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INTRODUCTION TO DATA
MINING
INTRODUCTION TO DATA
MINING
SECOND EDITION
PANG-NING TAN
Michigan State University
MICHAEL STEINBACH
University of Minnesota
ANUJ KARPATNE
University of Minnesota
VIPIN KUMAR
University of Minnesota
330 Hudson Street, NY NY 10013
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Library of Congress Cataloging-in-Publication Data on File
Names: Tan, Pang-Ning, author. | Steinbach, Michael, author. | Karpatne, Anuj,
author. | Kumar, Vipin, 1956- author.
Title: Introduction to Data Mining / Pang-Ning Tan, Michigan State University,
Michael Steinbach, University of Minnesota, Anuj Karpatne, University of
Minnesota, Vipin Kumar, University of Minnesota.
Description: Second edition. | New York, NY : Pearson Education, [2019] |
Includes bibliographical references and index.
Identifiers: LCCN 2017048641 | ISBN 9780133128901 | ISBN 0133128903
Subjects: LCSH: Data mining.
Classification: LCC QA76.9.D343 T35 2019 | DDC 006.3/12–dc23 LC record
available at https://lccn.loc.gov/2017048641
1 18
ISBN-10: 0133128903
ISBN-13: 9780133128901
To our families …
Preface to the Second Edition
Since the first edition, roughly 12 years ago, much has changed in the field of data
analysis. The volume and variety of data being collected continues to increase, as
has the rate (velocity) at which it is being collected and used to make decisions.
Indeed, the term, Big Data, has been used to refer to the massive and diverse data
sets now available. In addition, the term data science has been coined to describe
an emerging area that applies tools and techniques from various fields, such as
data mining, machine learning, statistics, and many others, to extract actionable
insights from data, often big data.
The growth in data has created numerous opportunities for all areas of data
analysis. The most dramatic developments have been in the area of predictive
modeling, across a wide range of application domains. For instance, recent
advances in neural networks, known as deep learning, have shown impressive
results in a number of challenging areas, such as image classification, speech
recognition, as well as text categorization and understanding. While not as
dramatic, other areas, e.g., clustering, association analysis, and anomaly detection
have also continued to advance. This new edition is in response to those advances.
Overview
As with the first edition, the second edition of the book provides a comprehensive
introduction to data mining and is designed to be accessible and useful to students,
instructors, researchers, and professionals. Areas covered include data
preprocessing, predictive modeling, association analysis, cluster analysis, anomaly
detection, and avoiding false discoveries. The goal is to present fundamental
concepts and algorithms for each topic, thus providing the reader with the
necessary background for the application of data mining to real problems. As
before, classification, association analysis and cluster analysis, are each covered in
a pair of chapters. The introductory chapter covers basic concepts, representative
algorithms, and evaluation techniques, while the more following chapter discusses
advanced concepts and algorithms. As before, our objective is to provide the
reader with a sound understanding of the foundations of data mining, while still
covering many important advanced topics. Because of this approach, the book is
useful both as a learning tool and as a reference.
To help readers better understand the concepts that have been presented, we
provide an extensive set of examples, figures, and exercises. The solutions to the
original exercises, which are already circulating on the web, will be made public.
The exercises are mostly unchanged from the last edition, with the exception of
new exercises in the chapter on avoiding false discoveries. New exercises for the
other chapters and their solutions will be available to instructors via the web.
Bibliographic notes are included at the end of each chapter for readers who are
interested in more advanced topics, historically important papers, and recent
trends. These have also been significantly updated. The book also contains a
comprehensive subject and author index.
What is New in the Second Edition?
Some of the most significant improvements in the text have been in the two
chapters on classification. The introductory chapter uses the decision tree classifier
for illustration, but the discussion on many topics—those that apply across all
classification approaches—has been greatly expanded and clarified, including
topics such as overfitting, underfitting, the impact of training size, model complexity,
model selection, and common pitfalls in model evaluation. Almost every section of
the advanced classification chapter has been significantly updated. The material on
Bayesian networks, support vector machines, and artificial neural networks has
been significantly expanded. We have added a separate section on deep networks
to address the current developments in this area. The discussion of evaluation,
which occurs in the section on imbalanced classes, has also been updated and
improved.
