classifier

formed using s* can be expected to show optimal performance on the test set

(with respect to the evaluation measure *E*). On the other hand, if *E *is sensitive

to the skew (e.g., precision or F1-measure), then we need to ensure that the

skew of the validation set used for selecting s* is similar to that of the test set,

so that the classifier formed using s* shows optimal test performance with

respect to *E*. Alternatively, given an estimate of the skew of the test data, α,

we can use it along with the TPR and TNR on the validation set to estimate

all entries of the confusion matrix (see Table 4.7), and thus the estimate of

any evaluation measure *E *on the test set. The score threshold s* selected

using this estimate of *E *can then be expected to produce optimal test

performance with respect to *E*. Furthermore, the methodology of selecting s*

on the validation set can help in comparing the test performance of different

classification algorithms, by using the optimal values of s* for each

algorithm.

4.11.4 Aggregate Evaluation of

Performance

Although the above approach helps in finding a score threshold s* that

provides optimal performance with respect to a desired evaluation measure

and a certain amount of skew, α, sometimes we are interested in evaluating

the performance of a classifier on a number of possible score thresholds, each

corresponding to a different choice of evaluation measure and skew value.

Assessing the performance of a classifier over a range of score thresholds is

called aggregate evaluation of performance. In this style of analysis, the

emphasis is not on evaluating the performance of a single classifier

corresponding to the optimal score threshold, but to assess the general quality

of *ranking *produced by the classification scores on the test set. In general,

this helps in obtaining robust estimates of classification performance that are

not sensitive to specific choices of score thresholds.

ROC Curve

One of the widely-used tools for aggregate evaluation is the receiver

operating characteristic (ROC) curve. An ROC curve is a graphical

approach for displaying the trade-off between TPR and FPR of a classifier,

over varying score thresholds. In an ROC curve, the TPR is plotted along the

*y*-axis and the FPR is shown on the *x*-axis. Each point along the curve

corresponds to a classification model generated by placing a threshold on the

test scores produced by the classifier. The following procedure describes the

generic approach for computing an ROC curve:

1. Sort the test instances in increasing order of their scores.

2. Select the lowest ranked test instance (i.e., the instance with lowest

score). Assign the selected instance and those ranked above it to the

positive class. This approach is equivalent to classifying all the test

instances as positive class. Because all the positive examples are

classified correctly and the negative examples are misclassified,

TPR=FPR=1.

3. Select the next test instance from the sorted list. Classify the selected

instance and those ranked above it as positive, while those ranked below

it as negative. Update the counts of *TP *and *FP *by examining the actual

class label of the selected instance. If this instance belongs to the

positive class, the *TP *count is decremented and the *FP *count remains

the same as before. If the instance belongs to the negative class, the *FP*

count is decremented and *TP *count remains the same as before.

4. Repeat Step 3 and update the *TP *and *FP *counts accordingly until the

highest ranked test instance is selected. At this final threshold,

TPR=FPR=0, as all instances are labeled as negative.

5. Plot the TPR against FPR of the classifier.

Example 4.10. [Generating ROC

Curve]

Figure 4.52 shows an example of how to compute the TPR and FPR values

for every choice of score threshold. There are five positive examples and five

negative examples in the test set. The class labels of the test instances are

shown in the first row of the table, while the second row corresponds to the

sorted score values for each instance. The next six rows contain the counts of

*TP *, *FP *, *TN*, and *FN*, along with their corresponding TPR and FPR. The

table is then filled from left to right. Initially, all the instances are predicted to

be positive. Thus, TP=FP=5 and TPR=FPR=1. Next, we assign the test

instance with the lowest score as the negative class. Because the selected

instance is actually a positive example, the *TP *count decreases from 5 to 4

and the *FP *count is the same as before. The FPR and TPR are updated

accordingly. This process is repeated until we reach the end of the list, where

TPR=0 and FPR=0. The ROC curve for this example is shown in Figure 4.53.

Figure 4.52.

Computing the TPR and FPR at every score threshold.

Figure 4.53.

ROC curve for the data shown in Figure 4.52.

Note that in an ROC curve, the TPR monotonically increases with FPR,

because the inclusion of a test instance in the set of predicted positives can

either increase the TPR or the FPR. The ROC curve thus has a staircase

pattern. Furthermore, there are several critical points along an ROC curve

that have well-known interpretations:

(TPR=0, FPR=0): Model predicts every instance to be a negative class.

(TPR=1, FPR=1): Model predicts every instance to be a positive class.

(TPR=1, FPR=0): The *perfect *model with zero misclassifications.

A good classification model should be located as close as possible to the

upper left corner of the diagram, while a model that makes random guesses

should reside along the main diagonal, connecting the points

(TPR=0, FPR=0) and (TPR=1, FPR=1). Random guessing means that an

instance is classified as a positive class with a fixed probability *p*, irrespective

of its attribute set. For example, consider a data set that contains n+ positive

instances and n− negative instances. The random classifier is expected to

correctly classify pn+ of the positive instances and to misclassify pn− of the

negative instances. Therefore, the TPR of the classifier is (pn+)/n+=p, while

its FPR is (pn−)/p=p. Hence, this random classifier will reside at the point (*p*,

*p*) in the ROC curve along the main diagonal.

Figure 4.54.

ROC curves for two different classifiers.

Figure 4.54. Full Alternative Text

Since every point on the ROC curve represents the performance of a classifier

generated using a particular score threshold, they can be viewed as different

operating points of the classifier. One may choose one of these operating

points depending on the requirements of the application. Hence, an ROC

curve facilitates the comparison of classifiers over a range of operating

points. For example, Figure 4.54 compares the ROC curves of two classifiers,

M1 and M2, generated by varying the score thresholds. We can see that M1

is better than M2 when FPR is less than 0.36, as M1 shows better TPR than

M2 for this range of operating points. On the other hand, M2 is superior

when FPR is greater than 0.36, since the TPR of M2 is higher than that of M1

for this range. Clearly, neither of the two classifiers dominates (is strictly

better than) the other, i.e., shows higher values of TPR and lower values of

FPR over all operating points.

To summarize the aggregate behavior across all operating points, one of the

commonly used measures is the area under the ROC curve (AUC). If the

classifier is perfect, then its area under the ROC curve will be equal 1. If the

algorithm simply performs random guessing, then its area under the ROC

curve will be equal to 0.5.

Although the AUC provides a useful summary of aggregate performance,

there are certain caveats in using the AUC for comparing classifiers. First,

even if the AUC of algorithm *A *is higher than the AUC of another algorithm

*B*, this does not mean that algorithm *A *is always better than *B*, i.e., the ROC

curve of *A *dominates that of *B *across all operating points. For example, even

though M1 shows a slightly lower AUC than M2 in Figure 4.54, we can see

that both M1 and M2 are useful over different ranges of operating points and

none of them are strictly better than the other across all possible operating

points. Hence, we cannot use the AUC to determine which algorithm is

better, unless we know that the ROC curve of one of the algorithms

dominates the other.

Second, although the AUC summarizes the aggregate performance over all

operating points, we are often interested in only a small range of operating

points in most applications. For example, even though M1 shows slightly

lower AUC than M2, it shows higher TPR values than M2 for small FPR

values (smaller than 0.36). In the presence of class imbalance, the behavior of

an algorithm over small FPR values (also termed as early retrieval) is often

more meaningful for comparison than the performance over all FPR values.

This is because, in many applications, it is important to assess the TPR

achieved by a classifier in the first few instances with highest scores, without

incurring a large FPR. Hence, in Figure 4.54, due to the high TPR values of

M1 during early retrieval (FPR<0.36), we may prefer M1 over M2 for

imbalanced test sets, despite the lower AUC of M1. Hence, care must be

taken while comparing the AUC values of different classifiers, usually by

visualizing their ROC curves rather than just reporting their AUC.

A key characteristic of ROC curves is that they are agnostic to the skew in

the test set, because both the evaluation measures used in constructing ROC

curves (TPR and FPR) are invariant to class imbalance. Hence, ROC curves

are not suitable for measuring the impact of skew on classification

performance. In particular, we will obtain the same ROC curve for two test

data sets that have very different skew.

Figure 4.55.

PR curves for two different classifiers.

Figure 4.55. Full Alternative Text

Precision-Recall Curve

An alternate tool for aggregate evaluation is the precision recall curve (PR

curve). The PR curve plots the precision and recall values of a classifier on

the *y *and *x *axes respectively, by varying the threshold on the test scores.

Figure 4.55 shows an example of PR curves for two hypothetical classifiers,

M1 and M2. The approach for generating a PR curve is similar to the

approach described above for generating an ROC curve. However, there are

some key distinguishing features in the PR curve:

1. PR curves are sensitive to the skew factor α=P/(P+N), and different PR

curves are generated for different values of α.

2. When the score threshold is lowest (every instance is labeled as

positive), the precision is equal to α while recall is 1. As we increase the

score threshold, the number of predicted positives can stay the same or

decrease. Hence, the recall monotonically declines as the score threshold

increases. In general, the precision may increase or decrease for the

same value of recall, upon addition of an instance into the set of

predicted positives. For example, if the *k*th ranked instance belongs to

the negative class, then including it will result in a drop in the precision

without affecting the recall. The precision may improve at the next step,

which adds the (k+1)th ranked instance, if this instance belongs to the

positive class. Hence, the PR curve is not a smooth, monotonically

increasing curve like the ROC curve, and generally has a zigzag pattern.

This pattern is more prominent in the left part of the curve, where even a

small change in the number of false positives can cause a large change

in precision.

3. As, as we increase the imbalance among the classes (reduce the value of

α), the rightmost points of all PR curves will move downwards. At and

near the leftmost point on the PR curve (corresponding to larger values

of score threshold), the recall is close to zero, while the precision is

equal to the fraction of positives in the top ranked instances of the

algorithm. Hence, different classifiers can have different values of

precision at the leftmost points of the PR curve. Also, if the

classification score of an algorithm monotonically varies with the

posterior probability of the positive class, we can expect the PR curve to

gradually decrease from a high value of precision on the leftmost point

to a constant value of α at the rightmost point, albeit with some ups and

downs. This can be observed in the PR curve of algorithm M1 in Figure

4.55, which starts from a higher value of precision on the left that

gradually decreases as we move towards the right. On the other hand,

the PR curve of algorithm M2 starts from a lower value of precision on

the left and shows more drastic ups and downs as we move right,

suggesting that the classification score of M2 shows a weaker

monotonic relationship with the posterior probability of the positive

class.

4. A random classifier that assigns an instance to be positive with a fixed

probability *p *has a precision of α and a recall of *p*. Hence, a classifier

that performs random guessing has a horizontal PR curve with y=α, as

shown using a dashed line in Figure 4.55. Note that the random baseline

in PR curves depends on the skew in the test set, in contrast to the fixed

main diagonal of ROC curves that represents random classifiers.

5. Note that the precision of an algorithm is impacted more strongly by

false positives in the top ranked test instances than the FPR of the

algorithm. For this reason, the PR curve generally helps to magnify the

differences between classifiers in the left portion of the PR curve.

Hence, in the presence of class imbalance in the test data, analyzing the

PR curves generally provides a deeper insight into the performance of

classifiers than the ROC curves, especially in the early retrieval range of

operating points.

6. The classifier corresponding to (precision=1, recall=1) represents the

perfect classifier. Similar to AUC, we can also compute the area under

the PR curve of an algorithm, known as AUC-PR. The AUC-PR of a

random classifier is equal to α, while that of a perfect algorithm is equal

to 1. Note that AUC-PR varies with changing skew in the test set, in

contrast to the area under the ROC curve, which is insensitive to the

skew. The AUC-PR helps in accentuating the differences between

classification algorithms in the early retrieval range of operating points.

Hence, it is more suited for evaluating classification performance in the

presence of class imbalance than the area under the ROC curve.

However, similar to ROC curves, a higher value of AUC-PR does not

guarantee the superiority of a classification algorithm over another. This

is because the PR curves of two algorithms can easily cross each other,

such that they both show better performances in different ranges of

operating points. Hence, it is important to visualize the PR curves before

comparing their AUC-PR values, in order to ensure a meaningful

evaluation.

4.12 Multiclass Problem

Some of the classification techniques described in this chapter are originally

designed for binary classification problems. Yet there are many real-world

problems, such as character recognition, face identification, and text

classification, where the input data is divided into more than two categories.

This section presents several approaches for extending the binary classifiers

to handle multiclass problems. To illustrate these approaches, let Y=

{y1, y2, … ,yK} be the set of classes of the input data.

The first approach decomposes the multiclass problem into *K *binary

problems. For each class yi∈Y, a binary problem is created where all

instances that belong to yi are considered positive examples, while the

remaining instances are considered negative examples. A binary classifier is

then constructed to separate instances of class yi from the rest of the classes.

This is known as the one-against-rest (1-r) approach.

The second approach, which is known as the one-against-one (1-1) approach,

constructs K(K −1)/2 binary classifiers, where each classifier is used to

distinguish between a pair of classes, (yi, yj). Instances that do not belong to

either yi or yj are ignored when constructing the binary classifier for (yi, yj).

In both 1-r and 1-1 approaches, a test instance is classified by combining the

predictions made by the binary classifiers. A voting scheme is typically

employed to combine the predictions, where the class that receives the

highest number of votes is assigned to the test instance. In the 1-r approach, if

an instance is classified as negative, then all classes except for the positive

class receive a vote. This approach, however, may lead to ties among the

different classes. Another possibility is to transform the outputs of the binary

classifiers into probability estimates and then assign the test instance to the

class that has the highest probability.

Example 4.11.

Consider a multiclass problem where Y={y1, y2, y3, y4}. Suppose a test

instance is classified as (+, −, −, −) according to the 1-r approach. In other

words, it is classified as positive when y1 is used as the positive class and

negative when y2, y3, and y4 are used as the positive class. Using a simple

majority vote, notice that y1 receives the highest number of votes, which is

four, while the remaining classes receive only three votes. The test instance is

therefore classified as y1.

Example 4.12.

Suppose the test instance is classified using the 1-1 approach as follows:

Binary pair of

classes +:y1−:y2 +:y1−:y3 +:y1−:y4 +:y2−:y3 +:y2−:y4 +:y3−:y4

Classification + + − + − +

The first two rows in this table correspond to the pair of classes (yi, yj)

chosen to build the classifier and the last row represents the predicted class

for the test instance. After combining the predictions, y1 and y4 each receive

two votes, while y2 and y3 each receives only one vote. The test instance is

therefore classified as either y1 or y4, depending on the tie-breaking

procedure.

Error-Correcting Output Coding

A potential problem with the previous two approaches is that they may be

sensitive to binary classification errors. For the 1-r approach given in

Example

4.12, if at least of one of the binary classifiers makes a mistake in its

prediction, then the classifier may end up declaring a tie between classes or

making a wrong prediction. For example, suppose the test instance is

classified as (+, −, +, −) due to misclassification by the third classifier. In this

case, it will be difficult to tell whether the instance should be classified as y1

or y3, unless the probability associated with each class prediction is taken

into account.

The error-correcting output coding (ECOC) method provides a more robust

way for handling multiclass problems. The method is inspired by an

information-theoretic approach for sending messages across noisy channels.

The idea behind this approach is to add redundancy into the transmitted

message by means of a codeword, so that the receiver may detect errors in the

received message and perhaps recover the original message if the number of

errors is small.

For multiclass learning, each class yi is represented by a unique bit string of

length *n *known as its codeword. We then train *n *binary classifiers to predict

each bit of the codeword string. The predicted class of a test instance is given

by the codeword whose Hamming distance is closest to the codeword

produced by the binary classifiers. Recall that the Hamming distance between

a pair of bit strings is given by the number of bits that differ.

Example 4.13.

Consider a multiclass problem where Y={y1, y2, y3, y4}. Suppose we

encode the classes using the following seven bit codewords:

Class Codeword

y1 1 1 1 1 1 1 1

y2 0 0 0 0 1 1 1

y3 0 0 1 1 0 0 1

y4 0 1 0 1 0 1 0

Each bit of the codeword is used to train a binary classifier. If a test instance

is classified as (0,1,1,1,1,1,1) by the binary classifiers, then the Hamming

distance between the codeword and y1 is 1, while the Hamming distance to

the remaining classes is 3. The test instance is therefore classified as y1.

An interesting property of an error-correcting code is that if the minimum

Hamming distance between any pair of codewords is *d*, then any ⌊ (d−1)/2) ⌋

errors in the output code can be corrected using its nearest codeword. In

Example 4.13, because the minimum Hamming distance between any pair of

codewords is 4, the classifier may tolerate errors made by one of the seven

binary classifiers. If there is more than one classifier that makes a mistake,

then the classifier may not be able to compensate for the error.

An important issue is how to design the appropriate set of codewords for

different classes. From coding theory, a vast number of algorithms have been

developed for generating *n*-bit codewords with bounded Hamming distance.

However, the discussion of these algorithms is beyond the scope of this book.

It is worthwhile mentioning that there is a significant difference between the

design of error-correcting codes for communication tasks compared to those

used for multiclass learning. For communication, the codewords should

maximize the Hamming distance between the rows so that error correction

can be performed. Multiclass learning, however, requires that both the rowwise and column-wise distances of the codewords must be well separated. A

larger column-wise distance ensures that the binary classifiers are mutually

independent, which is an important requirement for ensemble learning

methods.

4.13 Bibliographic Notes

Mitchell [278] provides excellent coverage on many classification techniques

from a machine learning perspective. Extensive coverage on classification

can also be found in Aggarwal [195], Duda et al. [229], Webb [307],

Fukunaga [237], Bishop [204], Hastie et al. [249], Cherkassky and Mulier

[215], Witten and Frank [310], Hand et al. [247], Han and Kamber [244], and

Dunham [230].

Direct methods for rule-based classifiers typically employ the sequential

covering scheme for inducing classification rules. Holte’s 1R [255] is the

simplest form of a rule-based classifier because its rule set contains only a

single rule. Despite its simplicity, Holte found that for some data sets that

exhibit a strong one-to-one relationship between the attributes and the class

label, 1R performs just as well as other classifiers. Other examples of rulebased classifiers include IREP [234], RIPPER [218], CN2 [216, 217], AQ

[276], RISE [224], and ITRULE [296]. Table 4.8 shows a comparison of the

characteristics of four of these classifiers.

Table 4.8. Comparison of

various rule-based classifiers.

RIPPER CN2

(unordered)

CN2

(ordered) AQR

Rulegrowing

strategy

General-tospecific

General-tospecific

General-tospecific

General-tospecific

(seeded by

a positive

example)

Evaluation

metric

FOIL’s Info

gain Laplace

Entropy and

likelihood

ratio

Number of

true

positives

Stopping

condition

forrulegrowing

All examples

belong to the

same class

No

performance

gain

No

performance

gain

Rules cover

only

positive

class

Rule pruning Reduced error

pruning None None None

Instance

elimination

Positive and

negative

Positive

only

Positive

only

Positive

and

negative

Stopping

condition for

adding rules

Error>50%

orbased on

MDL

No

performance

gain

No

performance

gain

All positive

examples

are covered

Rule setp

runing

Replace or

modify rules

Statistical

tests

None None

Search

strategy

Greedy Beam search Beam

search

Beam

search

For rule-based classifiers, the rule antecedent can be generalized to include

any propositional or first-order logical expression (e.g., Horn clauses).

Readers who are interested in first-order logic rule-based classifiers may refer

to references such as [278] or the vast literature on inductive logic

programming [279]. Quinlan [287] proposed the C4.5rules algorithm for

extracting classification rules from decision trees. An indirect method for

extracting rules from artificial neural networks was given by Andrews et al.

in [198].

Cover and Hart [220] presented an overview of the nearest neighbor

classification method from a Bayesian perspective. Aha provided both

theoretical and empirical evaluations for instance-based methods in [196].

PEBLS, which was developed by Cost and Salzberg [219], is a nearest

neighbor classifier that can handle data sets containing nominal attributes.

Each training example in PEBLS is also assigned a weight factor that

depends on the number of times the example helps make a correct prediction.

Han et al. [243] developed a weight-adjusted nearest neighbor algorithm, in

which the feature weights are learned using a greedy, hill-climbing

optimization algorithm. A more recent survey of *k*-nearest neighbor

classification is given by Steinbach and Tan [298].