The changes in association analysis are more localized. We have completely
reworked the section on the evaluation of association patterns (introductory
chapter), as well as the sections on sequence and graph mining (advanced
chapter). Changes to cluster analysis are also localized. The introductory chapter
added the K-means initialization technique and an updated the discussion of cluster
evaluation. The advanced clustering chapter adds a new section on spectral graph
clustering. Anomaly detection has been greatly revised and expanded. Existing
approaches—statistical, nearest neighbor/density-based, and clustering based—
have been retained and updated, while new approaches have been added:
reconstruction-based, one-class classification, and information-theoretic. The
reconstruction-based approach is illustrated using autoencoder networks that are
part of the deep learning paradigm. The data chapter has been updated to include
discussions of mutual information and kernel-based techniques.
The last chapter, which discusses how to avoid false discoveries and produce valid
results, is completely new, and is novel among other contemporary textbooks on
data mining. It supplements the discussions in the other chapters with a discussion
of the statistical concepts (statistical significance, p-values, false discovery rate,
permutation testing, etc.) relevant to avoiding spurious results, and then illustrates
these concepts in the context of data mining techniques. This chapter addresses
the increasing concern over the validity and reproducibility of results obtained from
data analysis. The addition of this last chapter is a recognition of the importance of
this topic and an acknowledgment that a deeper understanding of this area is
needed for those analyzing data.
The data exploration chapter has been deleted, as have the appendices, from the
print edition of the book, but will remain available on the web. A new appendix
provides a brief discussion of scalability in the context of big data.
To the Instructor
As a textbook, this book is suitable for a wide range of students at the advanced
undergraduate or graduate level. Since students come to this subject with diverse
backgrounds that may not include extensive knowledge of statistics or databases,
our book requires minimal prerequisites. No database knowledge is needed, and
we assume only a modest background in statistics or mathematics, although such a
background will make for easier going in some sections. As before, the book, and
more specifically, the chapters covering major data mining topics, are designed to
be as self-contained as possible. Thus, the order in which topics can be covered is
quite flexible. The core material is covered in chapters 2 (data), 3 (classification), 5
(association analysis), 7 (clustering), and 9 (anomaly detection). We recommend at
least a cursory coverage of Chapter 10 (Avoiding False Discoveries) to instill in
students some caution when interpreting the results of their data analysis. Although
the introductory data chapter (2) should be covered first, the basic classification (3),
association analysis (5), and clustering chapters (7), can be covered in any order.
Because of the relationship of anomaly detection (9) to classification (3) and
clustering (7), these chapters should precede Chapter 9. Various topics can be
selected from the advanced classification, association analysis, and clustering
chapters (4, 6, and 8, respectively) to fit the schedule and interests of the instructor
and students. We also advise that the lectures be augmented by projects or
practical exercises in data mining. Although they are time consuming, such handson assignments greatly enhance the value of the course.
Support Materials
Support materials available to all readers of this book are available at http://wwwusers.cs.umn.edu/~kumar/dmbook.
PowerPoint lecture slides
Suggestions for student projects
Data mining resources, such as algorithms and data sets
Online tutorials that give step-by-step examples for selected data mining
techniques described in the book using actual data sets and data analysis
software
Additional support materials, including solutions to exercises, are available only to
instructors adopting this textbook for classroom use. The book’s resources will be
mirrored at www.pearsonhighered.com/cs-resources. Comments and
suggestions, as well as reports of errors, can be sent to the authors through
dmbook@cs.umn.edu.
Acknowledgments
Many people contributed to the first and second editions of the book. We begin by
acknowledging our families to whom this book is dedicated. Without their patience
and support, this project would have been impossible.
We would like to thank the current and former students of our data mining groups at
the University of Minnesota and Michigan State for their contributions. Eui-Hong
(Sam) Han and Mahesh Joshi helped with the initial data mining classes. Some of
the exercises and presentation slides that they created can be found in the book
and its accompanying slides. Students in our data mining groups who provided
comments on drafts of the book or who contributed in other ways include Shyam
Boriah, Haibin Cheng, Varun Chandola, Eric Eilertson, Levent Ertöz, Jing Gao,
Rohit Gupta, Sridhar Iyer, Jung-Eun Lee, Benjamin Mayer, Aysel Ozgur, Uygar
Oztekin, Gaurav Pandey, Kashif Riaz, Jerry Scripps, Gyorgy Simon, Hui Xiong,
Jieping Ye, and Pusheng Zhang. We would also like to thank the students of our
data mining classes at the University of Minnesota and Michigan State University
who worked with early drafts of the book and provided invaluable feedback. We
specifically note the helpful suggestions of Bernardo Craemer, Arifin Ruslim,
Jamshid Vayghan, and Yu Wei.