Naïve Bayes classifiers have been investigated by many authors, including

Langley et al. [267], Ramoni and Sebastiani [288], Lewis [270], and

Domingos and Pazzani [227]. Although the independence assumption used in

naïve Bayes classifiers may seem rather unrealistic, the method has worked

surprisingly well for applications such as text classification. Bayesian

networks provide a more flexible approach by allowing some of the attributes

to be interdependent. An excellent tutorial on Bayesian networks is given by

Heckerman in [252] and Jensen in [258]. Bayesian networks belong to a

broader class of models known as probabilistic graphical models. A formal

introduction to the relationships between graphs and probabilities was

presented in Pearl [283]. Other great resources on probabilistic graphical

models include books by Bishop [205], and Jordan [259]. Detailed

discussions of concepts such as *d*-separation and Markov blankets are

provided in Geiger et al. [238] and Russell and Norvig [291].

Generalized linear models (GLM) are a rich class of regression models that

have been extensively studied in the statistical literature. They were

formulated by Nelder and Wedderburn in 1972 [280] to unify a number of

regression models such as linear regression, logistic regression, and Poisson

regression, which share some similarities in their formulations. An extensive

discussion of GLMs is provided in the book by McCullagh and Nelder [274].

Artificial neural networks (ANN) have witnessed a long and winding history

of developments, involving multiple phases of stagnation and resurgence.

The idea of a mathematical model of a neural network was first introduced in

1943 by McCulloch and Pitts [275]. This led to a series of computational

machines to simulate a neural network based on the theory of neural

plasticity [289]. The perceptron, which is the simplest prototype of modern

ANNs, was developed by Rosenblatt in 1958 [290]. The perceptron uses a

single layer of processing units that can perform basic mathematical

operations such as addition and multiplication. However, the perceptron can

only learn linear decision boundaries and is guaranteed to converge only

when the classes are linearly separable. Despite the interest in learning multilayer networks to overcome the limitations of perceptron, progress in this

area remain halted until the invention of the backpropagation algorithm by

Werbos in 1974 [309], which allowed for the quick training of multi-layer

ANNs using the gradient descent method. This led to an upsurge of interest in

the artificial intelligence (AI) community to develop multi-layer ANN

models, a trend that continued for more than a decade. Historically, ANNs

mark a paradigm shift in AI from approaches based on expert systems (where

knowledge is encoded using if-then rules) to machine learning approaches

(where the knowledge is encoded in the parameters of a computational

model). However, there were still a number of algorithmic and computational

challenges in learning large ANN models, which remained unresolved for a

long time. This hindered the development of ANN models to the scale

necessary for solving real-world problems. Slowly, ANNs started getting

outpaced by other classification models such as support vector machines,

which provided better performance as well as theoretical guarantees of

convergence and optimality. It is only recently that the challenges in learning

*deep *neural networks have been circumvented, owing to better computational

resources and a number of algorithmic improvements in ANNs since 2006.

This re-emergence of ANN has been dubbed as “deep learning,” which has

often outperformed existing classification models and gained wide-spread

popularity.

Deep learning is a rapidly evolving area of research with a number of

potentially impactful contributions being made every year. Some of the

landmark advancements in deep learning include the use of large-scale

restricted Boltzmann machines for learning generative models of data [201,

253], the use of autoencoders and its variants (denoising autoencoders) for

learning robust feature representations [199, 305, 306], and sophistical

architectures to promote sharing of parameters across nodes such as

convolutional neural networks for images [265, 268] and recurrent neural

networks for sequences [241, 242, 277]. Other major improvements include

the approach of unsupervised pretraining for initializing ANN models [232],

the dropout technique for regularization [254, 297], batch normalization for

fast learning of ANN parameters [256], and maxout networks for effective

usage of the dropout technique [240]. Even though the discussions in this

chapter on learning ANN models were centered around the gradient descent

method, most of the modern ANN models involving a large number of

hidden layers are trained using the stochastic gradient descent method since it

is highly scalable [207]. An extensive survey of deep learning approaches has

been presented in review articles by Bengio [200], LeCun et al. [269], and

Schmidhuber [293]. An excellent summary of deep learning approaches can

also be obtained from recent books by Goodfellow et al. [239] and Nielsen

[281].

Vapnik [303, 304] has written two authoritative books on Support Vector

Machines (SVM). Other useful resources on SVM and kernel methods

include the books by Cristianini and Shawe-Taylor [221] and Sch*ö*lkopf and

Smola [294]. There are several survey articles on SVM, including those

written by Burges [212], Bennet et al. [202], Hearst [251], and Mangasarian

[272]. SVM can also be viewed as an L2 norm regularizer of the hinge loss

function, as described in detail by Hastie et al. [249]. The L1 norm

regularizer of the square loss function can be obtained using the least

absolute shrinkage and selection operator (Lasso), which was introduced by

Tibshirani in 1996 [301]. The Lasso has several interesting properties such as

the ability to simultaneously perform feature selection as well as

regularization, so that only a subset of features are selected in the final model.

An excellent review of Lasso can be obtained from a book by Hastie et al.

[250].

A survey of ensemble methods in machine learning was given by Diet-terich

[222]. The bagging method was proposed by Breiman [209]. Freund and

Schapire [236] developed the AdaBoost algorithm. Arcing, which stands for

adaptive resampling and combining, is a variant of the boosting algorithm

proposed by Breiman [210]. It uses the non-uniform weights assigned to

training examples to resample the data for building an ensemble of training

sets. Unlike AdaBoost, the votes of the base classifiers are not weighted

when determining the class label of test examples. The random forest method

was introduced by Breiman in [211]. The concept of bias-variance

decomposition is explained in detail by Hastie et al. [249]. While the biasvariance decomposition was initially proposed for regression problems with

squared loss function, a unified framework for classification problems

involving 0–1 losses was introduced by Domingos [226].

Related work on mining rare and imbalanced data sets can be found in the

survey papers written by Chawla et al. [214] and Weiss [308]. Samplingbased methods for mining imbalanced data sets have been investigated by

many authors, such as Kubat and Matwin [266], Japkowitz [257], and

Drummond and Holte [228]. Joshi et al. [261] discussed the limitations of

boosting algorithms for rare class modeling. Other algorithms developed for

mining rare classes include SMOTE [213], PNrule [260], and CREDOS

[262].

Various alternative metrics that are well-suited for class imbalanced problems

are available. The precision, recall, and F1-measure are widely-used metrics

in information retrieval [302]. ROC analysis was originally used in signal

detection theory for performing aggregate evaluation over a range of score

thresholds. A method for comparing classifier performance using the convex

hull of ROC curves was suggested by Provost and Fawcett in [286]. Bradley

[208] investigated the use of area under the ROC curve (AUC) as a

performance metric for machine learning algorithms. Despite the vast body of

literature on optimizing the AUC measure in machine learning models, it is

well-known that AUC suffers from certain limitations. For example, the AUC

can be used to compare the quality of two classifiers only if the ROC curve of

one classifier strictly dominates the other. However, if the ROC curves of

two classifiers intersect at any point, then it is difficult to assess the relative

quality of classifiers using the AUC measure. An in-depth discussion of the

pitfalls in using AUC as a performance measure can be obtained in works by

Hand [245, 246], and Powers [284]. The AUC has also been considered to be

an *incoherent *measure of performance, i.e., it uses different scales while

comparing the performance of different classifiers, although a coherent

interpretation of AUC has been provided by Ferri et al. [235]. Berrar and

Flach [203] describe some of the common caveats in using the ROC curve for

clinical microarray research. An alternate approach for measuring the

aggregate performance of a classifier is the precision-recall (PR) curve,

which is especially useful in the presence of class imbalance [292].

An excellent tutorial on cost-sensitive learning can be found in a review

article by Ling and Sheng [271]. The properties of a cost matrix had been

studied by Elkan in [231]. Margineantu and Dietterich [273] examined

various methods for incorporating cost information into the C4.5 learning

algorithm, including wrapper methods, class distribution-based methods, and

loss-based methods. Other cost-sensitive learning methods that are algorithmindependent include AdaCost [233], MetaCost [225], and costing [312].

Extensive literature is also available on the subject of multiclass learning.

This includes the works of Hastie and Tibshirani [248], Allwein et al. [197],

Kong and Dietterich [264], and Tax and Duin [300]. The error-correcting

output coding (ECOC) method was proposed by Dietterich and Bakiri [223].

They had also investigated techniques for designing codes that are suitable

for solving multiclass problems.

Apart from exploring algorithms for traditional classification settings where

every instance has a single set of features with a unique categorical label,

there has been a lot of recent interest in non-traditional classification

paradigms, involving complex forms of inputs and outputs. For example, the

paradigm of multi-label learning allows for an instance to be assigned

multiple class labels rather than just one. This is useful in applications such

as object recognition in images, where a photo image may include more than

one classification object, such as, grass, sky, trees, and mountains. A survey

on multi-label learning can be found in [313]. As another example, the

paradigm of multi-instance learning considers the problem where the

instances are available in the form of groups called *bags*, and training labels

are available at the level of bags rather than individual instances. Multiinstance learning is useful in applications where an object can exist as

multiple instances in different states (e.g., the different isomers of a chemical

compound), and even if a single instance shows a specific characteristic, the

entire bag of instances associated with the object needs to be assigned the

relevant class. A survey on multi-instance learning is provided in [314].

In a number of real-world applications, it is often the case that the training

labels are scarce in quantity, because of the high costs associated with

obtaining gold-standard supervision. However, we almost always have

abundant access to unlabeled test instances, which do not have supervised

labels but contain valuable information about the structure or distribution of

instances. Traditional learning algorithms, which only make use of the

labeled instances in the training set for learning the decision boundary, are

unable to exploit the information contained in unlabeled instances. In

contrast, learning algorithms that make use of the structure in the unlabeled

data for learning the classification model are known as semi-supervised

learning algorithms [315, 316]. The use of unlabeled data is also explored in

the paradigm of multi-view learning [299, 311], where every object is

observed in multiple views of the data, involving diverse sets of features. A

common strategy used by multi-view learning algorithms is co-training [206],

where a different model is learned for every view of the data, but the model

predictions from every view are constrained to be identical to each other on

the unlabeled test instances.

Another learning paradigm that is commonly explored in the paucity of

training data is the framework of active learning, which attempts to seek the

smallest set of label annotations to learn a reasonable classification model.

Active learning expects the annotator to be involved in the process of model

learning, so that the labels are requested incrementally over the most relevant

set of instances, given a limited budget of label annotations. For example, it

may be useful to obtain labels over instances closer to the decision boundary

that can play a bigger role in fine-tuning the boundary. A review on active

learning approaches can be found in [285, 295].

In some applications, it is important to simultaneously solve multiple learning

tasks together, where some of the tasks may be similar to one another. For

example, if we are interested in translating a passage written in English into

different languages, the tasks involving lexically similar languages (such as

Spanish and Portuguese) would require similar learning of models. The

paradigm of multi-task learning helps in simultaneously learning across all

tasks while sharing the learning among related tasks. This is especially useful

when some of the tasks do not contain sufficiently many training samples, in

which case borrowing the learning from other related tasks helps in the

learning of robust models. A special case of multi-task learning is transfer

learning, where the learning from a source task (with sufficient number of

training samples) has to be *transferred *to a destination task (with paucity of

training data). An extensive survey of transfer learning approaches is

provided by Pan et al. [282].

Most classifiers assume every data instance must belong to a class, which is

not always true for some applications. For example, in malware detection,

due to the ease in which new malwares are created, a classifier trained on

existing classes may fail to detect new ones even if the features for the new

malwares are considerably different than those for existing malwares.

Another example is in critical applications such as medical diagnosis, where

prediction errors are costly and can have severe consequences. In this

situation, it would be better for the classifier to refrain from making any

prediction on a data instance if it is unsure of its class. This approach, known

as classifier with reject option, does not need to classify every data instance

unless it determines the prediction is reliable (e.g., if the class probability

exceeds a user-specified threshold). Instances that are unclassified can be

presented to domain experts for further determination of their true class

labels.

Classifiers can also be distinguished in terms of how the classification model

is trained. A batch classifier assumes all the labeled instances are available

during training. This strategy is applicable when the training set size is not

too large and for stationary data, where the relationship between the attributes

and classes does not vary over time. An online classifier, on the other hand,

trains an initial model using a subset of the labeled data [263]. The model is

then updated incrementally as more labeled instances become available. This

strategy is effective when the training set is too large or when there is *concept*

*drift *due to changes in the distribution of the data over time.

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4.14 Exercises

1. 1. Consider a binary classification problem with the following set of

attributes and attribute values:

Air Conditioner={Working, Broken}

Engine={Good, Bad}

Mileage={High, Medium, Low}

Rust={Yes, No}

Suppose a rule-based classifier produces the following rule set:

Mileage=High→Mileage=HighMileage=Low→Value=HighAir Conditioner

1. Are the rules mutually exclusive?

2. Is the rule set exhaustive?

3. Is ordering needed for this set of rules?

4. Do you need a default class for the rule set?

2. 2. The RIPPER algorithm (by Cohen [218]) is an extension of an earlier

algorithm called IREP (by Fürnkranz and Widmer [234]). Both

algorithms apply the reduced-error pruning method to determine

whether a rule needs to be pruned. The reduced error pruning method

uses a validation set to estimate the generalization error of a classifier.

Consider the following pair of rules:

R1:A→CR2:A∧B→C

R2 is obtained by adding a new conjunct, *B*, to the left-hand side of R1.

For this question, you will be asked to determine whether R2 is

preferred over R1 from the perspectives of rule-growing and rulepruning. To determine whether a rule should be pruned, IREP computes

the following measure:

vIREP=p+(N−n)P+N,

where *P *is the total number of positive examples in the validation set, *N*

is the total number of negative examples in the validation set, *p *is the

number of positive examples in the validation set covered by the rule,

and *n *is the number of negative examples in the validation set covered

by the rule. vIREP is actually similar to classification accuracy for the

validation set. IREP favors rules that have higher values of vIREP. On

the other hand, RIPPER applies the following measure to determine

whether a rule should be pruned:

vRIPPER=p−nP+n.

1. Suppose R1 is covered by 350 positive examples and 150 negative

examples, while R2 is covered by 300 positive examples and 50

negative examples. Compute the FOIL’s information gain for the

rule R2 with respect to R1.

2. Consider a validation set that contains 500 positive examples and

500 negative examples. For R1, suppose the number of positive

examples covered by the rule is 200, and the number of negative

examples covered bytheruleis50. For R2, suppose the number of

positive examples covered by the rule is 100 and the number of

negative examples is 5. Compute vIREP for both rules. Which rule

does IREP prefer?

3. Compute vRIPPER for the previous problem. Which rule does

RIPPER prefer?

3. 3. C4.5rules is an implementation of an indirect method for generating

rules from a decision tree. RIPPER is an implementation of a direct

method for generating rules directly from data.

1. Discuss the strengths and weaknesses of both methods.

2. Consider a data set that has a large difference in the class size (i.e.,

some classes are much bigger than others). Which method

(between C4.5rules and RIPPER) is better in terms of finding high

accuracy rules for the small classes?

4. 4. Consider a training set that contains 100 positive examples and 400

negative examples. For each of the following candidate rules,

R1:A→+(covers 4 positive and 1 negative examples),R2:B→+

(covers 30 positive and 10 negative examples),R3:C→+

(covers 100 positive and 90 negative examples),

determine which is the best and worst candidate rule according to:

1. Rule accuracy.

2. FOIL’s information gain.

3. The likelihood ratio statistic.

4. The Laplace measure.

5. The m-estimate measure (with k=2 and p+=0.2).

5. 5. Figure 4.3 illustrates the coverage of the classification rules *R*1, *R*2,

and *R*3. Determine which is the best and worst rule according to:

1. The likelihood ratio statistic.

2. The Laplace measure.

3. The m-estimate measure (with k=2 and p+=0.58).

4. The rule accuracy after *R*1 has been discovered, where none of the

examples covered by *R*1 are discarded.

5. The rule accuracy after *R*1 has been discovered, where only the

positive examples covered by *R*1 are discarded.

6. The rule accuracy after *R*1 has been discovered, where both

positive and negative examples covered by *R*1 are discarded.

6. 6.

1. Suppose the fraction of undergraduate students who smoke is 15%

and the fraction of graduate students who smoke is 23%. If onefifth of the college students are graduate students and the rest are

undergraduates, what is the probability that a student who smokes

is a graduate student?

2. Given the information in part (a), is a randomly chosen college

student more likely to be a graduate or undergraduate student?

3. Repeat part (b) assuming that the student is a smoker.

4. Suppose 30% of the graduate students live in a dorm but only 10%

of the undergraduate students live in a dorm. If a student smokes

and lives in the dorm, is he or she more likely to be a graduate or

undergraduate student? You can assume independence between

students who live in a dorm and those who smoke.

7. 7. Consider the data set shown in Table 4.9

Table 4.9. Data set for

Exercise 7.

Instance *A B C *Class

1 0 0 0 +

2 0 0 1 −

3 0 1 1 −

4 0 1 1 −

5 0 0 1 +

6 1 0 1 +

7 1 0 1 −

8 1 0 1 −

9 1 1 1 +

10 1 0 1 +

1. Estimate the conditional probabilities for

P(A|+), P(B|+), P(C|+), P(A|−), P(B|−), and P(C|−).

2. Use the estimate of conditional probabilities given in the previous

question to predict the class label for a test sample

(A=0, B=1, C=0) using the naïve Bayes approach.

3. Estimate the conditional probabilities using the m-estimate

approach, with p=1/2 and m=4.

4. Repeat part (b) using the conditional probabilities given in part (c).

5. Compare the two methods for estimating probabilities. Which

method is better and why?

8. 8. Consider the data set shown in Table 4.10.

Table 4.10. Data set for

Exercise 8.

Instance *A B C *Class

1 0 0 1 −

2 1 0 1 +

3 0 1 0 −

4 1 0 0 −

5 1 0 1 +

6 0 0 1 +

7 1 1 0 −

8 0 0 0 −

9 0 1 0 +

10 1 1 1 +

1. Estimate the conditional probabilities for

P(A=1|+), P(B=1|+), P(C=1|+), P(A=1|−), P(B=1|−), and P(C=1|−)

using the same approach as in the previous problem.

2. Use the conditional probabilities in part (a) to predict the class label

for a test sample (A=1, B=1, C=1) using the naïve Bayes approach.

3. Compare P(A=1), P(B=1), and P(A=1, B=1). State the relationships

between *A *and *B*.

4. Repeat the analysis in part (c) using P(A=1), P(B=0), and

P(A=1, B=0).

5. Compare P(A=1, B=1|Class=+) against P(A=1|Class=+) and

P(B=1|Class=+). Are the variables conditionally independent given

the class?

9. 9.

1. Explain how naïve Bayes performs on the data set shown in Figure

4.56.

2. If each class is further divided such that there are four classes (*A*1,

*A*2, *B*1, and *B*2), will naïve Bayes perform better?

3. How will a decision tree perform on this data set (for the two-class

problem)? What if there are four classes?

10. 10. Figure 4.57 illustrates the Bayesian network for the data set shown

in Table 4.11. (Assume that all the attributes are binary).

1. Draw the probability table for each node in the network.

2. Use the Bayesian network to compute

P(Engine=Bad, Air Conditioner=Broken).

11. 11. Given the Bayesian network shown in Figure 4.58, compute the

following probabilities:

Figure 4.56.

Data set for Exercise 9.

Figure 4.56. Full Alternative Text

Figure 4.57.

Bayesian network.

1. P(B=good,F=empty, G=empty, S=yes).

2. P(B=bad,F=empty, G=not empty, S=no).

3. Given that the battery is bad, compute the probability that the car

will start.

12. 12. Consider the one-dimensional data set shown in Table 4.12.

1. Classify the data point x=5.0 according to its 1-, 3-, 5-, and 9-

nearest neighbors (using majority vote).

2. Repeat the previous analysis using the distance-weighted voting

approach described in Section 4.3.1.

Table 4.11. Data set for

Exercise 10.

Mileage Engine Air

Conditioner

Number of

Instances

with

Car Value=Hi

Number of

Instances with

Car Value=Lo

Hi Good Working 3 4

Hi Good Broken 1 2

Hi Bad Working 1 5

Hi Bad Broken 0 4

Lo Good Working 9 0

Lo Good Broken 5 1

Lo Bad Working 1 2

Lo Bad Broken 0 2

Figure 4.58.

Bayesian network for Exercise 11.

Figure 4.58. Full Alternative Text

13. 13. The nearest neighbor algorithm described in Section 4.3 can be

extended to handle nominal attributes. A variant of the algorithm called

PEBLS (Parallel Exemplar-Based Learning System) by Cost and

Salzberg [219] measures the distance between two values of a nominal

attribute using the modified value difference metric (MVDM). Given a

pair of nominal attribute values, V1 and V2, the distance between them

is defined as follows:

d(V1, V2)=∑i=1k| ni1n1−ni2n2, | (4.108)

where nij is the number of examples from class *i *with attribute value Vj

and nj is the number of examples with attribute value Vj.

Table 4.12. Data set for

Exercise 12.

x 0.5 3.0 4.5 4.6 4.9 5.2 5.3 5.5 7.0 9.5

y − − + + + − − + − −

Consider the training set for the loan classification problem shown in

Figure 4.8. Use the MVDM measure to compute the distance between

every pair of attribute values for the Home Owner and Marital Status

attributes.

14. 14. For each of the Boolean functions given below, state whether the

problem is linearly separable.

1. *A *AND *B *AND *C*

2. NOT *A *AND *B*

3. (*A *OR *B*) AND (*A *OR *C*)

4. (*A *XOR *B*) AND (*A *OR *B*)

15. 15.

1. Demonstrate how the perceptron model can be used to represent the

AND and OR functions between a pair of Boolean variables.

2. Comment on the disadvantage of using linear functions as

activation functions for multi-layer neural networks.

16. 16. You are asked to evaluate the performance of two classification

models, M1 and M2. The test set you have chosen contains 26 binary

attributes, labeled as *A *through *Z*. Table 4.13 shows the posterior

probabilities obtained by applying the models to the test set. (Only the

posterior probabilities for the positive class are shown). As this is a twoclass problem, P(−)=1−P(+) and P(−|A, …, Z)=1−P(+|A, …, Z). Assume

that we are mostly interested in detecting instances from the positive

class.

1. Plot the ROC curve for both M1 and M2. (You should plot them on

the same graph.) Which model do you think is better? Explain your

reasons.

2. For model M1, suppose you choose the cutoff threshold to be t=0.5.

In other words, any test instances whose posterior probability is

greater than *t *will be classified as a positive example. Compute the

precision, recall, and *F*-measure for the model at this threshold

value.

3. Repeat the analysis for part (b) using the same cutoff threshold on

model M2. Compare the *F*-measure results for both models. Which

model is better? Are the results consistent with what you expect

from the ROC curve?

4. Repeat part (b) for model M1 using the threshold t=0.1. Which

threshold do you prefer, t=0.5 or t=0.1? Are the results consistent

with what you expect from the ROC curve?

Table 4.13. Posterior

probabilities for Exercise

16.

Instance True

Class P(+|A, …, Z, M1) P(+|A, …, Z, M2)

1 + 0.73 0.61

2 + 0.69 0.03

3 − 0.44 0.68

4 − 0.55 0.31

5 + 0.67 0.45

6 + 0.47 0.09

7 − 0.08 0.38

8 − 0.15 0.05

9 + 0.45 0.01

10 − 0.35 0.04

17. 17. Following is a data set that contains two attributes, *X *and *Y *, and two

class labels, “+” and “−”. Each attribute can take three different values:

0, 1, or 2.

*X Y *Number of Instances

+ −

0 0 0 100

1 0 0 0

2 0 0 100

0 1 10 100

1 1 10 0

2 1 10 100

0 2 0 100

1 2 0 0

2 2 0 100

The concept for the “+” class is Y=1 and the concept for the “−” class is

X=0∨X=2.

1. Build a decision tree on the data set. Does the tree capture the “+”

and “−” concepts?

2. What are the accuracy, precision, recall, and F1-measure of the

decision tree? (Note that precision, recall, and F1-measure are

defined with respect to the “+” class.)

3. Build a new decision tree with the following cost function:

C(i, j)={ 0,if i=j;1,if i=+, j=−;Number of

− instancesNumber of+ instancesif i=−, j=+;

(Hint: only the leaves of the old decision tree need to be changed.)

Does the decision tree capture the “+” concept?

4. What are the accuracy, precision, recall, and F1-measure of the new

decision tree?

18. 18. Consider the task of building a classifier from random data, where

the attribute values are generated randomly irrespective of the class

labels. Assume the data set contains instances from two classes, “+” and

“−.” Half of the data set is used for training while the remaining half is

used for testing.

1. Suppose there are an equal number of positive and negative

instances in the data and the decision tree classifier predicts every

test instance to be positive. What is the expected error rate of the

classifier on the test data?

2. Repeat the previous analysis assuming that the classifier predicts

each test instance to be positive class with probability 0.8 and

negative class with probability 0.2.

3. Suppose two-thirds of the data belong to the positive class and the

remaining one-third belong to the negative class. What is the

expected error of a classifier that predicts every test instance to be

positive?

4. Repeat the previous analysis assuming that the classifier predicts

each test instance to be positive class with probability 2/3 and

negative class with probability 1/3.

19. 19. Derive the dual Lagrangian for the linear SVM with non-separable

data where the objective function is

f(w)=ǁ w ǁ22+C(∑i=1Nξi)2.

20. 20. Consider the XOR problem where there are four training points:

(1, 1, −), (1, 0, +), (0, 1, +), (0, 0, −).

Transform the data into the following feature space:

φ=(1, 2×1, 2×2, 2x1x2, x12, x22).

Find the maximum margin linear decision boundary in the transformed

space.

21. 21. Given the data sets shown in Figures 4.59, explain how the decision

tree, naïve Bayes, and *k*-nearest neighbor classifiers would perform on

these data sets.

Figure 4.59.

Data set for Exercise 21.

Figure 4.59. Full Alternative Text

5 Association Analysis: Basic

Concepts and Algorithms

Many business enterprises accumulate large quantities of data from their dayto-day operations. For example, huge amounts of customer purchase data are

collected daily at the checkout counters of grocery stores. Table 5.1 gives an

example of such data, commonly known as market basket transactions.

Each row in this table corresponds to a transaction, which contains a unique

identifier labeled *TID *and a set of items bought by a given customer.

Retailers are interested in analyzing the data to learn about the purchasing

behavior of their customers. Such valuable information can be used to

support a variety of business-related applications such as marketing

promotions, inventory management, and customer relationship management.

Table 5.1. An example of

market basket transactions.

*TID *Items

1 {Bread, Milk}

2 {Bread, Diapers, Beer, Eggs}

3 {Milk, Diapers, Beer, Cola}

4 {Bread, Milk, Diapers, Beer}

5 {Bread, Milk, Diapers, Cola}

This chapter presents a methodology known as association analysis, which is

useful for discovering interesting relationships hidden in large data sets. The

uncovered relationships can be represented in the form of sets of items

present in many transactions, which are known as frequent itemsets, or

association rules, that represent relationships between two itemsets. For

example, the following rule can be extracted from the data set shown in Table

5.1:

{Diapers}→{Beer}.

The rule suggests a relationship between the sale of diapers and beer because

many customers who buy diapers also buy beer. Retailers can use these types

of rules to help them identify new opportunities for cross-selling their

products to the customers.

Besides market basket data, association analysis is also applicable to data

from other application domains such as bioinformatics, medical diagnosis,

web mining, and scientific data analysis. In the analysis of Earth science data,

for example, association patterns may reveal interesting connections among

the ocean, land, and atmospheric processes. Such information may help Earth

scientists develop a better understanding of how the different elements of the

Earth system interact with each other. Even though the techniques presented

here are generally applicable to a wider variety of data sets, for illustrative

purposes, our discussion will focus mainly on market basket data.

There are two key issues that need to be addressed when applying association

analysis to market basket data. First, discovering patterns from a large

transaction data set can be computationally expensive. Second, some of the

discovered patterns may be spurious (happen simply by chance) and even for

non-spurious patterns, some are more interesting than others. The remainder

of this chapter is organized around these two issues. The first part of the

chapter is devoted to explaining the basic concepts of association analysis

and the algorithms used to efficiently mine such patterns. The second part of

the chapter deals with the issue of evaluating the discovered patterns in order

to help prevent the generation of spurious results and to rank the patterns in

terms of some interestingness measure.

5.1 Preliminaries

This section reviews the basic terminology used in association analysis and

presents a formal description of the task.

Binary Representation

Market basket data can be represented in a binary format as shown in Table

5.2, where each row corresponds to a transaction and each column

corresponds to an item. An item can be treated as a binary variable whose

value is one if the item is present in a transaction and zero otherwise. Because

the presence of an item in a transaction is often considered more important

than its absence, an item is an asymmetric binary variable. This

representation is a simplistic view of real market basket data because it

ignores important aspects of the data such as the quantity of items sold or the

price paid to purchase them. Methods for handling such non-binary data will

be explained in Chapter 6.

Table 5.2. A binary 0/1

representation of market

basket data.

TID Bread Milk Diapers Beer Eggs Cola

1 1 1 0 0 0 0

2 1 0 1 1 1 0

3 0 1 1 1 0 1

4 1 1 1 1 0 0

5 1 1 1 0 0 1

Itemset and Support Count

Let I={i1, i2, … , id} be the set of all items in a market basket data and T=

{t1, t2, …, tN} be the set of all transactions. Each transaction, ti contains a

subset of items chosen from *I*. In association analysis, a collection of zero or

more items is termed an itemset. If an itemset contains *k *items, it is called a

*k*-itemset. For instance, {Beer, Diapers, Milk} is an example of a 3-itemset.

The null (or empty) set is an itemset that does not contain any items.

A transaction tj is said to contain an itemset *X *if *X *is a subset of tj. For

example, the second transaction shown in Table 5.2 contains the itemset

{Bread, Diapers} but not {Bread, Milk}. An important property of an

itemset is its support count, which refers to the number of transactions that

contain a particular itemset. Mathematically, the support count, σ(X), for an

itemset *X *can be stated as follows:

σ(X)=|{ti|X⊆ti, ti∈T}|,

where the symbol |⋅| denotes the number of elements in a set. In the data set

shown in Table 5.2, the support count for {Beer, Diapers, Milk} is equal to

two because there are only two transactions that contain all three items.

Often, the property of interest is the support, which is fraction of transactions

in which an itemset occurs:

s(X)=σ(X)/N.

An itemset *X *is called frequent if *s*(*X*) is greater than some user-defined

threshold, *minsup*.

Association Rule

An association rule is an implication expression of the form X→Y, where *X*

and *Y *are disjoint itemsets, i.e., X∩Y=∅. The strength of an association rule

can be measured in terms of its support and confidence. Support determines

how often a rule is applicable to a given data set, while confidence

determines how frequently items in *Y *appear in transactions that contain *X*.

The formal definitions of these metrics are

Support, s(X→Y)=σ(X∪Y)N; (5.1)

Confidence, c(X→Y)=σ(X∪Y)σ(X). (5.2)

Example 5.1.

Consider the rule {Milk, Diapers}→{Beer}. Because the support count for

{Milk, Diapers, Beer} is 2 and the total number of transactions is 5, the rule’s

support is 2/5=0.4. The rule’s confidence is obtained by dividing the support

count for {Milk, Diapers, Beer} by the support count for {Milk, Diapers}.

Since there are 3 transactions that contain milk and diapers, the confidence

for this rule is 2/3=0.67.

Why Use Support and Confidence?

Support is an important measure because a rule that has very low support

might occur simply by chance. Also, from a business perspective a low

support rule is unlikely to be interesting because it might not be profitable to

promote items that customers seldom buy together (with the exception of the

situation described in Section 5.8). For these reasons, we are interested in

finding rules whose support is greater than some user-defined threshold. As

will be shown in Section 5.2.1, support also has a desirable property that can

be exploited for the efficient discovery of association rules.

Confidence, on the other hand, measures the reliability of the inference made

by a rule. For a given rule X→Y, the higher the confidence, the more likely it

is for *Y *to be present in transactions that contain *X*. Confidence also provides

an estimate of the conditional probability of *Y *given *X*.

Association analysis results should be interpreted with caution. The inference

made by an association rule does not necessarily imply causality. Instead, it

can sometimes suggest a strong co-occurrence relationship between items in

the antecedent and consequent of the rule. Causality, on the other hand,

requires knowledge about which attributes in the data capture cause and

effect, and typically involves relationships occurring over time (e.g.,

greenhouse gas emissions lead to global warming). See Section 5.7.1 for

additional discussion.

Formulation of the Association Rule

Mining Problem

The association rule mining problem can be formally stated as follows:

Definition 5.1. (Association Rule

Discovery.)

Given a set of transactions *T *, find all the rules having support ≥ minsup and

confidence ≥ minconf, where *minsup *and *minconf *are the corresponding

support and confidence thresholds.

A brute-force approach for mining association rules is to compute the support

and confidence for every possible rule. This approach is prohibitively

expensive because there are exponentially many rules that can be extracted

from a data set. More specifically, assuming that neither the left nor the righthand side of the rule is an empty set, the total number of possible rules, *R*,

extracted from a data set that contains *d *items is

R=3d−2d+1+1. (5.3)

The proof for this equation is left as an exercise to the readers (see Exercise 5

on page 440). Even for the small data set shown in Table 5.1, this approach

requires us to compute the support and confidence for 36−27+1=602 rules.

More than 80% of the rules are discarded after applying minsup=20% and

mincof=50%, thus wasting most of the computations. To avoid performing

needless computations, it would be useful to prune the rules early without

having to compute their support and confidence values.

An initial step toward improving the performance of association rule mining

algorithms is to decouple the support and confidence requirements. From

Equation 5.1, notice that the support of a rule X→Y is the same as the

support of its corresponding itemset, X∪Y. For example, the following rules

have identical support because they involve items from the same itemset,

{Beer, Diapers, Milk}:

{Beer, Diapers}→{Milk},{Beer, Milk}→{Diapers},{Diapers, Milk}

→{Beer},{Beer}→{Diapers, Milk},{Milk}→{Beer, Diapers},{Diapers}

→{Beer, Milk}.

If the itemset is infrequent, then all six candidate rules can be pruned

immediately without our having to compute their confidence values.

Therefore, a common strategy adopted by many association rule mining

algorithms is to decompose the problem into two major subtasks:

1. Frequent Itemset Generation, whose objective is to find all the itemsets

that satisfy the *minsup *threshold.

2. Rule Generation, whose objective is to extract all the high confidence

rules from the frequent itemsets found in the previous step. These rules

are called strong rules.

The computational requirements for frequent itemset generation are generally

more expensive than those of rule generation. Efficient techniques for

generating frequent itemsets and association rules are discussed in Sections

5.2 and 5.3, respectively.

5.2 Frequent Itemset Generation

A lattice structure can be used to enumerate the list of all possible itemsets.

Figure 5.1 shows an itemset lattice for I={a, b, c, d, e}. In general, a data set

that contains *k *items can potentially generate up to 2k−1 frequent itemsets,

excluding the null set. Because *k *can be very large in many practical

applications, the search space of itemsets that need to be explored is

exponentially large.

Figure 5.1.

An itemset lattice.

Figure 5.1. Full Alternative Text

A brute-force approach for finding frequent itemsets is to determine the

support count for every candidate itemset in the lattice structure. To do this,

we need to compare each candidate against every transaction, an operation

that is shown in Figure 5.2. If the candidate is contained in a transaction, its

support count will be incremented. For example, the support for {Bread,

Milk} is incremented three times because the itemset is contained in

transactions 1, 4, and 5. Such an approach can be very expensive because it

requires *O*(*NMw*) comparisons, where *N *is the number of transactions, M=2k

−1 is the number of candidate itemsets, and *w *is the maximum transaction

width. Transaction width is the number of items present in a transaction.

Figure 5.2.

Counting the support of candidate itemsets.

There are three main approaches for reducing the computational complexity

of frequent itemset generation.

1. Reduce the number of candidate itemsets (*M*). The *Apriori *principle,

described in the next Section, is an effective way to eliminate some of

the candidate itemsets without counting their support values.

2. Reduce the number of comparisons. Instead of matching each candidate

itemset against every transaction, we can reduce the number of

comparisons by using more advanced data structures, either to store the

candidate itemsets or to compress the data set. We will discuss these

strategies in Sections 5.2.4 and 5.6, respectively.

3. Reduce the number of transactions (*N*). As the size of candidate itemsets

increases, fewer transactions will be supported by the itemsets. For

instance, since the width of the first transaction in Table 5.1 is 2, it

would be advantageous to remove this transaction before searching for

frequent itemsets of size 3 and larger. Algorithms that employ such a

strategy are discussed in the Bibliographic Notes.

5.2.1 The *Apriori *Principle

This Section describes how the support measure can be used to reduce the

number of candidate itemsets explored during frequent itemset generation.

The use of support for pruning candidate itemsets is guided by the following

principle.

Theorem 5.1 (*Apriori *Principle).

*If an itemset is frequent, then all of its subsets must also be frequent.*

To illustrate the idea behind the *Apriori *principle, consider the itemset lattice

shown in Figure 5.3. Suppose {*c*, *d*, *e*} is a frequent itemset. Clearly, any

transaction that contains {*c*, *d*, *e*} must also contain its subsets, {*c*, *d*}, {*c*, *e*},

{*d*, *e*}, {*c*}, {*d*}, and {*e*}. As a result, if {*c*, *d*, *e*} is frequent, then all subsets

of {*c*, *d*, *e*} (i.e., the shaded itemsets in this figure) must also be frequent.

Figure 5.3.

An illustration of the *Apriori *principle. If {*c*, *d*, *e*} is frequent, then

all subsets of this itemset are frequent.

Conversely, if an itemset such as {*a*, *b*} is infrequent, then all of its supersets

must be infrequent too. As illustrated in Figure 5.4, the entire subgraph

containing the supersets of {*a*, *b*} can be pruned immediately once {*a*, *b*} is

found to be infrequent. This strategy of trimming the exponential search

space based on the support measure is known as support-based pruning.

Such a pruning strategy is made possible by a key property of the support

measure, namely, that the support for an itemset never exceeds the support

for its subsets. This property is also known as the anti-monotone property of

the support measure.

Figure 5.4.

An illustration of support-based pruning. If {*a*, *b*} is infrequent,

then all supersets of {*a*, *b*} are infrequent.

Definition 5.2. (Anti-monotone

Property.)

A measure *f *possesses the anti-monotone property if for every itemset *X *that

is a proper subset of itemset *Y*, i.e. X⊂Y, we have f(Y)≤f(X).

More generally, a large number of measures—see Section 5.7.1—can be

applied to itemsets to evaluate various properties of itemsets. As will be

shown in the next Section, any measure that has the anti-monotone property

can be incorporated directly into an itemset mining algorithm to effectively

prune the exponential search space of candidate itemsets.

5.2.2 Frequent Itemset Generation

in the *Apriori *Algorithm

*Apriori *is the first association rule mining algorithm that pioneered the use of

support-based pruning to systematically control the exponential growth of

candidate itemsets. Figure 5.5 provides a high-level illustration of the

frequent itemset generation part of the *Apriori *algorithm for the transactions

shown in Table 5.1. We assume that the support threshold is 60%, which is

equivalent to a minimum support count equal to 3.

Figure 5.5.

Illustration of frequent itemset generation using the *Apriori*

algorithm.

Figure 5.5. Full Alternative Text

Initially, every item is considered as a candidate 1-itemset. After counting

their supports, the candidate itemsets {Cola} and {Eggs} are discarded

because they appear in fewer than three transactions. In the next iteration,

candidate 2-itemsets are generated using only the frequent 1-itemsets because

the *Apriori *principle ensures that all supersets of the infrequent 1-itemsets

must be infrequent. Because there are only four frequent 1-itemsets, the

number of candidate 2-itemsets generated by the algorithm is (42)=6. Two of

these six candidates, {Beer, Bread} and {Beer, Milk}, are subsequently

found to be infrequent after computing their support values. The remaining

four candidates are frequent, and thus will be used to generate candidate 3-

itemsets. Without support-based pruning, there are(63)=20 candidate 3-

itemsets that can be formed using the six items given in this example. With

the *Apriori *principle, we only need to keep candidate 3-itemsets whose

subsets are frequent. The only candidate that has this property is {Bread,

Diapers, Milk}. However, even though the subsets of {Bread, Diapers,

Milk} are frequent, the itemset itself is not.

The effectiveness of the *Apriori *pruning strategy can be shown by counting

the number of candidate itemsets generated. A brute-force strategy of

enumerating all itemsets (up to size 3) as candidates will produce

(61)+(62)+(63)=6+15+20=41

candidates. With the *Apriori *principle, this number decreases to

(61)+(42)+1=6+6+1=13

candidates, which represents a 68% reduction in the number of candidate

itemsets even in this simple example.

The pseudocode for the frequent itemset generation part of the *Apriori*

algorithm is shown in Algorithm 5.1. Let Ck denote the set of candidate *k*itemsets and Fk denote the set of frequent *k*-itemsets:

The algorithm initially makes a single pass over the data set to

determine the support of each item. Upon completion of this step, the set

of all frequent 1-itemsets, F1, will be known (steps 1 and 2).