Joydeep Ghosh (University of Texas) and Sanjay Ranka (University of Florida)
class tested early versions of the book. We also received many useful suggestions
directly from the following UT students: Pankaj Adhikari, Rajiv Bhatia, Frederic
Bosche, Arindam Chakraborty, Meghana Deodhar, Chris Everson, David Gardner,
Saad Godil, Todd Hay, Clint Jones, Ajay Joshi, Joonsoo Lee, Yue Luo, Anuj
Nanavati, Tyler Olsen, Sunyoung Park, Aashish Phansalkar, Geoff Prewett, Michael
Ryoo, Daryl Shannon, and Mei Yang.
Ronald Kostoff (ONR) read an early version of the clustering chapter and offered
numerous suggestions. George Karypis provided invaluable LATEX assistance in
creating an author index. Irene Moulitsas also provided assistance with LATEX and
reviewed some of the appendices. Musetta Steinbach was very helpful in finding
errors in the figures.
We would like to acknowledge our colleagues at the University of Minnesota and
Michigan State who have helped create a positive environment for data mining
research. They include Arindam Banerjee, Dan Boley, Joyce Chai, Anil Jain, Ravi
Janardan, Rong Jin, George Karypis, Claudia Neuhauser, Haesun Park, William F.
Punch, György Simon, Shashi Shekhar, and Jaideep Srivastava. The collaborators
on our many data mining projects, who also have our gratitude, include Ramesh
Agrawal, Maneesh Bhargava, Steve Cannon, Alok Choudhary, Imme Ebert-Uphoff,
Auroop Ganguly, Piet C. de Groen, Fran Hill, Yongdae Kim, Steve Klooster, Kerry
Long, Nihar Mahapatra, Rama Nemani, Nikunj Oza, Chris Potter, Lisiane Pruinelli,
Nagiza Samatova, Jonathan Shapiro, Kevin Silverstein, Brian Van Ness, Bonnie
Westra, Nevin Young, and Zhi-Li Zhang.
The departments of Computer Science and Engineering at the University of
Minnesota and Michigan State University provided computing resources and a
supportive environment for this project. ARDA, ARL, ARO, DOE, NASA, NOAA,
and NSF provided research support for Pang-Ning Tan, Michael Stein-bach, Anuj
Karpatne, and Vipin Kumar. In particular, Kamal Abdali, Mitra Basu, Dick Brackney,
Jagdish Chandra, Joe Coughlan, Michael Coyle, Stephen Davis, Frederica
Darema, Richard Hirsch, Chandrika Kamath, Tsengdar Lee, Raju Namburu, N.
Radhakrishnan, James Sidoran, Sylvia Spengler, Bhavani Thuraisingham, Walt
Tiernin, Maria Zemankova, Aidong Zhang, and Xiaodong Zhang have been
supportive of our research in data mining and high-performance computing.
It was a pleasure working with the helpful staff at Pearson Education. In particular,
we would like to thank Matt Goldstein, Kathy Smith, Carole Snyder, and Joyce
Wells. We would also like to thank George Nichols, who helped with the art work
and Paul Anagnostopoulos, who provided LATEX support.
We are grateful to the following Pearson reviewers: Leman Akoglu (Carnegie
Mellon University), Chien-Chung Chan (University of Akron), Zhengxin Chen
(University of Nebraska at Omaha), Chris Clifton (Purdue University), Joy-deep
Ghosh (University of Texas, Austin), Nazli Goharian (Illinois Institute of
Technology), J. Michael Hardin (University of Alabama), Jingrui He (Arizona State
University), James Hearne (Western Washington University), Hillol Kargupta
(University of Maryland, Baltimore County and Agnik, LLC), Eamonn Keogh
(University of California-Riverside), Bing Liu (University of Illinois at Chicago),
Mariofanna Milanova (University of Arkansas at Little Rock), Srinivasan
Parthasarathy (Ohio State University), Zbigniew W. Ras (University of North
Carolina at Charlotte), Xintao Wu (University of North Carolina at Charlotte), and
Mohammed J. Zaki (Rensselaer Polytechnic Institute).