Next, the algorithm will iteratively generate new candidate *k*-itemsets

and prune unnecessary candidates that are guaranteed to be infrequent

given the frequent (k−1)-itemsets found in the previous iteration (steps 5

and 6). Candidate generation and pruning is implemented using the

functions candidate-gen and candidate-prune, which are described in

Section 5.2.3.

To count the support of the candidates, the algorithm needs to make an

additional pass over the data set (steps 7–12). The subset function is

used to determine all the candidate itemsets in Ck that are contained in

each transaction *t*. The implementation of this function is described in

Section 5.2.4.

After counting their supports, the algorithm eliminates all candidate

itemsets whose support counts are less than N×minsup (step 13).

The algorithm terminates when there are no new frequent itemsets

generated, i.e., Fk=∅ (step 14).

The frequent itemset generation part of the *Apriori *algorithm has two

important characteristics. First, it is a level-wise algorithm; i.e., it traverses

the itemset lattice one level at a time, from frequent 1-itemsets to the

maximum size of frequent itemsets. Second, it employs a generate-and-test

strategy for finding frequent itemsets. At each iteration (level), new candidate

itemsets are generated from the frequent itemsets found in the previous

iteration. The support for each candidate is then counted and tested against

the *minsup *threshold. The total number of iterations needed by the algorithm

is kmax+1, where kmax is the maximum size of the frequent itemsets.

5.2.3 Candidate Generation and

Pruning

The candidate-gen and candidate-prune functions shown in Steps 5 and 6 of

Algorithm 5.1 generate candidate itemsets and prunes unnecessary ones by

performing the following two operations, respectively:

1. Candidate Generation. This operation generates new candidate *k*itemsets based on the frequent (k−1)-itemsets found in the previous

iteration.

Algorithm 5.1 Frequent itemset

generation of the *Apriori*

algorithm.

1: *k *= 1.

2: *F*

*k *= {*i *| *i *∈ *I *∧ σ({*i*}) ≥ *N *× *minsup*}. {Find all frequent 1-

3: repeat

4: *k *= *k *+ 1.

5: *C*

*k *= candidate-gen(*F**k *− 1). {Generate candidate itemsets.}

6: *C*

*k *= candidate-prune(*C**k*, *F**k *− 1). {Prune candidate itemsets.}

7: for each transaction *t *∈ *T *do

8: *C*

*t *= subset(*C**k*, *t*). {Identify all candidates that belong to

9: for each candidate itemset *c *∈ *C*

*t *do

10: σ(*c*) = σ(*c*) + 1. {Increment support count.}

11: end for

12: end for

13: *F*

*k *= {*c *| *c *∈ *C**k *∧ σ(*c*) ≥ *N *× *minsup*}. {Extract the frequent

14: until *F*

*k *= ∅

15: Result = ∪*F*

*k*.

2. Candidate Pruning. This operation eliminates some of the candidate *k*itemsets using support-based pruning, i.e. by removing *k*-itemsets whose

subsets are known to be infrequent in previous iterations. Note that this

pruning is done without computing the actual support of these *k*-itemsets

(which could have required comparing them against each transaction).

Candidate Generation

In principle, there are many ways to generate candidate itemsets. An effective

candidate generation procedure must be complete and non-redundant. A

candidate generation procedure is said to be *complete *if it does not omit any

frequent itemsets. To ensure completeness, the set of candidate itemsets must

subsume the set of all frequent itemsets, i.e., ∀k:Fk⊆Ck. A candidate

generation procedure is *non-redundant *if it does not generate the same

candidate itemset more than once. For example, the candidate itemset {*a*, *b*,

*c*, *d*} can be generated in many ways—by merging {*a*, *b*, *c*} with {*d*}, {*b*, *d*}

with {*a*, *c*}, {*c*} with {*a*, *b*, *d*}, etc. Generation of duplicate candidates leads

to wasted computations and thus should be avoided for efficiency reasons.

Also, an effective candidate generation procedure should avoid generating

too many unnecessary candidates. A candidate itemset is unnecessary if at

least one of its subsets is infrequent, and thus, eliminated in the candidate

pruning step.

Next, we will briefly describe several candidate generation procedures,

including the one used by the candidate-gen function.

Brute-Force Method

The brute-force method considers every *k*-itemset as a potential candidate and

then applies the candidate pruning step to remove any unnecessary candidates

whose subsets are infrequent (see Figure 5.6). The number of candidate

itemsets generated at level *k *is equal to (dk), where *d *is the total number of

items. Although candidate generation is rather trivial, candidate pruning

becomes extremely expensive because a large number of itemsets must be

examined.

Figure 5.6.

A brute-force method for generating candidate 3-itemsets.

Fk−1×F1 Method

An alternative method for candidate generation is to extend each frequent (k

−1)-itemset with frequent items that are not part of the (k−1)-itemset. Figure

5.7 illustrates how a frequent 2-itemset such as {Beer, Diapers} can be

augmented with a frequent item such as Bread to produce a candidate 3-

itemset {Beer, Diapers, Bread}.

Figure 5.7.

Generating and pruning candidate *k*-itemsets by merging a frequent

(k−1)-itemset with a frequent item. Note that some of the

candidates are unnecessary because their subsets are infrequent.

The procedure is complete because every frequent *k*-itemset is composed of a

frequent (k−1)-itemset and a frequent 1-itemset. Therefore, all frequent *k*itemsets are part of the candidate *k*-itemsets generated by this procedure.

Figure 5.7 shows that the Fk−1×F1 candidate generation method only

produces four candidate 3-itemsets, instead of the

(63)=20 itemsets produced by the brute-force method. The Fk−1×F1 method

generates lower number of candidates because every candidate is guaranteed

to contain at least one frequent (k−1)-itemset. While this procedure is a

substantial improvement over the brute-force method, it can still produce a

large number of unnecessary candidates, as the remaining subsets of a

candidate itemset can still be infrequent.

Note that the approach discussed above does not prevent the same candidate

itemset from being generated more than once. For instance, {Bread, Diapers,

Milk} can be generated by merging {Bread, Diapers} with {Milk}, {Bread,

Milk} with {Diapers}, or {Diapers, Milk} with {Bread}. One way to avoid

generating duplicate candidates is by ensuring that the items in each frequent

itemset are kept sorted in their lexicographic order. For example, itemsets

such as {Bread, Diapers}, {Bread, Diapers, Milk}, and {Diapers, Milk}

follow lexicographic order as the items within every itemset are arranged

alphabetically. Each frequent (k−1)-itemset *X *is then extended with frequent

items that are lexicographically larger than the items in *X*. For example, the

itemset {Bread, Diapers} can be augmented with {Milk} because Milk is

lexicographically larger than Bread and Diapers. However, we should not

augment {Diapers, Milk} with {Bread} nor {Bread, Milk} with {Diapers}

because they violate the lexicographic ordering condition. Every candidate *k*itemset is thus generated exactly once, by merging the lexicographically

largest item with the remaining k−1 items in the itemset. If the Fk−1×F1

method is used in conjunction with lexicographic ordering, then only two

candidate 3-itemsets will be produced in the example illustrated in Figure 5.7.

{Beer, Bread, Diapers} and {Beer, Bread, Milk} will not be generated

because {Beer, Bread} is not a frequent 2-itemset.

Fk−1×Fk−1 Method

This candidate generation procedure, which is used in the candidate-gen

function of the *Apriori *algorithm, merges a pair of frequent (k−1)-itemsets

only if their first k−2 items, arranged in lexicographic order, are identical. Let

A={a1, a2, …, ak−1} and B={b1, b2, …, bk−1} be a pair of frequent (k−1)-

itemsets, arranged lexicographically. *A *and *B *are merged if they satisfy the

following conditions:

ai=bi (for i=1, 2, …, k−2).

Note that in this case, ak−1≠bk−1 because *A *and *B *are two distinct itemsets.

The candidate *k*-itemset generated by merging *A *and *B *consists of the first k

−2 common items followed by ak−1 and bk−1 in lexicographic order. This

candidate generation procedure is complete, because for every

lexicographically ordered frequent *k*-itemset, there exists two

lexicographically ordered frequent (k−1)-itemsets that have identical items in

the first k−2 positions.

In Figure 5.8, the frequent itemsets {Bread, Diapers} and {Bread, Milk} are

merged to form a candidate 3-itemset {Bread, Diapers, Milk}. The algorithm

does not have to merge {Beer, Diapers} with {Diapers, Milk} because the

first item in both itemsets is different. Indeed, if {Beer, Diapers, Milk} is a

viable candidate, it would have been obtained by merging {Beer, Diapers}

with {Beer, Milk} instead. This example illustrates both the completeness of

the candidate generation procedure and the advantages of using lexicographic

ordering to prevent duplicate candidates. Also, if we order the frequent (k−1)-

itemsets according to their lexicographic rank, itemsets with identical first k

−2 items would take consecutive ranks. As a result, the Fk−1×Fk−1 candidate

generation method would consider merging a frequent itemset only with ones

that occupy the next few ranks in the sorted list, thus saving some

computations.

Figure 5.8.

Generating and pruning candidate *k*-itemsets by merging pairs of

frequent (k−1)-itemsets.

Figure 5.8 shows that the Fk−1×Fk−1 candidate generation procedure results

in only one candidate 3-itemset. This is a considerable reduction from the

four candidate 3-itemsets generated by the Fk−1×F1 method. This is because

the Fk−1×Fk−1 method ensures that every candidate *k*-itemset contains at

least two frequent (k−1)-itemsets, thus greatly reducing the number of

candidates that are generated in this step.

Note that there can be multiple ways of merging two frequent (k−1)-itemsets

in the Fk−1×Fk−1 procedure, one of which is merging if their first k−2 items

are identical. An alternate approach could be to merge two frequent (k−1)-

itemsets *A *and *B *if the last k−2 items of *A *are identical to the first k−2

itemsets of *B*. For example, {Bread, Diapers} and {Diapers, Milk} could be

merged using this approach to generate the candidate 3-itemset {Bread,

Diapers, Milk}. As we will see later, this alternate Fk−1×Fk−1 procedure is

useful in generating sequential patterns, which will be discussed in Chapter 6.

Candidate Pruning

To illustrate the candidate pruning operation for a candidate *k*-itemset, X=

{i1, i2, …, ik}, consider its *k *proper subsets, X−{ij}(∀j=1, 2, …, k). If any

of them are infrequent, then *X *is immediately pruned by using the *Apriori*

principle. Note that we don’t need to explicitly ensure that all subsets of *X *of

size less than k−1 are frequent (see Exercise 7). This approach greatly

reduces the number of candidate itemsets considered during support

counting. For the brute-force candidate generation method, candidate pruning

requires checking only *k *subsets of size k−1 for each candidate *k*-itemset.

However, since the Fk−1×F1 candidate generation strategy ensures that at

least one of the (k−1)-size subsets of every candidate *k*-itemset is frequent,

we only need to check for the remaining k−1 subsets. Likewise, the Fk−1×Fk

−1 strategy requires examining only k−2 subsets of every candidate *k*-itemset,

since two of its (k−1)-size subsets are already known to be frequent in the

candidate generation step.

5.2.4 Support Counting

Support counting is the process of determining the frequency of occurrence

for every candidate itemset that survives the candidate pruning step. Support

counting is implemented in steps 6 through 11 of Algorithm 5.1. A bruteforce approach for doing this is to compare each transaction against every

candidate itemset (see Figure 5.2) and to update the support counts of

candidates contained in a transaction. This approach is computationally

expensive, especially when the numbers of transactions and candidate

itemsets are large.

An alternative approach is to enumerate the itemsets contained in each

transaction and use them to update the support counts of their respective

candidate itemsets. To illustrate, consider a transaction *t *that contains five

items, {1, 2, 3, 5, 6}. There are (53)=10 itemsets of size 3 contained in this

transaction. Some of the itemsets may correspond to the candidate 3-itemsets

under investigation, in which case, their support counts are incremented.

Other subsets of *t *that do not correspond to any candidates can be ignored.

Figure 5.9 shows a systematic way for enumerating the 3-itemsets contained

in *t*. Assuming that each itemset keeps its items in increasing lexicographic

order, an itemset can be enumerated by specifying the smallest item first,

followed by the larger items. For instance, givent={1, 2, 3, 5, 6}, all the 3-

itemsets contained in *t *must begin with item 1, 2, or 3. It is not possible to

construct a 3-itemset that begins with items 5 or 6 because there are only two

items in *t *whose labels are greater than or equal to 5. The number of ways to

specify the first item of a 3-itemset contained in *t *is illustrated by the Level 1

prefix tree structure depicted in Figure 5.9. For instance, 1 2 3 5 6 represents

a 3-itemset that begins with item 1, followed by two more items chosen from

the set {2, 3, 5, 6}.

Figure 5.9.

Enumerating subsets of three items from a transaction *t*.

Figure 5.9. Full Alternative Text

After fixing the first item, the prefix tree structure at Level 2 represents the

number of ways to select the second item. For example, 1 2 3 5 6 corresponds

to itemsets that begin with the prefix (1 2) and are followed by the items 3, 5,

or 6. Finally, the prefix tree structure at Level 3 represents the complete set of

3-itemsets contained in *t*. For example, the 3-itemsets that begin with prefix

{1 2} are {1, 2, 3}, {1, 2, 5}, and {1, 2, 6}, while those that begin with prefix

{2 3} are {2, 3, 5} and {2, 3, 6}.

The prefix tree structure shown in Figure 5.9 demonstrates how itemsets

contained in a transaction can be systematically enumerated, i.e., by

specifying their items one by one, from the leftmost item to the rightmost

item. We still have to determine whether each enumerated 3-itemset

corresponds to an existing candidate itemset. If it matches one of the

candidates, then the support count of the corresponding candidate is

incremented. In the next Section, we illustrate how this matching operation

can be performed efficiently using a hash tree structure.

Support Counting Using a Hash

Tree*

In the *Apriori *algorithm, candidate itemsets are partitioned into different

buckets and stored in a hash tree. During support counting, itemsets

contained in each transaction are also hashed into their appropriate buckets.

That way, instead of comparing each itemset in the transaction with every

candidate itemset, it is matched only against candidate itemsets that belong to

the same bucket, as shown in Figure 5.10.

Figure 5.10.

Counting the support of itemsets using hash structure.

Figure 5.11 shows an example of a hash tree structure. Each internal node of

the tree uses the following hash function, h(p)=(p−1) mod 3,, where mode

refers to the modulo (remainder) operator, to determine which branch of the

current node should be followed next. For example, items 1, 4, and 7 are

hashed to the same branch (i.e., the leftmost branch) because they have the

same remainder after dividing the number by 3. All candidate itemsets are

stored at the leaf nodes of the hash tree. The hash tree shown in Figure 5.11

contains 15 candidate 3-itemsets, distributed across 9 leaf nodes.

Figure 5.11.

Hashing a transaction at the root node of a hash tree.

Figure 5.11. Full Alternative Text

Consider the transaction, t={1, 2, 3, 4, 5, 6}. To update the support counts of

the candidate itemsets, the hash tree must be traversed in such a way that all

the leaf nodes containing candidate 3-itemsets belonging to *t *must be visited

at least once. Recall that the 3-itemsets contained in *t *must begin with items

1, 2, or 3, as indicated by the Level 1 prefix tree structure shown in Figure

5.9. Therefore, at the root node of the hash tree, the items 1, 2, and 3 of the

transaction are hashed separately. Item 1 is hashed to the left child of the root

node, item 2 is hashed to the middle child, and item 3 is hashed to the right

child. At the next level of the tree, the transaction is hashed on the second

item listed in the Level 2 tree structure shown in Figure 5.9. For example,

after hashing on item 1 at the root node, items 2, 3, and 5 of the transaction

are hashed. Based on the hash function, items 2 and 5 are hashed to the

middle child, while item 3 is hashed to the right child, as shown in Figure

5.12. This process continues until the leaf nodes of the hash tree are reached.

The candidate itemsets stored at the visited leaf nodes are compared against

the transaction. If a candidate is a subset of the transaction, its support count

is incremented. Note that not all the leaf nodes are visited while traversing the

hash tree, which helps in reducing the computational cost. In this example, 5

out of the 9 leaf nodes are visited and 9 out of the 15 itemsets are compared

against the transaction.

Figure 5.12.

Subset operation on the leftmost subtree of the root of a candidate

hash tree.

5.2.5 Computational Complexity

The computational complexity of the *Apriori *algorithm, which includes both

its runtime and storage, can be affected by the following factors.

Support Threshold

Lowering the support threshold often results in more itemsets being declared

as frequent. This has an adverse effect on the computational complexity of

the algorithm because more candidate itemsets must be generated and

counted at every level, as shown in Figure 5.13. The maximum size of

frequent itemsets also tends to increase with lower support thresholds. This

increases the total number of iterations to be performed by the *Apriori*

algorithm, further increasing the computational cost.

Figure 5.13.

Effect of support threshold on the number of candidate and

frequent itemsets obtained from a benchmark data set.

Figure 5.13. Full Alternative Text

Number of Items (Dimensionality)

As the number of items increases, more space will be needed to store the

support counts of items. If the number of frequent items also grows with the

dimensionality of the data, the runtime and storage requirements will increase

because of the larger number of candidate itemsets generated by the

algorithm.

Number of Transactions

Because the *Apriori *algorithm makes repeated passes over the transaction

data set, its run time increases with a larger number of transactions.

Average Transaction Width

For dense data sets, the average transaction width can be very large. This

affects the complexity of the *Apriori *algorithm in two ways. First, the

maximum size of frequent itemsets tends to increase as the average

transaction width increases. As a result, more candidate itemsets must be

examined during candidate generation and support counting, as illustrated in

Figure 5.14. Second, as the transaction width increases, more itemsets are

contained in the transaction. This will increase the number of hash tree

traversals performed during support counting.

A detailed analysis of the time complexity for the *Apriori *algorithm is

presented next.

Figure 5.14.

Effect of average transaction width on the number of candidate and

frequent itemsets obtained from a synthetic data set.

Figure 5.14. Full Alternative Text

Generation of frequent 1-itemsets

For each transaction, we need to update the support count for every item

present in the transaction. Assuming that *w *is the average transaction width,

this operation requires *O*(*Nw*) time, where *N *is the total number of

transactions.

Candidate generation

To generate candidate *k*-itemsets, pairs of frequent (k−1)-itemsets are merged

to determine whether they have at least k−2 items in common. Each merging

operation requires at most k−2 equality comparisons. Every merging step can

produce at most one viable candidate *k*-itemset, while in the worst-case, the

algorithm must try to merge every pair of frequent (k−1)-itemsets found in

the previous iteration. Therefore, the overall cost of merging frequent

itemsets is

∑k=2w(k−2)|Ck|<Cost of merging<∑k=2w(k−2)|Fk−1|2,

where *w *is the maximum transaction width. A hash tree is also constructed

during candidate generation to store the candidate itemsets. Because the

maximum depth of the tree is *k*, the cost for populating the hash tree with

candidate itemsets is O(∑k=2wk|Ck|). During candidate pruning, we need to

verify that the k−2 subsets of every candidate *k*-itemset are frequent. Since

the cost for looking up a candidate in a hash tree is *O*(*k*), the candidate

pruning step requires O(∑k=2wk(k−2)|Ck|) time.

Support counting

Each transaction of width |t| produces (|t|k) itemsets of size *k*. This is also the

effective number of hash tree traversals performed for each transaction. The

cost for support counting is O(N∑k(wk)αk), where *w *is the maximum

transaction width and αk is the cost for updating the support count of a

candidate *k*-itemset in the hash tree.

5.3 Rule Generation

This Section describes how to extract association rules efficiently from a

given frequent itemset. Each frequent *k*-itemset, *Y*, can produce up to 2k−2

association rules, ignoring rules that have empty antecedents or consequents

∅→Y or Y→∅). An association rule can be extracted by partitioning the

itemset *Y *into two non-empty subsets, *X *and Y−X, such that X→Y−X

satisfies the confidence threshold. Note that all such rules must have already

met the support threshold because they are generated from a frequent itemset.

Example 5.2.

Let X={a, b, c} be a frequent itemset. There are six candidate association

rules that can be generated from X:{a, b}→{c}, {a, c}→{b}, {b, c}

→{a}, {a}→{b, c}, {b}→{a, c}, and {c}→{a, b}. As each of their support is

identical to the support for *X*, all the rules satisfy the support threshold.