Over the years since the first edition, we have also received numerous comments
from readers and students who have pointed out typos and various other issues.
We are unable to mention these individuals by name, but their input is much
appreciated and has been taken into account for the second edition.
Contents
Preface to the Second Edition v
1 Introduction 1
1.1 What Is Data Mining? 4
1.2 Motivating Challenges 5
1.3 The Origins of Data Mining 7
1.4 Data Mining Tasks 9
1.5 Scope and Organization of the Book 13
1.6 Bibliographic Notes 15
1.7 Exercises 21
2 Data 23
2.1 Types of Data 26
2.1.1 Attributes and Measurement 27
2.1.2 Types of Data Sets 34
2.2 Data Quality 42
2.2.1 Measurement and Data Collection Issues 42
2.2.2 Issues Related to Applications 49
2.3 Data Preprocessing 50
2.3.1 Aggregation 51
2.3.2 Sampling 52
2.3.3 Dimensionality Reduction 56
2.3.4 Feature Subset Selection 58
2.3.5 Feature Creation 61
2.3.6 Discretization and Binarization 63
2.3.7 Variable Transformation 69
2.4 Measures of Similarity and Dissimilarity 71
2.4.1 Basics 72
2.4.2 Similarity and Dissimilarity between Simple Attributes 74
2.4.3 Dissimilarities between Data Objects 76
2.4.4 Similarities between Data Objects 78
2.4.5 Examples of Proximity Measures 79
2.4.6 Mutual Information 88
2.4.7 Kernel Functions* 90
2.4.8 Bregman Divergence* 94
2.4.9 Issues in Proximity Calculation 96
2.4.10 Selecting the Right Proximity Measure 98
2.5 Bibliographic Notes 100
2.6 Exercises 105
3 Classification: Basic Concepts and Techniques 113
3.1 Basic Concepts 114
3.2 General Framework for Classification 117
3.3 Decision Tree Classifier 119
3.3.1 A Basic Algorithm to Build a Decision Tree 121
3.3.2 Methods for Expressing Attribute Test Conditions 124
3.3.3 Measures for Selecting an Attribute Test Condition 127
3.3.4 Algorithm for Decision Tree Induction 136
3.3.5 Example Application: Web Robot Detection 138
3.3.6 Characteristics of Decision Tree Classifiers 140
3.4 Model Overfitting 147
3.4.1 Reasons for Model Overfitting 149
3.5 Model Selection 156
3.5.1 Using a Validation Set 156
3.5.2 Incorporating Model Complexity 157
3.5.3 Estimating Statistical Bounds 162
3.5.4 Model Selection for Decision Trees 162
3.6 Model Evaluation 164
3.6.1 Holdout Method 165
3.6.2 Cross-Validation 165
3.7 Presence of Hyper-parameters 168
3.7.1 Hyper-parameter Selection 168
3.7.2 Nested Cross-Validation 170
3.8 Pitfalls of Model Selection and Evaluation 172
3.8.1 Overlap between Training and Test Sets 172
3.8.2 Use of Validation Error as Generalization Error 172
3.9 Model Comparison* 173
3.9.1 Estimating the Confidence Interval for Accuracy 174
3.9.2 Comparing the Performance of Two Models 175
3.10 Bibliographic Notes 176
3.11 Exercises 185
4 Classification: Alternative Techniques 193
4.1 Types of Classifiers 193
4.2 Rule-Based Classifier 195
4.2.1 How a Rule-Based Classifier Works 197
4.2.2 Properties of a Rule Set 198
4.2.3 Direct Methods for Rule Extraction 199
4.2.4 Indirect Methods for Rule Extraction 204
4.2.5 Characteristics of Rule-Based Classifiers 206
4.3 Nearest Neighbor Classifiers 208
4.3.1 Algorithm 209
4.3.2 Characteristics of Nearest Neighbor Classifiers 210
4.4 Naïve Bayes Classifier 212
4.4.1 Basics of Probability Theory 213
4.4.2 Naïve Bayes Assumption 218
4.5 Bayesian Networks 227
4.5.1 Graphical Representation 227
4.5.2 Inference and Learning 233
4.5.3 Characteristics of Bayesian Networks 242
4.6 Logistic Regression 243
4.6.1 Logistic Regression as a Generalized Linear Model 244
4.6.2 Learning Model Parameters 245
4.6.3 Characteristics of Logistic Regression 248
4.7 Artificial Neural Network (ANN) 249
4.7.1 Perceptron 250