Computing the confidence of an association rule does not require additional

scans of the transaction data set. Consider the rule {1, 2}→{3}, which is

generated from the frequent itemset X={1, 2, 3}. The confidence for this rule

is σ{(1, 2, 3})/σ({1, 2}). Because {1, 2, 3} is frequent, the anti-monotone

property of support ensures that {1, 2} must be frequent, too. Since the

support counts for both itemsets were already found during frequent itemset

generation, there is no need to read the entire data set again.

5.3.1 Confidence-Based Pruning

Confidence does not show the anti-monotone property in the same way as the

support measure. For example, the confidence for X→Y can be larger,

smaller, or equal to the confidence for another rule X˜→Y˜, where X˜⊆X

and Y˜⊆Y (see Exercise 3 on page 439). Nevertheless, if we compare rules

generated from the same frequent itemset *Y*, the following theorem holds for

the confidence measure.

Theorem 5.2.

*Let Y be an itemset and X is a subset of Y. If a rule *X→Y−X *does not satisfy*

*the confidence threshold, then any rule *X˜→Y−X˜, *where *X˜ *is a subset of X,*

*must not satisfy the confidence threshold as well.*

To prove this theorem, consider the following two rules: X˜→Y−X˜ and

X→Y−X, where X˜⊂X. The confidence of the rules are σ(Y)/σ(X˜) and

σ(Y)/σ(X), respectively. Since X˜ is a subset of *X*, σ(X˜)/σ(X). Therefore, the

former rule cannot have a higher confidence than the latter rule.

5.3.2 Rule Generation in *Apriori*

Algorithm

The *Apriori *algorithm uses a level-wise approach for generating association

rules, where each level corresponds to the number of items that belong to the

rule consequent. Initially, all the high confidence rules that have only one

item in the rule consequent are extracted. These rules are then used to

generate new candidate rules. For example, if {acd}→{b} and {abd}→{c}

are high confidence rules, then the candidate rule {ad}→{bc} is generated by

merging the consequents of both rules. Figure 5.15 shows a lattice structure

for the association rules generated from the frequent itemset {*a*, *b*, *c*, *d*}. If

any node in the lattice has low confidence, then according to Theorem 5.2,

the entire subgraph spanned by the node can be pruned immediately. Suppose

the confidence for {bcd}→{a} is low. All the rules containing item *a *in its

consequent, including {cd}→{ab}, {bd}→{ac}, {bc}→{ad}, and {d}

→{abc} can be discarded.

Figure 5.15.

Pruning of association rules using the confidence measure.

Figure 5.15. Full Alternative Text

A pseudocode for the rule generation step is shown in Algorithms 5.2 and

5.3. Note the similarity between the ap-genrules procedure given in

Algorithm 5.3 and the frequent itemset generation procedure given in

Algorithm 5.1. The only difference is that, in rule generation, we do not have

to make additional passes over the data set to compute the confidence of the

candidate rules. Instead, we determine the confidence of each rule by using

the support counts computed during frequent itemset generation.

Algorithm 5.2 Rule generation of

the *Apriori *algorithm.

1: for each frequent *k*-itemset *f**k*, *k *≥ 2 do

2: *H*

1 = {*i *| *i *∈ *f**k*} {1-item consequents of the rule.}

3: call ap-genrules(*f**k*, *H*1.)

4: end for

Algorithm 5.3 Procedure apgenrules (fk, Hm).

1: *k *= |*f**k*| {size of frequent itemset.}

2: *m *= |*H**m*| {size of rule consequent.}

3: if *k *> *m *+ 1 then

4: *H*

*m *+ 1 = candidate-gen(*H**m*).

5: *H*

*m *+ 1 = candidate-prune(*H**m *+ 1,*H**m*).

6: for each *h*

*m *+ 1 ∈ *H**m *+ 1 do

7: *conf *= σ(*f**k*)/σ(*f**k *− *h**m *+ 1).

8: if *conf *≥ *minconf *then

9: output the rule (*f**k *− *h**m *+ 1) → *h**m *+ 1.

10: else

11: delete *h*

*m *+ 1 from *H**m *+ 1.

12: end if

13: end for

14: call ap-genrules(*f**k*, *H**m *+ 1.)

15: end if

5.3.3 An Example: Congressional

Voting Records

This Section demonstrates the results of applying association analysis to the

voting records of members of the United States House of Representatives.

The data is obtained from the 1984 Congressional Voting Records Database,

which is available at the UCI machine learning data repository. Each

transaction contains information about the party affiliation for a

representative along with his or her voting record on 16 key issues. There are

435 transactions and 34 items in the data set. The set of items are listed in

Table 5.3.

Table 5.3. List of binary

attributes from the 1984 United

States Congressional Voting

Records. Source: The UCI

machine learning repository.

1. Republican

2. Democrat

3. handicapped-infants=yes

4. handicapped-infants=no

5. water project cost sharing=yes

6. water project cost sharing=no

7. budget-resolution=yes

8. budget-resolution=no

9. physician fee freeze=yes

10. physician fee freeze=no

11. aid to El Salvador=yes

12. aid to El Salvador=no

13. religious groups in schools=yes

14. religious groups in schools=no

15. anti-satellite test ban=yes

16. anti-satellite test ban=no

17. aid to Nicaragua=yes

18. aid to Nicaragua=no

19. MX-missile=yes

20. MX-missile=no

21. immigration=yes

22. immigration=no

23. synfuel corporation cutback=yes

24. synfuel corporation cutback=no

25. education spending=yes

26. education spending=no

27. right-to-sue=yes

28. right-to-sue=no

29. crime=yes

30. crime=no

31. duty-free-exports=yes

32. duty-free-exports=no

33. export administration act=yes

34. export administration act=no

The *Apriori *algorithm is then applied to the data set with minsup=30% and

minconf=90%. Some of the high confidence rules extracted by the algorithm

are shown in Table 5.4. The first two rules suggest that most of the members

who voted yes for aid to El Salvador and no for budget resolution and MX

missile are Republicans; while those who voted no for aid to El Salvador and

yes for budget resolution and MX missile are Democrats. These high

confidence rules show the key issues that divide members from both political

parties.

Table 5.4. Association rules

extracted from the 1984 United

States Congressional Voting

Records.

Association Rule Confidence

{budget resolution=no, MXmissile=no, aid to El Salvador=yes }→{Republican} 91.0%

{budget resolution=yes, MXmissile=yes, aid to El Salvador=no }→{Democrat} 97.5%

{crime=yes, right-tosue=yes, physician fee freeze=yes }→{Republican} 93.5%

{crime=no, right-to-sue=no, physician fee freeze=no }

→{Democrat} 100%

5.4 Compact Representation of

Frequent Itemsets

In practice, the number of frequent itemsets produced from a transaction data

set can be very large. It is useful to identify a small representative set of

frequent itemsets from which all other frequent itemsets can be derived. Two

such representations are presented in this Section in the form of maximal and

closed frequent itemsets.

5.4.1 Maximal Frequent Itemsets

Definition 5.3. (Maximal Frequent

Itemset.)

A frequent itemset is maximal if none of its immediate supersets are frequent.

To illustrate this concept, consider the itemset lattice shown in Figure 5.16.

The itemsets in the lattice are divided into two groups: those that are frequent

and those that are infrequent. A frequent itemset border, which is represented

by a dashed line, is also illustrated in the diagram. Every itemset located

above the border is frequent, while those located below the border (the

shaded nodes) are infrequent. Among the itemsets residing near the border,

{*a*, *d*}, {*a*, *c*, *e*}, and {*b*, *c*, *d*, *e*} are maximal frequent itemsets because all of

their immediate supersets are infrequent. For example, the itemset {*a*, *d*} is

maximal frequent because all of its immediate supersets, {*a*, *b*, *d*}, {*a*, *c*, *d*},

and {*a*, *d*, *e*}, are infrequent. In contrast, {*a*, *c*} is non-maximal because one

of its immediate supersets, {*a*, *c*, *e*}, is frequent.

Figure 5.16.

Maximal frequent itemset.

Figure 5.16. Full Alternative Text

Maximal frequent itemsets effectively provide a compact representation of

frequent itemsets. In other words, they form the smallest set of itemsets from

which all frequent itemsets can be derived. For example, every frequent

itemset in Figure 5.16 is a subset of one of the three maximal frequent

itemsets, {*a*, *d*}, {*a*, *c*, *e*}, and {*b*, *c*, *d*, *e*}. If an itemset is not a proper subset

of any of the maximal frequent itemsets, then it is either infrequent (e.g., {*a*,

*d*, *e*}) or maximal frequent itself (e.g., {*b*, *c*, *d*, *e*}). Hence, the maximal

frequent itemsets {*a*, *c*, *e*}, {*a*, *d*}, and {*b*, *c*, *d*, *e*} provide a compact

representation of the frequent itemsets shown in Figure 5.16. Enumerating all

the subsets of maximal frequent itemsets generates the complete list of all

frequent itemsets.

Maximal frequent itemsets provide a valuable representation for data sets that

can produce very long, frequent itemsets, as there are exponentially many

frequent itemsets in such data. Nevertheless, this approach is practical only if

an efficient algorithm exists to explicitly find the maximal frequent itemsets.

We briefly describe one such approach in Section 5.5.

Despite providing a compact representation, maximal frequent itemsets do

not contain the support information of their subsets. For example, the support

of the maximal frequent itemsets {*a*, *c*, *e*}, {*a*, *d*}, and {*b*, *c*, *d*, *e*} do not

provide any information about the support of their subsets except that it

meets the support threshold. An additional pass over the data set is therefore

needed to determine the support counts of the non-maximal frequent itemsets.

In some cases, it is desirable to have a minimal representation of itemsets that

preserves the support information. We describe such a representation in the

next Section.

5.4.2 Closed Itemsets

Closed itemsets provide a minimal representation of all itemsets without

losing their support information. A formal definition of a closed itemset is

presented below.

Definition 5.4. (Closed Itemset.)

An itemset *X *is closed if none of its immediate supersets has exactly the same

support count as *X*.

Put another way, *X *is not closed if at least one of its immediate supersets has

the same support count as *X*. Examples of closed itemsets are shown in

Figure 5.17. To better illustrate the support count of each itemset, we have

associated each node (itemset) in the lattice with a list of its corresponding

transaction IDs. For example, since the node {*b*, *c*} is associated with

transaction IDs 1, 2, and 3, its support count is equal to three. From the

transactions given in this diagram, notice that the support for {*b*} is identical

to {*b*, *c*}.This is because every transaction that contains *b *also contains *c*.

Hence, {*b*} is not a closed itemset. Similarly, since *c *occurs in every

transaction that contains both *a *and *d*, the itemset {*a*, *d*} is not closed as it

has the same support as its superset {*a*, *c*, *d*}. On the other hand, {*b*, *c*} is a

closed itemset because it does not have the same support count as any of its

supersets.

Figure 5.17.

An example of the closed frequent itemsets (with minimum support

equal to 40%).

Figure 5.17. Full Alternative Text

An interesting property of closed itemsets is that if we know their support

counts, we can derive the support count of every other itemset in the itemset

lattice without making additional passes over the data set. For example,

consider the 2-itemset {*b*, *e*} in Figure 5.17. Since {*b*, *e*} is not closed, its

support must be equal to the support of one of its immediate supersets, {*a*, *b*,

*e*}, {*b*, *c*, *e*}, and {*b*, *d*, *e*}. Further, none of the supersets of {*b*, *e*} can have a

support greater than the support of {*b*, *e*}, due to the anti-monotone nature of

the support measure. Hence, the support of {*b*, *e*} can be computed by

examining the support counts of all of its immediate supersets of size three

and taking their maximum value. If an immediate superset is closed (e.g., {*b*,

*c*, *e*}), we would know its support count. Otherwise, we can recursively

compute its support by examining the supports of its immediate supersets of

size four. In general, the support count of any non-closed (k−1)-itemset can

be determined as long as we know the support counts of all *k*-itemsets.

Hence, one can devise an iterative algorithm that computes the support

counts of itemsets at level k−1 using the support counts of itemsets at level *k*,

starting from the level kmax, where kmax is the size of the largest closed

itemset.

Even though closed itemsets provide a compact representation of the support

counts of all itemsets, they can still be exponentially large in number.

Moreover, for most practical applications, we only need to determine the

support count of all frequent itemsets. In this regard, closed frequent itemsets provide a compact representation of the support counts of all frequent

itemsets, which can be defined as follows.

Definition 5.5. (Closed Frequent

Itemset.)

An itemset is a closed frequent itemset if it is closed and its support is greater

than or equal to *minsup*.

In the previous example, assuming that the support threshold is 40%, {*b*, *c*} is

a closed frequent itemset because its support is 60%. In Figure 5.17, the

closed frequent itemsets are indicated by the shaded nodes.

Algorithms are available to explicitly extract closed frequent itemsets from a

given data set. Interested readers may refer to the Bibliographic Notes at the

end of this chapter for further discussions of these algorithms. We can use

closed frequent itemsets to determine the support counts for all non-closed

frequent itemsets. For example, consider the frequent itemset {*a*, *d*} shown in

Figure 5.17. Because this itemset is not closed, its support count must be

equal to the maximum support count of its immediate supersets, {*a*, *b*, *d*}, {*a*,

*c*, *d*}, and {*a*, *d*, *e*}. Also, since {*a*, *d*} is frequent, we only need to consider

the support of its frequent supersets. In general, the support count of every

non-closed frequent *k*-itemset can be obtained by considering the support of

all its frequent supersets of size k+1. For example, since the only frequent

superset of {*a*, *d*} is {*a*, *c*, *d*}, its support is equal to the support of {*a*, *c*, *d*},

which is 2. Using this methodology, an algorithm can be developed to

compute the support for every frequent itemset. The pseudocode for this

algorithm is shown in Algorithm 5.4. The algorithm proceeds in a specific-togeneral fashion, i.e., from the largest to the smallest frequent itemsets. This is

because, in order to find the support for a non-closed frequent itemset, the

support for all of its supersets must be known. Note that the set of all frequent

itemsets can be easily computed by taking the union of all subsets of frequent

closed itemsets.

Algorithm 5.4 Support counting

using closed frequent itemsets.

1: Let *C *denote the set of closed frequent itemsets and *F *denote the set of all frequent itemsets.

2: Let *k*

max denote the maximum size of closed frequent itemsets

3: *F*

*k*max = {*f*|*f *∈ *C*, |*f*| = *k*max} {Find all frequent itemsets of size

4: for *k *= *k*

max − 1 down to 1 do

5: *F*

*k *= {*f*|*f *∈ *F*, |*f*| = *k*} {Find all frequent itemsets of size

6: for each *f *∈ *F*

*k *do

7: if *f *∉ *C *then

8: f⋅support=max{f′⋅support|f′∈Fk+1, f⊂f′}

9: end if

10: end for

11: end for

To illustrate the advantage of using closed frequent itemsets, consider the

data set shown in Table 5.5, which contains ten transactions and fifteen items.

The items can be divided into three groups: (1) Group *A*, which contains

items a1 through a5; (2) Group *B*, which contains items b1 through b5; and

(3) Group *C*, which contains items c1 through c5. Assuming that the support

threshold is 20%, itemsets involving items from the same group are frequent,

but itemsets involving items from different groups are infrequent. The total

number of frequent itemsets is thus 3×(25−1)=93. However, there are only

four closed frequent itemsets in the data:

({a3, a4}, {a1, a2, a3, a4, a5}, {b1,b2,b3,b4,b5}, and {c1, c2, c3, c4, c5}). It

is often sufficient to present only the closed frequent itemsets to the analysts

instead of the entire set of frequent itemsets.

Table 5.5. A transaction data

set for mining closed itemsets.

TID a1 a2 a3 a4 a5 b1 b2 b3 b4 b5 c1 c2 c3 c4 c5

1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

2 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

3 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

4 0 0 1 1 0 1 1 1 1 1 0 0 0 0 0

5 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0

6 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0

7 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1

8 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1

9 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1

10 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1

Finally, note that all maximal frequent itemsets are closed because none of

the maximal frequent itemsets can have the same support count as their

immediate supersets. The relationships among frequent, closed, closed

frequent, and maximal frequent itemsets are shown in Figure 5.18.

Figure 5.18.

Relationships among frequent, closed, closed frequent, and

maximal frequent itemsets.

5.5 Alternative Methods for

Generating Frequent Itemsets*

*Apriori *is one of the earliest algorithms to have successfully addressed the

combinatorial explosion of frequent itemset generation. It achieves this by

applying the *Apriori *principle to prune the exponential search space. Despite

its significant performance improvement, the algorithm still incurs

considerable I/O overhead since it requires making several passes over the

transaction data set. In addition, as noted in Section 5.2.5, the performance of

the *Apriori *algorithm may degrade significantly for dense data sets because

of the increasing width of transactions. Several alternative methods have been

developed to overcome these limitations and improve upon the efficiency of

the *Apriori *algorithm. The following is a high-level description of these

methods.

Traversal of Itemset Lattice

A search for frequent itemsets can be conceptually viewed as a traversal on

the itemset lattice shown in Figure 5.1. The search strategy employed by an

algorithm dictates how the lattice structure is traversed during the frequent

itemset generation process. Some search strategies are better than others,

depending on the configuration of frequent itemsets in the lattice. An

overview of these strategies is presented next.

General-to-Specific versus Specific-to-General: The *Apriori *algorithm

uses a general-to-specific search strategy, where pairs of frequent (k−1)-

itemsets are merged to obtain candidate *k*-itemsets. This general-tospecific search strategy is effective, provided the maximum length of a

frequent itemset is not too long. The configuration of frequent itemsets

that works best with this strategy is shown in Figure 5.19(a), where the

darker nodes represent infrequent itemsets. Alternatively, a specifictogeneral search strategy looks for more specific frequent itemsets first,

before finding the more general frequent itemsets. This strategy is useful

to discover maximal frequent itemsets in dense transactions, where the

frequent itemset border is located near the bottom of the lattice, as

shown in Figure 5.19(b). The *Apriori *principle can be applied to prune

all subsets of maximal frequent itemsets. Specifically, if a candidate *k*itemset is maximal frequent, we do not have to examine any of its

subsets of size k−1. However, if the candidate *k*-itemset is infrequent,

we need to check all of its k−1 subsets in the next iteration. Another

approach is to combine both general-to-specific and specific-to-general

search strategies. This bidirectional approach requires more space to

store the candidate itemsets, but it can help to rapidly identify the

frequent itemset border, given the configuration shown in Figure

5.19(c).

Figure 5.19.

General-to-specific, specific-to-general, and bidirectional

search.

Figure 5.19. Full Alternative Text

Equivalence Classes: Another way to envision the traversal is to first

partition the lattice into disjoint groups of nodes (or equivalence

classes). A frequent itemset generation algorithm searches for frequent

itemsets within a particular equivalence class first before moving to

another equivalence class. As an example, the level-wise strategy used

in the *Apriori *algorithm can be considered to be partitioning the lattice

on the basis of itemset sizes; i.e., the algorithm discovers all frequent 1-

itemsets first before proceeding to larger-sized itemsets. Equivalence

classes can also be defined according to the prefix or suffix labels of an

itemset. In this case, two itemsets belong to the same equivalence class

if they share a common prefix or suffix of length *k*. In the prefix-based

approach, the algorithm can search for frequent itemsets starting with

the prefix *a *before looking for those starting with prefixes *b*, *c*, and so

on. Both prefix-based and suffix-based equivalence classes can be

demonstrated using the tree-like structure shown in Figure 5.20.

Figure 5.20.

Equivalence classes based on the prefix and suffix labels of

itemsets.

Figure 5.20. Full Alternative Text

Breadth-First versus Depth-First: The *Apriori *algorithm traverses the

lattice in a breadth-first manner, as shown in Figure 5.21(a). It first

discovers all the frequent 1-itemsets, followed by the frequent 2-

itemsets, and so on, until no new frequent itemsets are generated. The

itemset lattice can also be traversed in a depth-first manner, as shown in

Figures 5.21(b) and 5.22. The algorithm can start from, say, node *a *in

Figure 5.22, and count its support to determine whether it is frequent. If

so, the algorithm progressively expands the next level of nodes, i.e., *ab*,

*abc*, and so on, until an infrequent node is reached, say, *abcd*. It then

backtracks to another branch, say, *abce*, and continues the search from

there.

Figure 5.21.

Breadth-first and depth-first traversals.

Figure 5.21. Full Alternative Text

Figure 5.22.

Generating candidate itemsets using the depth-first approach.

Figure 5.22. Full Alternative Text

The depth-first approach is often used by algorithms designed to find

maximal frequent itemsets. This approach allows the frequent itemset

border to be detected more quickly than using a breadth-first approach.