4.7.2 Multi-layer Neural Network 254
4.7.3 Characteristics of ANN 261
4.8 Deep Learning 262
4.8.1 Using Synergistic Loss Functions 263
4.8.2 Using Responsive Activation Functions 266
4.8.3 Regularization 268
4.8.4 Initialization of Model Parameters 271
4.8.5 Characteristics of Deep Learning 275
4.9 Support Vector Machine (SVM) 276
4.9.1 Margin of a Separating Hyperplane 276
4.9.2 Linear SVM 278
4.9.3 Soft-margin SVM 284
4.9.4 Nonlinear SVM 290
4.9.5 Characteristics of SVM 294
4.10 Ensemble Methods 296
4.10.1 Rationale for Ensemble Method 297
4.10.2 Methods for Constructing an Ensemble Classifier 297
4.10.3 Bias-Variance Decomposition 300
4.10.4 Bagging 302
4.10.5 Boosting 305
4.10.6 Random Forests 310
4.10.7 Empirical Comparison among Ensemble Methods 312
4.11 Class Imbalance Problem 313
4.11.1 Building Classifiers with Class Imbalance 314
4.11.2 Evaluating Performance with Class Imbalance 318
4.11.3 Finding an Optimal Score Threshold 322
4.11.4 Aggregate Evaluation of Performance 323
4.12 Multiclass Problem 330
4.13 Bibliographic Notes 333
4.14 Exercises 345
5 Association Analysis: Basic Concepts and Algorithms 357
5.1 Preliminaries 358
5.2 Frequent Itemset Generation 362
5.2.1 The Apriori Principle 363
5.2.2 Frequent Itemset Generation in the Apriori Algorithm 364
5.2.3 Candidate Generation and Pruning 368
5.2.4 Support Counting 373
5.2.5 Computational Complexity 377
5.3 Rule Generation 380
5.3.1 Confidence-Based Pruning 380
5.3.2 Rule Generation in Apriori Algorithm 381
5.3.3 An Example: Congressional Voting Records 382
5.4 Compact Representation of Frequent Itemsets 384
5.4.1 Maximal Frequent Itemsets 384
5.4.2 Closed Itemsets 386
5.5 Alternative Methods for Generating Frequent Itemsets* 389
5.6 FP-Growth Algorithm* 393
5.6.1 FP-Tree Representation 394
5.6.2 Frequent Itemset Generation in FP-Growth Algorithm 397
5.7 Evaluation of Association Patterns 401
5.7.1 Objective Measures of Interestingness 402
5.7.2 Measures beyond Pairs of Binary Variables 414
5.7.3 Simpson’s Paradox 416
5.8 Effect of Skewed Support Distribution 418
5.9 Bibliographic Notes 424
5.10 Exercises 438
6 Association Analysis: Advanced Concepts 451
6.1 Handling Categorical Attributes 451
6.2 Handling Continuous Attributes 454
6.2.1 Discretization-Based Methods 454
6.2.2 Statistics-Based Methods 458
6.2.3 Non-discretization Methods 460
6.3 Handling a Concept Hierarchy 462
6.4 Sequential Patterns 464
6.4.1 Preliminaries 465
6.4.2 Sequential Pattern Discovery 468
6.4.3 Timing Constraints∗ 473
6.4.4 Alternative Counting Schemes∗ 477
6.5 Subgraph Patterns 479
6.5.1 Preliminaries 480
6.5.2 Frequent Subgraph Mining 483
6.5.3 Candidate Generation 487
6.5.4 Candidate Pruning 493
6.5.5 Support Counting 493
6.6 Infrequent Patterns∗ 493
6.6.1 Negative Patterns 494
6.6.2 Negatively Correlated Patterns 495
6.6.3 Comparisons among Infrequent Patterns, Negative Patterns, and
Negatively Correlated Patterns 496
6.6.4 Techniques for Mining Interesting Infrequent Patterns 498
6.6.5 Techniques Based on Mining Negative Patterns 499
6.6.6 Techniques Based on Support Expectation 501
6.7 Bibliographic Notes 505
6.8 Exercises 510
7 Cluster Analysis: Basic Concepts and Algorithms 525
7.1 Overview 528
7.1.1 What Is Cluster Analysis? 528
7.1.2 Different Types of Clusterings 529
7.1.3 Different Types of Clusters 531
7.2 K-means 534
7.2.1 The Basic K-means Algorithm 535
7.2.2 K-means: Additional Issues 544
7.2.3 Bisecting K-means 547
7.2.4 K-means and Different Types of Clusters 548
7.2.5 Strengths and Weaknesses 549
7.2.6 K-means as an Optimization Problem 549
7.3 Agglomerative Hierarchical Clustering 554