Once a maximal frequent itemset is found, substantial pruning can be

performed on its subsets. For example, if the node *bcde *shown in Figure

5.22 is maximal frequent, then the algorithm does not have to visit the

subtrees rooted at *bd*, *be*, *c*, *d*, and *e *because they will not contain any

maximal frequent itemsets. However, if *abc *is maximal frequent, only

the nodes such as *ac *and *bc *are not maximal frequent (but the subtrees

of *ac *and *bc *may still contain maximal frequent itemsets). The depthfirst approach also allows a different kind of pruning based on the

support of itemsets. For example, suppose the support for {*a*, *b*, *c*} is

identical to the support for {*a*, *b*}. The subtrees rooted at *abd *and *abe*

can be skipped because they are guaranteed not to have any maximal

frequent itemsets. The proof of this is left as an exercise to the readers.

Representation of Transaction Data

Set

There are many ways to represent a transaction data set. The choice of

representation can affect the I/O costs incurred when computing the support

of candidate itemsets. Figure 5.23 shows two different ways of representing

market basket transactions. The representation on the left is called a

horizontal data layout, which is adopted by many association rule mining

algorithms, including *Apriori*. Another possibility is to store the list of

transaction identifiers (TID-list) associated with each item. Such a

representation is known as the vertical data layout. The support for each

candidate itemset is obtained by intersecting the TID-lists of its subset items.

The length of the TID-lists shrinks as we progress to larger sized itemsets.

However, one problem with this approach is that the initial set of TID-lists

might be too large to fit into main memory, thus requiring more sophisticated

techniques to compress the TID-lists. We describe another effective approach

to represent the data in the next Section.

Figure 5.23.

Horizontal and vertical data format.

Horizontal Data Layout

Figure 5.23. Full Alternative Text

5.6 FP-Growth Algorithm*

This Section presents an alternative algorithm called FP-growth that takes a

radically different approach to discovering frequent itemsets. The algorithm

does not subscribe to the generate-and-test paradigm of *Apriori*. Instead, it

encodes the data set using a compact data structure called an FP-tree and

extracts frequent itemsets directly from this structure. The details of this

approach are presented next.

5.6.1 FP-Tree Representation

An FP-tree is a compressed representation of the input data. It is constructed

by reading the data set one transaction at a time and mapping each transaction

onto a path in the FP-tree. As different transactions can have several items in

common, their paths might overlap. The more the paths overlap with one

another, the more compression we can achieve using the FP-tree structure. If

the size of the FP-tree is small enough to fit into main memory, this will

allow us to extract frequent itemsets directly from the structure in memory

instead of making repeated passes over the data stored on disk.

Figure 5.24 shows a data set that contains ten transactions and five items. The

structures of the FP-tree after reading the first three transactions are also

depicted in the diagram. Each node in the tree contains the label of an item

along with a counter that shows the number of transactions mapped onto the

given path. Initially, the FP-tree contains only the root node represented by

the null symbol. The FP-tree is subsequently extended in the following way:

Figure 5.24.

Construction of an FP-tree.

Figure 5.24. Full Alternative Text

1. The data set is scanned once to determine the support count of each

item. Infrequent items are discarded, while the frequent items are sorted

in decreasing support counts inside every transaction of the data set. For

the data set shown in Figure 5.24, *a *is the most frequent item, followed

by *b*, *c*, *d*, and *e*.

2. The algorithm makes a second pass over the data to construct the FPtree. After reading the first transaction, {*a*, *b*}, the nodes labeled as *a*

and *b *are created. A path is then formed from null →a→b to encode the

transaction. Every node along the path has a frequency count of 1.

3. After reading the second transaction, {*b*, *c*, *d*}, a new set of nodes is

created for items *b*, *c*, and *d*. A path is then formed to represent the

transaction by connecting the nodes null →b→c→d. Every node along

this path also has a frequency count equal to one. Although the first two

transactions have an item in common, which is *b*, their paths are disjoint

because the transactions do not share a common prefix.

4. The third transaction, {*a*, *c*, *d*, *e*}, shares a common prefix item (which

is *a*) with the first transaction. As a result, the path for the third

transaction, null →a→c→d→e, overlaps with the path for the first

transaction, null →a→b. Because of their overlapping path, the

frequency count for node *a *is incremented to two, while the frequency

counts for the newly created nodes, *c*, *d*, and *e*, are equal to one.

5. This process continues until every transaction has been mapped onto one

of the paths given in the FP-tree. The resulting FP-tree after reading all

the transactions is shown at the bottom of Figure 5.24.

The size of an FP-tree is typically smaller than the size of the uncompressed

data because many transactions in market basket data often share a few items

in common. In the best-case scenario, where all the transactions have the

same set of items, the FP-tree contains only a single branch of nodes. The

worst-case scenario happens when every transaction has a unique set of

items. As none of the transactions have any items in common, the size of the

FP-tree is effectively the same as the size of the original data. However, the

physical storage requirement for the FP-tree is higher because it requires

additional space to store pointers between nodes and counters for each item.

The size of an FP-tree also depends on how the items are ordered. The notion

of ordering items in decreasing order of support counts relies on the

possibility that the high support items occur more frequently across all paths

and hence must be used as most commonly occurring prefixes. For example,

if the ordering scheme in the preceding example is reversed, i.e., from lowest

to highest support item, the resulting FP-tree is shown in Figure 5.25. The

tree appears to be denser because the branching factor at the root node has

increased from 2 to 5 and the number of nodes containing the high support

items such as *a *and *b *has increased from 3 to 12. Nevertheless, ordering by

decreasing support counts does not always lead to the smallest tree,

especially when the high support items do not occur frequently together with

the other items. For example, suppose we augment the data set given in

Figure 5.24 with 100 transactions that contain {*e*}, 80 transactions that

contain {*d*}, 60 transactions that contain {*c*}, and 40 transactions that contain

{*b*}. Item *e *is now most frequent, followed by *d*, *c*, *b*, and *a*. With the

augmented transactions, ordering by decreasing support counts will result in

an FP-tree similar to Figure 5.25, while a scheme based on increasing support

counts produces a smaller FP-tree similar to Figure 5.24(iv).

Figure 5.25.

An FP-tree representation for the data set shown in Figure 5.24

with a different item ordering scheme.

Figure 5.25. Full Alternative Text

An FP-tree also contains a list of pointers connecting nodes that have the

same items. These pointers, represented as dashed lines in Figures 5.24 and

5.25, help to facilitate the rapid access of individual items in the tree. We

explain how to use the FP-tree and its corresponding pointers for frequent

itemset generation in the next Section.

5.6.2 Frequent Itemset Generation

in FP-Growth Algorithm

FP-growth is an algorithm that generates frequent itemsets from an FP-tree

by exploring the tree in a bottom-up fashion. Given the example tree shown

in Figure 5.24, the algorithm looks for frequent itemsets ending in *e *first,

followed by *d*, *c*, *b*, and finally, *a*. This bottom-up strategy for finding

frequent itemsets ending with a particular item is equivalent to the suffixbased approach described in Section 5.5. Since every transaction is mapped

onto a path in the FP-tree, we can derive the frequent itemsets ending with a

particular item, say, *e*, by examining only the paths containing node *e*. These

paths can be accessed rapidly using the pointers associated with node *e*. The

extracted paths are shown in Figure 5.26 (a). Similar paths for itemsets

ending in *d*, *c*, *b*, and *a *are shown in Figures 5.26 (b), (c), (d), and (e),

respectively.

Figure 5.26.

Decomposing the frequent itemset generation problem into multiple

subproblems, where each subproblem involves finding frequent

itemsets ending in *e*, *d*, *c*, *b*, and *a*.

Figure 5.26. Full Alternative Text

FP-growth finds all the frequent itemsets ending with a particular suffix by

employing a divide-and-conquer strategy to split the problem into smaller

subproblems. For example, suppose we are interested in finding all frequent

itemsets ending in *e*. To do this, we must first check whether the itemset {*e*}

itself is frequent. If it is frequent, we consider the subproblem of finding

frequent itemsets ending in *de*,followedby *ce*, *be*,and *ae*. In turn, each of

these subproblems are further decomposed into smaller subproblems. By

merging the solutions obtained from the subproblems, all the frequent

itemsets ending in *e *can be found. Finally, the set of all frequent itemsets can

be generated by merging the solutions to the subproblems of finding frequent

itemsets ending in *e*, *d*, *c*, *b*, and *a*. This divide-and-conquer approach is the

key strategy employed by the FP-growth algorithm.

For a more concrete example on how to solve the subproblems, consider the

task of finding frequent itemsets ending with *e*.

1. The first step is to gather all the paths containing node *e*. These initial

paths are called prefix paths and are shown in Figure 5.27(a).

Figure 5.27.

Example of applying the FP-growth algorithm to find frequent

itemsets ending in *e*.

Figure 5.27. Full Alternative Text

2. From the prefix paths shown in Figure 5.27(a), the support count for *e *is

obtained by adding the support counts associated with node *e*. Assuming

that the minimum support count is 2, {*e*} is declared a frequent itemset

because its support count is 3.

3. Because {*e*} is frequent, the algorithm has to solve the subproblems of

finding frequent itemsets ending in *de*, *ce*, *be*,and *ae*. Before solving

these subproblems, it must first convert the prefix paths into a

conditional FP-tree, which is structurally similar to an FP-tree, except

it is used to find frequent itemsets ending with a particular suffix. A

conditional FP-tree is obtained in the following way:

1. First, the support counts along the prefix paths must be updated

because some of the counts include transactions that do not contain

item *e*. For example, the rightmost path shown in Figure 5.27(a),

null → b:2 → c:2 → e:1, includes a transaction {*b*, *c*} that does

not contain item *e*. The counts along the prefix path must therefore

be adjusted to 1 to reflect the actual number of transactions

containing {*b*, *c*, *e*}.

2. The prefix paths are truncated by removing the nodes for *e*. These

nodes can be removed because the support counts along the prefix

paths have been updated to reflect only transactions that contain *e*

and the subproblems of finding frequent itemsets ending in *de*, *ce*,

*be*, and *ae *no longer need information about node *e*.

3. After updating the support counts along the prefix paths, some of

the items may no longer be frequent. For example, the node *b*

appears only once and has a support count equal to 1, which means

that there is only one transaction that contains both *b *and *e*.Item *b*

can be safely ignored from subsequent analysis because all itemsets

ending in *be *must be infrequent.

The conditional FP-tree for *e *is shown in Figure 5.27(b). The tree looks

different than the original prefix paths because the frequency counts

have been updated and the nodes *b *and *e *have been eliminated.

4. FP-growth uses the conditional FP-tree for *e *to solve the subproblems of

finding frequent itemsets ending in *de*, *ce*,and *ae*. To find the frequent

itemsets ending in *de*, the prefix paths for *d *are gathered from the

conditional FP-tree for *e *(Figure 5.27(c)). By adding the frequency

counts associated with node *d*, we obtain the support count for {*d*, *e*}.

Since the support count is equal to 2, {*d*, *e*} is declared a frequent

itemset. Next, the algorithm constructs the conditional FP-tree for *de*

using the approach described in step 3. After updating the support

counts and removing the infrequent item *c*, the conditional FP-tree for

*de *is shown in Figure 5.27(d). Since the conditional FP-tree contains

only one item, *a*, whose support is equal to *minsup*, the algorithm

extracts the frequent itemset {*a*, *d*, *e*} and moves on to the next

subproblem, which is to generate frequent itemsets ending in *ce*. After

processing the prefix paths for *c*, {*c*, *e*} is found to be frequent.

However, the conditional FP-tree for *ce *will have no frequent items and

thus will be eliminated. The algorithm proceeds to solve the next

subproblem and finds {*a*, *e*} to be the only frequent itemset remaining.

This example illustrates the divide-and-conquer approach used in the FPgrowth algorithm. At each recursive step, a conditional FP-tree is constructed

by updating the frequency counts along the prefix paths and removing all

infrequent items. Because the subproblems are disjoint, FP-growth will not

generate any duplicate itemsets. In addition, the counts associated with the

nodes allow the algorithm to perform support counting while generating the

common suffix itemsets.

FP-growth is an interesting algorithm because it illustrates how a compact

representation of the transaction data set helps to efficiently generate frequent

itemsets. In addition, for certain transaction data sets, FP-growth outperforms

the standard *Apriori *algorithm by several orders of magnitude. The run-time

performance of FP-growth depends on the compaction factor of the data set.

If the resulting conditional FP-trees are very bushy (in the worst case, a full

prefix tree), then the performance of the algorithm degrades significantly

because it has to generate a large number of subproblems and merge the

results returned by each subproblem.

5.7 Evaluation of Association

Patterns

Although the *Apriori *principle significantly reduces the exponential search

space of candidate itemsets, association analysis algorithms still have the

potential to generate a large number of patterns. For example, although the

data set shown in Table 5.1 contains only six items, it can produce hundreds

of association rules at particular support and confidence thresholds. As the

size and dimensionality of real commercial databases can be very large, we

can easily end up with thousands or even millions of patterns, many of which

might not be interesting. Identifying the most interesting patterns from the

multitude of all possible ones is not a trivial task because “one person’s trash

might be another person’s treasure.” It is therefore important to establish a set

of well-accepted criteria for evaluating the quality of association patterns.

The first set of criteria can be established through a data-driven approach to

define objective interestingness measures. These measures can be used to

rank patterns—itemsets or rules—and thus provide a straightforward way of

dealing with the enormous number of patterns that are found in a data set.

Some of the measures can also provide statistical information, e.g., itemsets

that involve a set of unrelated items or cover very few transactions are

considered uninteresting because they may capture spurious relationships in

the data and should be eliminated. Examples of objective interestingness

measures include support, confidence, and correlation.

The second set of criteria can be established through subjective arguments. A

pattern is considered subjectively uninteresting unless it reveals unexpected

information about the data or provides useful knowledge that can lead to

profitable actions. For example, the rule {Butter}→{Bread} may not be

interesting, despite having high support and confidence values, because the

relationship represented by the rule might seem rather obvious. On the other

hand, the rule {Diapers}→{Beer} is interesting because the relationship is

quite unexpected and may suggest a new cross-selling opportunity for

retailers. Incorporating subjective knowledge into pattern evaluation is a

difficult task because it requires a considerable amount of prior information

from domain experts. Readers interested in subjective interestingness

measures may refer to resources listed in the bibliography at the end of this

chapter.

5.7.1 Objective Measures of

Interestingness

An objective measure is a data-driven approach for evaluating the quality of

association patterns. It is domain-independent and requires only that the user

specifies a threshold for filtering low-quality patterns. An objective measure

is usually computed based on the frequency counts tabulated in a

contingency table. Table 5.6 shows an example of a contingency table for a

pair of binary variables, *A *and *B*.We use the notation A¯(B¯) to indicate that

*A *(*B*)isabsent from a transaction. Each entry fij in this 2×2 table denotes a

frequency count. For example, f11 is the number of times *A *and *B *appear

together in the same transaction, while f01 is the number of transactions that

contain *B *but not *A*. The row sum f1+ represents the support count for *A*,

while the column sum f+1 represents the support count for *B*. Finally, even

though our discussion focuses mainly on asymmetric binary variables, note

that contingency tables are also applicable to other attribute types such as

symmetric binary, nominal, and ordinal variables.

Table 5.6. A 2-way contingency

table for variables *A *and *B*.

*B *B¯

*A *f11 f10 f1+

A¯ f01 f00 f0+

f+1 f+0 *N*

Limitations of the SupportConfidence Framework

The classical association rule mining formulation relies on the support and

confidence measures to eliminate uninteresting patterns. The drawback of

support, which is described more fully in Section 5.8, is that many potentially

interesting patterns involving low support items might be eliminated by the

support threshold. The drawback of confidence is more subtle and is best

demonstrated with the following example.

Example 5.3.

Suppose we are interested in analyzing the relationship between people who

drink tea and coffee. We may gather information about the beverage

preferences among a group of people and summarize their responses into a

contingency table such as the one shown in Table 5.7.

Table 5.7. Beverage preferences

among a group of 1000 people.

*Coffee *Coffee¯

*Tea *150 50 200

Tea¯ 650 150 800

800 200 1000

The information given in this table can be used to evaluate the association

rule {Tea}→{Coffee}. At first glance, it may appear that people who drink

tea also tend to drink coffee because the rule’s support (15%) and confidence

(75%) values are reasonably high. This argument would have been acceptable

except that the fraction of people who drink coffee, regardless of whether

they drink tea, is 80%, while the fraction of tea drinkers who drink coffee is

only 75%. Thus knowing that a person is a tea drinker actually decreases her

probability of being a coffee drinker from 80% to 75%! The rule {Tea}

→{Coffee} is therefore misleading despite its high confidence value.

Now consider a similar problem where we are interested in analyzing the

relationship between people who drink tea and people who use honey in their

beverage. Table 5.8 summarizes the information gathered over the same

group of people about their preferences for drinking tea and using honey. If

we evaluate the association rule {Tea}→{Honey} using this information, we

will find that the confidence value of this rule is merely 50%, which might be

easily rejected using a reasonable threshold on the confidence value, say

70%. One thus might consider that the preference of a person for drinking tea

has no influence on her preference for using honey. However, the fraction of

people who use honey, regardless of whether they drink tea, is only 12%.

Hence, knowing that a person drinks tea significantly increases her

probability of using honey from 12% to 50%. Further, the fraction of people

who do not drink tea but use honey is only 2.5%! This suggests that there is

definitely some information in the preference of a person of using honey

given that she drinks tea. The rule {Tea}→{Honey} may therefore be falsely

rejected if confidence is used as the evaluation measure.

Table 5.8. Information about

people who drink tea and

people who use honey in their

beverage.

*Honey *Honey¯

*Tea *100 100 200

Tea¯ 20 780 800

120 880 1000

Note that if we take the support of coffee drinkers into account, we would not

be surprised to find that many of the people who drink tea also drink coffee,

since the overall number of coffee drinkers is quite large by itself. What is

more surprising is that the fraction of tea drinkers who drink coffee is

actually less than the overall fraction of people who drink coffee, which

points to an inverse relationship between tea drinkers and coffee drinkers.

Similarly, if we account for the fact that the support of using honey is

inherently small, it is easy to understand that the fraction of tea drinkers who

use honey will naturally be small. Instead, what is important to measure is the

change in the fraction of honey users, given the information that they drink

tea.

The limitations of the confidence measure are well-known and can be

understood from a statistical perspective as follows. The support of a variable

measures the probability of its occurrence, while the support *s*(*A, B*) of a pair

of a variables *A *and *B *measures the probability of the two variables occurring

together. Hence, the joint probability *P *(*A, B*) can be written as

P(A, B)=s(A, B)=f11N.

If we assume *A *and *B *are statistically independent, i.e. there is no relationship

between the occurrences of *A *and *B*, then P(A, B)=P(A)×P(B). Hence, under

the assumption of statistical independence between *A *and *B*, the support

*s*indep(*A, B*) of *A *and *B *can be written as

sindep(A, B)=s(A)×s(B)or equivalently,sindep(A, B)=f1+N×f+1N. (5.4)

If the support between two variables, *s*(*A, B*) is equal to sindep(A, B), then *A*

and *B *can be considered to be unrelated to each other. However, if *s*(*A, B*) is

widely different from sindep(A, B), then *A *and *B *are most likely dependent.

Hence, any deviation of *s*(*A, B*) from s(A)×s(B) can be seen as an indication

of a statistical relationship between *A *and *B*. Since the confidence measure

only considers the deviance of *s*(*A, B*) from *s*(*A*) and not from s(A)×s(B), it

fails to account for the support of the consequent, namely *s*(*B*). This results in

the detection of spurious patterns (e.g., {Tea}→{Coffee}) and the rejection

of truly interesting patterns (e.g., {Tea}→{Honey}), as illustrated in the

previous example.

Various objective measures have been used to capture the deviance of *s*(*A, B*)

from sindep(A, B), that are not susceptible to the limitations of the

confidence measure. Below, we provide a brief description of some of these

measures and discuss some of their properties.

Interest Factor

The interest factor, which is also called as the “lift,” can be defined as

follows:

I(A, B)=s(A, B)s(A)×s(B)=Nf11f1+f+1. (5.5)

Notice that s(A)×s(B)=sindep(A, B). Hence, the interest factor measures the

ratio of the support of a pattern *s*(*A, B*) against its baseline support sindep(*A,*

*B*) computed under the statistical independence assumption. Using Equations

5.5 and 5.4, we can interpret the measure as follows:

I(A, B)={=1,if A and B are independent;>1,if A and B are positively related;

<1,if A and B are negatively related. (5.6)

For the tea-coffee example shown in Table 5.7,I=0.150.2×0.8=0.9375, thus

suggesting a slight negative relationship between tea drinkers and coffee

drinkers. Also, for the tea-honey example shown in Table 5.8,

I=0.10.12×0.2=4.1667, suggesting a strong positive relationship between

people who drink tea and people who use honey in their beverage. We can

thus see that the interest factor is able to detect meaningful patterns in the teacoffee and tea-honey examples. Indeed, the interest factor has a number of

statistical advantages over the confidence measure that make it a suitable

measure for analyzing statistical independence between variables.

Piatesky-Shapiro (PS) Measure

Instead of computing the ratio between *s*(*A, B*) and sindep(A, B)=s(A)×s(B),

the *PS *measure considers the difference between *s*(*A, B*) and s(A)×s(B) as

follows.

PS=s(A, B)−s(A)×s(B)=f11N−f1+f+1N2 (5.7)

The PS value is 0 when *A *and *B *are mutually independent of each other.

Otherwise, PS>0 when there is a positive relationship between the two

variables, and PS<0 when there is a negative relationship.

Correlation Analysis

Correlation analysis is one of the most popular techniques for analyzing

relationships between a pair of variables. For continuous variables,

correlation is defined using Pearson’s correlation coefficient (see Equation

2.10 on page 83). For binary variables, correlation can be measured using the

ϕ-coefficient, which is defined as

ϕ=f11f00−f01f10f1+f+1f0+f+0. (5.8)

If we rearrange the terms in 5.8, we can show that the ϕ-coefficient can be

rewritten in terms of the support measures of *A*, *B*, and {*A*, *B*} as follows:

ϕ=s(A, B)−s(A)×s(B)s(A)×(1−s(A))×s(B)×(1−s(B)). (5.9)

Note that the numerator in the above equation is identical to the *PS *measure.

Hence, the ϕ-coefficient can be understood as a normalized version of the *PS*

measure, where that the value of the ϕ-coefficient ranges from −1 to +1.

From a statistical viewpoint, the correlation captures the normalized

difference between *s*(*A, B*) and sindep(*A, B*). A correlation value of 0 means

no relationship, while a value of +1 suggests a perfect positive relationship

and a value of −1 suggests a perfect negative relationship. The correlation

measure has a statistical meaning and hence is widely used to evaluate the

strength of statistical independence among variables. For instance, the

correlation between tea and coffee drinkers in Table 5.7 is −0.0625 which is

slightly less than 0. On the other hand, the correlation between people who

drink tea and people who use honey in Table 5.8 is 0.5847, suggesting a

positive relationship.

IS Measure

*IS *is an alternative measure for capturing the relationship between *s*(*A, B*) and

s(A)×s(B). The IS measure is defined as follows:

IS(A, B)=I(A, B)×s(A, B)=s(A, B)s(A)s(B)=f11f1+f+1. (5.10)

Although the definition of *IS *looks quite similar to the interest factor, they

share some interesting differences. Since *IS *is the geometric mean between

the interest factor and the support of a pattern, *IS *is large when both the

interest factor and support are large. Hence, if the interest factor of two

patterns are identical, the *IS *has a preference of selecting the pattern with

higher support. It is also possible to show that *IS *is mathematically equivalent

to the cosine measure for binary variables (see Equation 2.6 on page 81). The

value of *IS *thus varies from 0 to 1, where an *IS *value of 0 corresponds to no

co-occurrence of the two variables, while an *IS *value of 1 denotes perfect

relationship, since they occur in exactly the same transactions. For the teacoffee example shown in Table 5.7, the value of *IS *is equal to 0.375, while

the value of *IS *for the tea-honey example in Table 5.8 is 0.6455. The *IS*

measure thus gives a higher value for the {Tea}→{Honey} rule than the

{Tea}→{Coffee} rule, which is consistent with our understanding of the two

rules.

Alternative Objective

Interestingness Measures

Note that all of the measures defined in the previous Section use different

techniques to capture the deviance between *s*(*A, B*) and sindep=s(A)×s(B).

Some measures use the ratio between *s*(*A, B*) and sindep(*A, B*), e.g., the

interest factor and *IS*, while some other measures consider the difference

between the two, e.g., the *PS *and the ϕ-coefficient. Some measures are

bounded in a particular range, e.g., the *IS *and the ϕ-coefficient, while others

are unbounded and do not have a defined maximum or minimum value, e.g.,

the Interest Factor. Because of such differences, these measures behave

differently when applied to different types of patterns. Indeed, the measures

defined above are not exhaustive and there exist many alternative measures

for capturing different properties of relationships between pairs of binary

variables. Table 5.9 provides the definitions for some of these measures in

terms of the frequency counts of a 2×2 contingency table.

Table 5.9. Examples of

objective measures for the

itemset {*A*, *B*}.

Measure

(Symbol) Definition

Correlation

(ϕ) Nf11−f1+f+1f1+f+1f0+f+0

Odds ratio

(α) (f11f00)/(f10f01)

Kappa (κ) Nf11+Nf00−f1+f+1−f0+f+0N2−f1+f+1−f0+f+0

Interest (*I*) (Nf11)/(f1+f+1)

Cosine (*IS*) (f11)/(f1+f+1)

PiatetskyShapiro

(*PS*)

f11N−f1+f+1N2

Collective

strength (*S*) f11+f00f1+f+1+f0+f+0×N−f1+f+1−f0+f+0N−f11−f00

Jaccard (ζ) f11/(f1++f+1−f11)

Allconfidence

(*h*)

min[f11f1+, f11f+1]

Consistency among Objective

Measures

Given the wide variety of measures available, it is reasonable to question

whether the measures can produce similar ordering results when applied to a

set of association patterns. If the measures are consistent, then we can choose

any one of them as our evaluation metric. Otherwise, it is important to

understand what their differences are in order to determine which measure is

more suitable for analyzing certain types of patterns.

Suppose the measures defined in Table 5.9 are applied to rank the ten

contingency tables shown in Table 5.10. These contingency tables are chosen

to illustrate the differences among the existing measures. The ordering

produced by these measures is shown in Table 5.11 (with 1 as the most

interesting and 10 as the least interesting table). Although some of the

measures appear to be consistent with each other, others produce quite

different ordering results. For example, the rankings given by the ϕ-

coefficient agrees mostly with those provided by κ and collective strength,

but are quite different than the rankings produced by interest factor.

Furthermore, a contingency table such as E10 is ranked lowest according to

the ϕ-coefficient, but highest according to interest factor.

Table 5.10. Example of

contingency tables.

Example f11 f10 f01 f00

E1 8123 83 424 1370

E2 8330 2 622 1046

E3 3954 3080 5 2961

E4 2886 1363 1320 4431

E5 1500 2000 500 6000

E6 4000 2000 1000 3000

E7 9481 298 127 94

E8 4000 2000 2000 2000

E9 7450 2483 4 63

E10 61 2483 4 7452

Table 5.11. Rankings of

contingency tables using the

measures given in Table 5.9.

ϕ α κ *I IS PS S *ζ *h*

E1 1 3 1 6 2 2 1 2 2

E2 2 1 2 7 3 5 2 3 3

E3 3 2 4 4 5 1 3 6 8

E4 4 8 3 3 7 3 4 7 5

E5 5 7 6 2 9 6 6 9 9

E6 6 9 5 5 6 4 5 5 7

E7 7 6 7 9 1 8 7 1 1

E8 8 10 8 8 8 7 8 8 7

E9 9 4 9 10 4 9 9 4 4

E10 10 5 10 1 10 10 10 10 10

Properties of Objective Measures

The results shown in Table 5.11 suggest that the measures greatly differ from

each other and can provide conflicting information about the quality of a

pattern. In fact, no measure is universally best for all applications. In the

following, we describe some properties of the measures that play an

important role in determining if they are suited for a certain application.

Inversion Property

Consider the binary vectors shown in Figure 5.28. The 0/1 value in each

column vector indicates whether a transaction (row) contains a particular item

(column). For example, the vector *A *indicates that the item appears in the

first and last transactions, whereas the vector *B *indicates that the item is

contained only in the fifth transaction. The vectors A¯ and B¯ are the

inverted versions of *A *and *B*, i.e., their 1 values have been changed to 0

values (absence to presence) and vice versa. Applying this transformation to

a binary vector is called inversion. If a measure is invariant under the

inversion operation, then its value for the vector pair {A¯, B¯} should be

identical to its value for {*A*, *B*}. The inversion property of a measure can be

tested as follows.

Figure 5.28.

Effect of the inversion operation. The vectors A¯ and E¯ are

inversions of vectors *A *and *B*, respectively.

Figure 5.28. Full Alternative Text

Definition 5.6. (Inversion Property.)

An objective measure *M *is invariant under the inversion operation if its value

remains the same when exchanging the frequency counts f11 with f00 and

f10 with f01.

Measures that are invariant to the inversion property include the correlation

(ϕ-coefficient), odds ratio, κ, and collective strength. These measures are

especially useful in scenarios where the presence (1’s) of a variable is as

important as its absence (0’s). For example, if we compare two sets of

answers to a series of true/false questions where 0’s (true) and 1’s (false) are

equally meaningful, we should use a measure that gives equal importance to

occurrences of 0–0’s and 1–1’s. For the vectors shown in Figure 5.28, the ϕ-

coefficient is equal to -0.1667 regardless of whether we consider the pair {*A*,

*B*} or pair {A¯, B¯}. Similarly, the odds ratio for both pairs of vectors is

equal to a constant value of 0. Note that even though the ϕ-coefficient and the

odds ratio are invariant to inversion, they can still show different results, as

will be shown later.

Measures that do not remain invariant under the inversion operation include

the interest factor and the *IS *measure. For example, the *IS *value for the pair

{A¯, B¯} in Figure 5.28 is 0.825, which reflects the fact that the 1’s in A¯

and B¯ occur frequently together. However, the *IS *value of its inverted pair

{*A*, *B*} is equal to 0, since *A *and *B *do not have any co-occurrence of 1’s. For

asymmetric binary variables, e.g., the occurrence of words in documents, this

is indeed the desired behavior. A desired similarity measure between

asymmetric variables should not be invariant to inversion, since for these

variables, it is more meaningful to capture relationships based on the

presence of a variable rather than its absence. On the other hand, if we are

dealing with symmetric binary variables where the relationships between 0’s

and 1’s are equally meaningful, care should be taken to ensure that the chosen

measure is invariant to inversion.

Although the values of the interest factor and *IS *change with the inversion

operation, they can still be inconsistent with each other. To illustrate this,

consider Table 5.12, which shows the contingency tables for two pairs of

variables, {*p*, *q*} and {*r*, *s*}. Note that *r *and *s *are inverted transformations of

*p *and *q*, respectively, where the roles of 0’s and 1’s have just been reversed.

The interest factor for {*p*, *q*} is 1.02 and for {*r*, *s*} is 4.08, which means that

the interest factor finds the inverted pair {*r*, *s*} more related than the original

pair {*p*, *q*}. On the contrary, the *IS *value decreases upon inversion from

0.9346 for {*p*, *q*} to 0.286 for {*r*, *s*}, suggesting quite an opposite trend to

that of the interest factor. Even though these measures conflict with each

other for this example, they may be the right choice of measure in different

applications.

Table 5.12. Contingency tables

for the pairs {*p*,*q*} and {*r*,*s*}.

*p *p¯

*q *880 50 930

q¯ 50 20 70

930 70 1000

*r *r¯

*s *20 50 70

s¯ 50 880 930

70 930 1000

Scaling Property

Table 5.13 shows two contingency tables for gender and the grades achieved

by students enrolled in a particular course. These tables can be used to study

the relationship between gender and performance in the course. The second

contingency table has data from the same population but has twice as many

males and three times as many females. The actual number of males or

females can depend upon the samples available for study, but the relationship

between gender and grade should not change just because of differences in

sample sizes. Similarly, if the number of students with high and low grades

are changed in a new study, a measure of association between gender and

grades should remain unchanged. Hence, we need a measure that is invariant

to scaling of rows or columns. The process of multiplying a row or column of

a contingency table by a constant value is called a row or column scaling

operation. A measure that is invariant to scaling does not change its value

after any row or column scaling operation.

Table 5.13. The gradegender example. (a)

Sample data of size 100.

Male Female

High 30 20 50

Low 40 10 50

70 30 100

(b) Sample data of size 230.

Male Female

High 60 60 120

Low 80 30 110

140 90 230

Definition 5.7. (Scaling Invariance

Property.)

Let *T *be a contingency table with frequency counts [f11; f10; f01; f00]. Let T

′ be the transformed a contingency table with scaled frequency counts

[k1k3f11; k2k3f10; k1k4f01; k2k4f00], where k1, k2, k3, k4 are positive

constants used to scale the two rows and the two columns of *T *. An objective

measure *M *is invariant under the row/column scaling operation if M(T)=M(T

′).

Note that the use of the term ‘scaling’ here should not be confused with the

scaling operation for continuous variables introduced in Chapter 2 on page

23, where all the values of a variable were being multiplied by a constant

factor, instead of scaling a row or column of a contingency table.

Scaling of rows and columns in contingency tables occurs in multiple ways in

different applications. For example, if we are measuring the effect of a

particular medical procedure on two sets of subjects, *healthy *and *diseased*,

the ratio of healthy and diseased subjects can widely vary across different

studies involving different groups of participants. Further, the fraction of

healthy and diseased subjects chosen for a controlled study can be quite

different from the true fraction observed in the complete population. These

differences can result in a row or column scaling in the contingency tables for

different populations of subjects. In general, the frequencies of items in a

contingency table closely depends on the sample of transactions used to

generate the table. Any change in the sampling procedure may affect a row or

column scaling transformation. A measure that is expected to be invariant to

differences in the sampling procedure must not change with row or column

scaling.

Of all the measures introduced in Table 5.9, only the odds ratio (α) is

invariant to row and column scaling operations. For example, the value of

odds ratio for both the tables in Table 5.13 is equal to 0.375. All other

measures such as the ϕ-coefficient, κ, *IS*, interest factor, and collective

strength (*S*) change their values when the rows and columns of the

contingency table are rescaled. Indeed, the odds ratio is a preferred choice of

measure in the medical domain, where it is important to find relationships

that do not change with differences in the population sample chosen for a

study.

Null Addition Property

Suppose we are interested in analyzing the relationship between a pair of

words, such as data and mining, in a set of documents. If a collection of

articles about ice fishing is added to the data set, should the association

between data and mining be affected? This process of adding unrelated data

(in this case, documents) to a given data set is known as the null addition

operation.

Definition 5.8. (Null Addition

Property.)

An objective measure *M *is invariant under the null addition operation if it is

not affected by increasing f00, while all other frequencies in the contingency

table stay the same.

For applications such as document analysis or market basket analysis, we

would like to use a measure that remains invariant under the null addition

operation. Otherwise, the relationship between words can be made to change

simply by adding enough documents that do not contain both words!

Examples of measures that satisfy this property include cosine (*IS*) and

Jaccard (ξ) measures, while those that violate this property include interest

factor, *PS*, odds ratio, and the ϕ-coefficient.

To demonstrate the effect of null addition, consider the two contingency

tables T1 and T2 shown in Table 5.14. Table T2 has been obtained from T1

by adding 1000 extra transactions with both *A *and *B *absent. This operation

only affects the f00 entry of Table T2, which has increased from 100 to 1100,

whereas all the other frequencies in the table (f11, f10, and f01) remain the

same. Since *IS *is invariant to null addition, it gives a constant value of 0.875

to both the tables. However, the addition of 1000 extra transactions with

occurrences of 0–0’s changes the value of interest factor from 0.972 for T1

(denoting a slightly negative correlation) to 1.944 for T2 (positive

correlation). Similarly, the value of odds ratio increases from 7 for T1 to 77

for T2. Hence, when the interest factor or odds ratio are used as the

association measure, the relationships between variables changes by the

addition of null transactions where both the variables are absent. In contrast,

the *IS *measure is invariant to null addition, since it considers two variables to

be related only if they frequently occur together. Indeed, the *IS *measure

(cosine measure) is widely used to measure similarity among documents,

which is expected to depend only on the joint occurrences (1’s) of words in

documents, but not their absences (0’s).

Table 5.14. An example

demonstrating the effect of

null addition.

(a) Table T1.

*B *B¯

*A *700 100 800

A¯ 100 100 200

800 200 1000

(b) Table T2.

*B *B¯

*A *700 100 800

A¯ 10 1100 1200

800 1200 2000

Table 5.15 provides a summary of properties for the measures defined in

Table 5.9. Even though this list of properties is not exhaustive, it can serve as

a useful guide for selecting the right choice of measure for an application.

Ideally, if we know the specific requirements of a certain application, we can

ensure that the selected measure shows properties that adhere to those

requirements. For example, if we are dealing with asymmetric variables, we

would prefer to use a measure that is not invariant to null addition or

inversion. On the other hand, if we require the measure to remain invariant to

changes in the sample size, we would like to use a measure that does not

change with scaling.

Table 5.15. Properties of

symmetric measures.

Symbol Measure Inversion Null Addition Scaling

ϕ ϕ-coefficient Yes No No

α odds ratio Yes No Yes

κ Cohen’s Yes No No

*I *Interest No No No

*IS *Cosine No Yes No

*PS *Piatetsky-Shapiro’s Yes No No

*S *Collective strength Yes No No

ζ Jaccard No Yes No

*h *All-confidence No Yes No

*s *Support No No No

Asymmetric Interestingness

Measures

Note that in the discussion so far, we have only considered measures that do

not change their value when the order of the variables are reversed. More

specifically, if *M *is a measure and *A *and *B *are two variables, then *M*(*A, B*) is

equal to *M*(*B*, *A*) if the order of the variables does not matter. Such measures

are called symmetric. On the other hand, measures that depend on the order

of variables (M(A, B)≠M(B, A)) are called asymmetric measures. For

example, the interest factor is a symmetric measure because its value is

identical for the rules A→B and B→A. In contrast, confidence is an

asymmetric measure since the confidence for A→B and B→A may not be

the same. Note that the use of the term ‘asymmetric’ to describe a particular

type of measure of relationship—one in which the order of the variables is

important—should not be confused with the use of ‘asymmetric’ to describe a

binary variable for which only 1’s are important. Asymmetric measures are

more suitable for analyzing association rules, since the items in a rule do

have a specific order. Even though we only considered symmetric measures

to discuss the different properties of association measures, the above

discussion is also relevant for the asymmetric measures. See Bibliographic

Notes for more information about different kinds of asymmetric measures

and their properties.

5.7.2 Measures beyond Pairs of

Binary Variables

The measures shown in Table 5.9 are defined for pairs of binary variables

(e.g., 2-itemsets or association rules). However, many of them, such as

support and all-confidence, are also applicable to larger-sized itemsets. Other

measures, such as interest factor, *IS*, *PS*, and Jaccard coefficient, can be

extended to more than two variables using the frequency tables tabulated in a

multidimensional contingency table. An example of a three-dimensional

contingency table for *a*, *b*, and *c *is shown in Table 5.16. Each entry fijk in

this table represents the number of transactions that contain a particular

combination of items *a*, *b*, and *c*. For example, f101 is the number of

transactions that contain *a *and *c*, but not *b*. On the other hand, a marginal

frequency such as f1+1 is the number of transactions that contain *a *and *c*,

irrespective of whether *b *is present in the transaction.

Table 5.16. Example of a threedimensional contingency table.

*c b *b¯

*a *f111 f101 f1+1

a¯ f011 f001 f0+1

f+11 f+01 f++1

*c b *b¯

*a *f110 f100 f1+0

a¯ f010 f000 f0+0

f+10 f+00 f++0

Given a *k*-itemset {i1, i2, …, ik}, the condition for statistical independence

can be stated as follows:

fi1i2…ik=fi1+…+×f+i2…+×…×f++…ikNk−1. (5.11)

With this definition, we can extend objective measures such as interest factor

and *PS*, which are based on deviations from statistical independence, to more

than two variables:

I=Nk−1×fi1i2…ikfi1+…+×f+i2…+×…×f++…ikPS=fi1i2…ikN−fi1+

…+×f+i2…+×…×f++…ikNk

Another approach is to define the objective measure as the maximum,

minimum, or average value for the associations between pairs of items in a

pattern. For example, given a *k*-itemset X={i1, i2, …, ik}, we may define the

ϕ-coefficient for *X *as the average ϕ-coefficient between every pair of items

(ip, iq) in *X*. However, because the measure considers only pairwise

associations, it may not capture all the underlying relationships within a

pattern. Also, care should be taken in using such alternate measures for more

than two variables, since they may not always show the anti-monotone

property in the same way as the support measure, making them unsuitable for

mining patterns using the *Apriori *principle.