7.3.1 Basic Agglomerative Hierarchical Clustering Algorithm 555
7.3.2 Specific Techniques 557
7.3.3 The Lance-Williams Formula for Cluster Proximity 562
7.3.4 Key Issues in Hierarchical Clustering 563
7.3.5 Outliers 564
7.3.6 Strengths and Weaknesses 565
7.4 DBSCAN 565
7.4.1 Traditional Density: Center-Based Approach 565
7.4.2 The DBSCAN Algorithm 567
7.4.3 Strengths and Weaknesses 569
7.5 Cluster Evaluation 571
7.5.1 Overview 571
7.5.2 Unsupervised Cluster Evaluation Using Cohesion and Separation
574
7.5.3 Unsupervised Cluster Evaluation Using the Proximity Matrix
582
7.5.4 Unsupervised Evaluation of Hierarchical Clustering 585
7.5.5 Determining the Correct Number of Clusters 587
7.5.6 Clustering Tendency 588
7.5.7 Supervised Measures of Cluster Validity 589
7.5.8 Assessing the Significance of Cluster Validity Measures 594
7.5.9 Choosing a Cluster Validity Measure 596
7.6 Bibliographic Notes 597
7.7 Exercises 603
8 Cluster Analysis: Additional Issues and Algorithms 613
8.1 Characteristics of Data, Clusters, and Clustering Algorithms 614
8.1.1 Example: Comparing K-means and DBSCAN 614
8.1.2 Data Characteristics 615
8.1.3 Cluster Characteristics 617
8.1.4 General Characteristics of Clustering Algorithms 619
8.2 Prototype-Based Clustering 621
8.2.1 Fuzzy Clustering 621
8.2.2 Clustering Using Mixture Models 627
8.2.3 Self-Organizing Maps (SOM) 637
8.3 Density-Based Clustering 644
8.3.1 Grid-Based Clustering 644
8.3.2 Subspace Clustering 648
8.3.3 DENCLUE: A Kernel-Based Scheme for Density-Based Clustering
652
8.4 Graph-Based Clustering 656
8.4.1 Sparsification 657
8.4.2 Minimum Spanning Tree (MST) Clustering 658
8.4.3 OPOSSUM: Optimal Partitioning of Sparse Similarities Using
METIS 659
8.4.4 Chameleon: Hierarchical Clustering with Dynamic Modeling
660
8.4.5 Spectral Clustering 666
8.4.6 Shared Nearest Neighbor Similarity 673
8.4.7 The Jarvis-Patrick Clustering Algorithm 676
8.4.8 SNN Density 678
8.4.9 SNN Density-Based Clustering 679
8.5 Scalable Clustering Algorithms 681
8.5.1 Scalability: General Issues and Approaches 681
8.5.2 BIRCH 684
8.5.3 CURE 686
8.6 Which Clustering Algorithm? 690
8.7 Bibliographic Notes 693
8.8 Exercises 699
9 Anomaly Detection 703
9.1 Characteristics of Anomaly Detection Problems 705
9.1.1 A Definition of an Anomaly 705
9.1.2 Nature of Data 706
9.1.3 How Anomaly Detection is Used 707
9.2 Characteristics of Anomaly Detection Methods 708
9.3 Statistical Approaches 710
9.3.1 Using Parametric Models 710
9.3.2 Using Non-parametric Models 714
9.3.3 Modeling Normal and Anomalous Classes 715
9.3.4 Assessing Statistical Significance 717
9.3.5 Strengths and Weaknesses 718
9.4 Proximity-based Approaches 719
9.4.1 Distance-based Anomaly Score 719
9.4.2 Density-based Anomaly Score 720
9.4.3 Relative Density-based Anomaly Score 722
9.4.4 Strengths and Weaknesses 723
9.5 Clustering-based Approaches 724
9.5.1 Finding Anomalous Clusters 724
9.5.2 Finding Anomalous Instances 725
9.5.3 Strengths and Weaknesses 728
9.6 Reconstruction-based Approaches 728
9.6.1 Strengths and Weaknesses 731
9.7 One-class Classification 732
9.7.1 Use of Kernels 733
9.7.2 The Origin Trick 734
9.7.3 Strengths and Weaknesses 738
9.8 Information Theoretic Approaches 738
9.8.1 Strengths and Weaknesses 740
9.9 Evaluation of Anomaly Detection 740
9.10 Bibliographic Notes 742
9.11 Exercises 749
10 Avoiding False Discoveries 755
10.1 Preliminaries: Statistical Testing 756
10.1.1 Significance Testing 756
10.1.2 Hypothesis Testing 761
10.1.3 Multiple Hypothesis Testing 767
10.1.4 Pitfalls in Statistical Testing 776
10.2 Modeling Null and Alternative Distributions 778
10.2.1 Generating Synthetic Data Sets 781
10.2.2 Randomizing Class Labels 782