Analysis of multidimensional contingency tables is more complicated

because of the presence of partial associations in the data. For example, some

associations may appear or disappear when conditioned upon the value of

certain variables. This problem is known as Simpson’s paradox and is

described in Section 5.7.3. More sophisticated statistical techniques are

available to analyze such relationships, e.g., loglinear models, but these

techniques are beyond the scope of this book.

5.7.3 Simpson’s Paradox

It is important to exercise caution when interpreting the association between

variables because the observed relationship may be influenced by the

presence of other confounding factors, i.e., hidden variables that are not

included in the analysis. In some cases, the hidden variables may cause the

observed relationship between a pair of variables to disappear or reverse its

direction, a phenomenon that is known as Simpson’s paradox. We illustrate

the nature of this paradox with the following example.

Consider the relationship between the sale of high-definition televisions

(HDTV) and exercise machines, as shown in Table 5.17. The rule

{HDTV=Yes}→{Exercise machine=Yes} has a confidence of 99/180=55%

and the rule {HDTV=No}→{Exercise machine=Yes} has a confidence of

54/120=45%. Together, these rules suggest that customers who buy highdefinition televisions are more likely to buy exercise machines than those

who do not buy high-definition televisions.

Table 5.17. A two-way

contingency table between the

sale of high-definition television

and exercise machine.

Buy

HDTV

Buy Exercise Machine

Yes No

Yes 99 81 180

No 54 66 120

153 147 300

However, a deeper analysis reveals that the sales of these items depend on

whether the customer is a college student or a working adult. Table 5.18

summarizes the relationship between the sale of HDTVs and exercise

machines among college students and working adults. Notice that the support

counts given in the table for college students and working adults sum up to

the frequencies shown in Table 5.17. Furthermore, there are more working

adults than college students who buy these items. For college students:

Table 5.18. Example of a threeway contingency table.

Customer

Group

Buy

HDTV

Buy Exercise Machine

Total

Yes No

College Students Yes 1 9 10

No 4 30 34

Working Adult Yes 98 72 170

No 50 36 86

c({HDTV=Yes}→{Exercise machine=Yes})=1/10=10%,c({HDTV=No}

→{Exercise machine=Yes})=4/34=11.8%.

while for working adults:

c({HDTV=Yes}→{Exercise machine=Yes})=98/170=57.7%,c({HDTV=No}

→{Exercise machine=Yes})=50/86=58.1%.

The rules suggest that, for each group, customers who do not buy highdefinition televisions are more likely to buy exercise machines, which

contradicts the previous conclusion when data from the two customer groups

are pooled together. Even if alternative measures such as correlation, odds

ratio, or interest are applied, we still find that the sale of HDTV and exercise

machine is positively related in the combined data but is negatively related in

the stratified data (see Exercise 21 on page 449). The reversal in the direction

of association is known as Simpson’s paradox.

The paradox can be explained in the following way. First, notice that most

customers who buy HDTVs are working adults. This is reflected in the high

confidence of the rule {HDTV=Yes}→{Working Adult}(170/180=94.4%).

Second, the high confidence of the rule {Exercise machine=Yes}

→{Working Adult}(148/153=96.7%) suggests that most customers who buy

exercise machines are also working adults. Since working adults form the

largest fraction of customers for both HDTVs and exercise machines, they

both look related and the rule {HDTV=Yes}→{Exercise machine=Yes}

turns out to be stronger in the combined data than what it would have been if

the data is stratified. Hence, customer group acts as a *hidden *variable that

affects both the fraction of customers who buy HDTVs and those who buy

exercise machines. If we factor out the effect of the hidden variable by

stratifying the data, we see that the relationship between buying HDTVs and

buying exercise machines is not direct, but shows up as an indirect

consequence of the effect of the hidden variable.

The Simpson’s paradox can also be illustrated mathematically as follows.

Suppose

a/b<c/dandp/q<r/s,

where *a*/*b *and *p*/*q *may represent the confidence of the rule A→B in two

different strata, while *c*/*d *and *r*/*s *may represent the confidence of the rule

A¯→B in the two strata. When the data is pooled together, the confidence

values of the rules in the combined data are (a+p)/(b+q) and (c+r)/(d+s),

respectively. Simpson’s paradox occurs when

a+pb+q>c+rd+s,

thus leading to the wrong conclusion about the relationship between the

variables. The lesson here is that proper stratification is needed to avoid

generating spurious patterns resulting from Simpson’s paradox. For example,

market basket data from a major supermarket chain should be stratified

according to store locations, while medical records from various patients

should be stratified according to confounding factors such as age and gender.

5.8 Effect of Skewed Support

Distribution

The performances of many association analysis algorithms are influenced by

properties of their input data. For example, the computational complexity of

the *Apriori *algorithm depends on properties such as the number of items in

the data, the average transaction width, and the support threshold used. This

Section examines another important property that has significant influence on

the performance of association analysis algorithms as well as the quality of

extracted patterns. More specifically, we focus on data sets with skewed

support distributions, where most of the items have relatively low to

moderate frequencies, but a small number of them have very high

frequencies.

Figure 5.29.

A transaction data set containing three items, *p*, *q*, and *r*, where *p *is

a high support item and *q *and *r *are low support items.

Figure 5.29shows an illustrative example of a data set that has a skewed

support distribution of its items. While *p *has a high support of 83.3% in the

data, *q *and *r *are low-support items with a support of 16.7%. Despite their

low support, *q *and *r *always occur together in the limited number of

transactions that they appear and hence are strongly related. A pattern mining

algorithm therefore should report {*q*, *r*} as interesting.

However, note that choosing the right support threshold for mining item-sets

such as {*q*, *r*} can be quite tricky. If we set the threshold too high (e.g., 20%),

then we may miss many interesting patterns involving low-support items

such as {*q*, *r*}. Conversely, setting the support threshold too low can be

detrimental to the pattern mining process for the following reasons. First, the

computational and memory requirements of existing association analysis

algorithms increase considerably with low support thresholds. Second, the

number of extracted patterns also increases substantially with low support

thresholds, which makes their analysis and interpretation difficult. In

particular, we may extract many spurious patterns that relate a highfrequency item such as *p *to a low-frequency item such as *q*. Such patterns,

which are called cross-support patterns, are likely to be spurious because the

association between *p *and *q *is largely influenced by the frequent occurrence

of *p *instead of the joint occurrence of *p *and *q *together. Because the support

of {*p*, *q*} is quite close to the support of {*q*, *r*}, we may easily select {*p*, *q*} if

we set the support threshold low enough to include {*q*, *r*}.

An example of a real data set that exhibits a skewed support distribution is

shown in Figure 5.30. The data, taken from the PUMS (Public Use Microdata

Sample) census data, contains 49,046 records and 2113 asymmetric binary

variables. We shall treat the asymmetric binary variables as items and records

as transactions. While more than 80% of the items have support less than 1%,

a handful of them have support greater than 90%. To understand the effect of

skewed support distribution on frequent itemset mining, we divide the items

into three groups, G1, G2, and G3, according to their support levels, as

shown in Table 5.19. We can see that more than 82% of items belong to G1

and have a support less than 1%. In market basket analysis, such low support

items may correspond to expensive products (such as jewelry) that are

seldom bought by customers, but whose patterns are still interesting to

retailers. Patterns involving such low-support items, though meaningful, can

easily be rejected by a frequent pattern mining algorithm with a high support

threshold. On the other hand, setting a low support threshold may result in the

extraction of spurious patterns that relate a high-frequency item in G3 to a

low-frequency item in G1. For example, at a support threshold equal to

0.05%, there are 18,847 frequent pairs involving items from G1 and G3. Out

of these, 93% of them are cross-support patterns; i.e., the patterns contain

items from both G1 and G3.

Figure 5.30.

Support distribution of items in the census data set.

Figure 5.30. Full Alternative Text

Table 5.19. Grouping the items

in the census data set based on

their support values.

Group G1 G2 G3

Support <1% 1%−90% >90%

Number of Items 1735 358 20

This example shows that a large number of weakly related cross-support

patterns can be generated when the support threshold is sufficiently low. Note

that finding interesting patterns in data sets with skewed support distributions

is not just a challenge for the support measure, but similar statements can be

made about many other objective measures discussed in the previous

Sections. Before presenting a methodology for finding interesting patterns

and pruning spurious ones, we formally define the concept of cross-support

patterns.

Definition 5.9. (Cross-support

Pattern.)

Let us define the support ratio, *r*(*X*), of an itemset X={i1, i2, …, ik} as

r(X)=min[s(i1), s(i2), …, s(ik)}max[s(i1), s(i2), …, s(ik)} (5.12)

Given a user-specified threshold hc, an itemset *X *is a cross-support pattern if

r(X)<hc.

Example 5.4.

Suppose the support for milk is 70%, while the support for sugar is 10% and

caviar is 0.04%. Given hc=0.01, the frequent itemset {milk, sugar, caviar} is

a cross-support pattern because its support ratio is

r=min[0.7, 0.1, 0.0004]max[0.7, 0.1, 0.0004]=0.0040.7=0.00058<0.01.

Existing measures such as support and confidence may not be sufficient to

eliminate cross-support patterns. For example, if we assume hc=0.3 for the

data set presented in Figure 5.29, the itemsets {*p*, *q*}, {*p*, *r*}, and {*p*, *q*, *r*} are

cross-support patterns because their support ratios, which are equal to 0.2, are

less than the threshold hc. However, their supports are comparable to that of

{*q*, *r*}, making it difficult to eliminate cross-support patterns without loosing

interesting ones using a support-based pruning strategy. Confidence pruning

also does not help because the confidence of the rules extracted from crosssupport patterns can be very high. For example, the confidence for {q}→{p}

is 80% even though {*p*, *q*} is a cross-support pattern. The fact that the crosssupport pattern can produce a high confidence rule should not come as a

surprise because one of its items (*p*) appears very frequently in the data.

Therefore, *p *is expected to appear in many of the transactions that contain *q*.

Meanwhile, the rule {q}→{r} also has high confidence even though {*q*, *r*} is

not a cross-support pattern. This example demonstrates the difficulty of using

the confidence measure to distinguish between rules extracted from crosssupport patterns and interesting patterns involving strongly connected but

low-support items.

Even though the rule {q}→{p} has very high confidence, notice that the rule

{p}→{q} has very low confidence because most of the transactions that

contain *p *do not contain *q*. In contrast, the rule {r}→{q}, which is derived

from {*q*, *r*}, has very high confidence. This observation suggests that crosssupport patterns can be detected by examining the lowest confidence rule that

can be extracted from a given itemset. An approach for finding the rule with

the lowest confidence given an itemset can be described as follows.

1. Recall the following anti-monotone property of confidence:

conf({i1i2}→{i3, i4, …, ik})≤conf({i1i2i3}→{i4, i5, …, ik}).

This property suggests that confidence never increases as we shift more

items from the left- to the right-hand side of an association rule. Because

of this property, the lowest confidence rule extracted from a frequent

itemset contains only one item on its left-hand side. We denote the set of

all rules with only one item on its left-hand side as R1.

2. Given a frequent itemset {i1, i2, …, ik}, the rule

{ij}→{i1, i2, …, ij−1, ij+1, …,ik}

has the lowest confidence in R1 if s(ij)=max[s(i1), s(i2), …, s(ik)].This

follows directly from the definition of confidence as the ratio between

the rule’s support and the support of the rule antecedent. Hence, the

confidence of a rule will be lowest when the support of the antecedent is

highest.

3. Summarizing the previous points, the lowest confidence attainable from

a frequent itemset {i1, i2, …, ik} is

s({i1, i2, …, ik})max[s(i1), s(i2), …, s(ik)].

This expression is also known as the h-confidence or all-confidence

measure. Because of the anti-monotone property of support, the

numerator of the h-confidence measure is bounded by the minimum

support of any item that appears in the frequent itemset. In other words,

the h-confidence of an itemset X={i1, i2, …, ik} must not exceed the

following expression:

h-confidence(X)≤min[s(i1), s(i2), …, s(ik)]max[s(i1), s(i2), …, s(ik)].

Note that the upper bound of h-confidence in the above equation is exactly

same as support ratio (*r*) given in Equation 5.12. Because the support ratio for

a cross-support pattern is always less than hc, the h-confidence of the pattern

is also guaranteed to be less than hc. Therefore, cross-support patterns can be

eliminated by ensuring that the h-confidence values for the patterns exceed

hc. As a final note, the advantages of using h-confidence go beyond

eliminating cross-support patterns. The measure is also anti-monotone, i.e.,

h-confidence({i1, i2,

EFFAT UNIVERSITY

COLLEGE OF ENGINEERING

DEPARTMENT OF INFORMATION SYSTEMS

1.1. Introduction/background/overview/significance. 5

1.2. Related research/Literature Review.. 5

2.2. Research Question/Hypothesis. 6

Phase 1: Starting the project. 9

Phase 2: Development of an analytical framework for efficient bank marketing.. 9

Phase 3: Development of the efficient classifier. 10

Phase 4: Verification and optimization of the model. 10

6.2. Population and Study Sample. 11

6.3. Sample Size and Selection of Sample. 11

6.8. Data Analysis Strategies. 11

6.9. Research Design and Prototype. 11

6.10. Clients/Stakeholders. 11

- RESULTS. 12
- DISCUSSION… 12
- CONCLUSION… 12
- ETHICS AND HUMAN SUBJECTS ISSUES. 12
- TIMEFRAMES/PROJECT PLAN/MILESTONES. 12

11.1. Time Schedule (Ghantt Chart) 13

11.2. Activity Sequencing.. 14

13.1. Resource requirements. 16

This project focuses on design and development of an intelligent bank marketing system for a better bank marketing. The project will be carried out in four phases with significant milestone at the end of each phase. In the first phase, we will download and prepare the bank marketing data. Then we will search for information to have clear knowledge about the bank marketing problem and its solutions.

The second phase of the project would concentrate on an analytical framework to analyze the performance of different data mining algorithms that would be developed identifying the main components of the bank marketing.

The third phase would concentrate on the design and the implementation of efficient, reliable and robust data mining tool that achieves a better classification accuracy. Moreover, an ensemble of classifiers would be developed to increase classification performance of an efficient bank marketing.

The fourth and final phase would concentrate on preparing and submitting a conference paper. More details on the different phases are as follows:

First of all, we will download and prepare the bank marketing data for the implementation with WEKA data mining tool. Since the data is unbalanced we try to balance the by using SMOTE technique to have an efficient and accurate classification. Then we will check previous studies done on the same field to have clear knowledge about the bank marketing problem and its solutions and applied methods before.

In this phase, efficient, intelligent and robust bank marketing system will be developed. The system will employ the bank marketing data with robust attribute selection. The extracted attributes will be employed by means of the Data mining techniques for intelligent and more efficient bank marketing environment. This platform will serve to design and develop application oriented bank marketing system. The downloaded data can be processed by employing a variety of feature extraction and attribute selection techniques. Then comparison of different feature extraction, attribute selection techniques, and data mining algorithms for bank marketing will be made. The employed feature extraction and attribute selection techniques should be robust and intelligent, so that a reliable bank marketing can be realized. The proposed platform will be used to understand and implement the intelligent system for a Bank Marketing. The finding of robust features and attributes of bank marketing data would be very meaningful for determining important insights into the effects of various parameters on the performance of bank marketing system to enlighten which feature extraction and attribute selection a is the most effective and less time consuming.

A classification problem is referred to as imbalanced when the instances in one or several classes, known as the majority classes, out number the instances of the other classes, called the minority classes. The synthetic minority over-sampling technique (SMOTE) (Chawla et al., 2002) is a well acknowledged over-sampling method. In the SMOTE, instead of mere data oriented duplicating, the positive classis over-sampled by creating synthetic instances in the feature space formed by the positive instances and their K-nearest neighbours.

In this step, feature extraction, attribute selection and data mining algorithms will be implemented as off-line. The feature extraction and attribute selection methods are in the set of data processing tools which extract features from the bank marketing data. Feature extraction and attribute selection are also one of the most important steps in data classification. It is highly effective technique in selection of attributes and is frequently applied to complex, high dimensional, multivariate data. When the features are not appropriate for the given classification problem, obtained performances will be unsatisfactory. In this case, even the classification algorithm is optimally determined for the problem, because of the improper features/attributes; the algorithm cannot generate high performance. Therefore, it is mandatory to find and extract suitable features from the raw data to be able to obtain good classification results. Many feature extraction and attribute selection techniques will be applied. These are CFS Subset Evaluator, and Infogain Attribute Evaluator etc. will be applied.

The marketing data data is of multi-dimensional nature. Therefore, it is difficult to find a robust feature extraction, attribute selection and data mining algorithm for Bank marketing. Data mining algorithms have ability to distinguish different type of data. In this phase the focus will be to make a comparison of different data mining algorithms in the field of Bank marketing. The outcome will be the analysis and choice of most appropriate sets of features extraction, attribute selection and data mining techniques, for the Bank marketing. After applying different feature extraction/attribute selection algorithms, the existing data mining algorithms (such as ANN, k-NN, SVM, decision tree algorithms, etc.) capable of dealing with network data will be applied, implemented and tested. For this purpose, various classification schemes for a particular data classification task will be developed. The aim of using data mining techniques is to make a better Bank marketing. Even though one of the algorithms would produce the best overall performance, ensemble classifier approach, where the idea is to consider more than one classification scheme can give better classification accuracy. Classifier ensembles are multiple classifier systems trained on different data or feature subsets, will be used to get better performance and accuracy.

The fourth and final phase would concentrate on preparing and submitting a conference paper. Here, the performance metrics such as Area under ROC curve, F-measure, kappa statistic and total classification accuracy would be quantified with the help of WEKA software. Obtained results from experimentation would then be used to verify the accuracy, reliability and robustness of the developed models and would provide feedback for improvement and fine-tuning the models in phase 1, phase 2 and phase 3. The obtained results of the ensemble classifiers would be benchmarked with classical single classifiers. Moreover efficient, intelligent and robust methods will be achieved for bank marketing.

Use headings 2 and 3 as appropriate, and use these headings if appropriate.

- Use case Diagrams
- Database Diagrams
- Relation Diagrams

- Give the tables and results of your research project one by one
- Give the explanation on the results.

- Give discussion on the results of your research project
- Give comments on the results.

- Conclude and summarize the results of your research project.

Change the activities and relate it to the topic which is covid-19

Proposed steps schedule is planned like in the table.

Activity | Months | |||||||||||||||||||||||||||||||||||

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | |

ANN Stock Market Prediction | ||||||||||||||||||||||||||||||||||||

Literature Survey | ||||||||||||||||||||||||||||||||||||

Literature Search | ||||||||||||||||||||||||||||||||||||

Literature Review | ||||||||||||||||||||||||||||||||||||

Completed Literature Review | ||||||||||||||||||||||||||||||||||||

Get Stock Market Data | ||||||||||||||||||||||||||||||||||||

Develop ANN | ||||||||||||||||||||||||||||||||||||

Investigate and Evaluate ANN | ||||||||||||||||||||||||||||||||||||

Design ANN | ||||||||||||||||||||||||||||||||||||

Develop and Test ANN | ||||||||||||||||||||||||||||||||||||

Evaluation | ||||||||||||||||||||||||||||||||||||

Train ANN | ||||||||||||||||||||||||||||||||||||

Use Stock Market Models | ||||||||||||||||||||||||||||||||||||

Analysis | ||||||||||||||||||||||||||||||||||||

Review Statistical Tests | ||||||||||||||||||||||||||||||||||||

Analyse and Evaluate | ||||||||||||||||||||||||||||||||||||

Complete Report | ||||||||||||||||||||||||||||||||||||

Project Completed |

*Activity-on-the-node *diagram represents the tasks you are performing in your project as nodes connected by arrows (Dawson, 2005)

Check the related slides and Rubric for the project report details

…, ik})≥h-confidence({i1, i2, …, ik+1}),

and thus can be incorporated directly into the mining algorithm. Furthermore,

h-confidence ensures that the items contained in an itemset are strongly

associated with each other. For example, suppose the h-confidence of an

itemset *X *is 80%. If one of the items in *X *is present in a transaction, there is

at least an 80% chance that the rest of the items in *X *also belong to the same

transaction. Such strongly associated patterns involving low-support items

are called hyperclique patterns.

Definition 5.10. (Hyperclique

Pattern.)

An itemset *X *is a hyperclique pattern if h-confidence(X)>hc, where hc is a

user-specified threshold.

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