10.2.3 Resampling Instances 782
10.2.4 Modeling the Distribution of the Test Statistic 783
10.3 Statistical Testing for Classification 783
10.3.1 Evaluating Classification Performance 783
10.3.2 Binary Classification as Multiple Hypothesis Testing 785
10.3.3 Multiple Hypothesis Testing in Model Selection 786
10.4 Statistical Testing for Association Analysis 787
10.4.1 Using Statistical Models 788
10.4.2 Using Randomization Methods 794
10.5 Statistical Testing for Cluster Analysis 795
10.5.1 Generating a Null Distribution for Internal Indices 796
10.5.2 Generating a Null Distribution for External Indices 798
10.5.3 Enrichment 798
10.6 Statistical Testing for Anomaly Detection 800
10.7 Bibliographic Notes 803
10.8 Exercises 808
Author Index 816
Subject Index 829
Copyright Permissions 839
1 Introduction
Rapid advances in data collection and storage technology,
coupled with the ease with which data can be generated and
disseminated, have triggered the explosive growth of data,
leading to the current age of big data. Deriving actionable
insights from these large data sets is increasingly important in
decision making across almost all areas of society, including
business and industry; science and engineering; medicine and
biotechnology; and government and individuals. However, the
amount of data (volume), its complexity (variety), and the rate at
which it is being collected and processed (velocity) have simply
become too great for humans to analyze unaided. Thus, there is a
great need for automated tools for extracting useful information
from the big data despite the challenges posed by its enormity
and diversity.
Data mining blends traditional data analysis methods with
sophisticated algorithms for processing this abundance of data. In
this introductory chapter, we present an overview of data mining
and outline the key topics to be covered in this book. We start with
a description of some applications that require more advanced
techniques for data analysis.
Business and Industry Point-of-sale data collection (bar code scanners, radio
frequency identification (RFID), and smart card technology) have allowed retailers
to collect up-to-the-minute data about customer purchases at the checkout
counters of their stores. Retailers can utilize this information, along with other
business-critical data, such as web server logs from e-commerce websites and
customer service records from call centers, to help them better understand the
needs of their customers and make more informed business decisions.
Data mining techniques can be used to support a wide range of business
intelligence applications, such as customer profiling, targeted marketing, workflow
management, store layout, fraud detection, and automated buying and selling. An
example of the last application is high-speed stock trading, where decisions on
buying and selling have to be made in less than a second using data about
financial transactions. Data mining can also help retailers answer important
business questions, such as “Who are the most profitable customers?” “What
products can be cross-sold or up-sold?” and “What is the revenue outlook of the
company for next year?” These questions have inspired the development of such
and 6 ).
data mining techniques as association analysis (Chapters 5
As the Internet continues to revolutionize the way we interact and make decisions
in our everyday lives, we are generating massive amounts of data about our online
experiences, e.g., web browsing, messaging, and posting on social networking
websites. This has opened several opportunities for business applications that use
web data. For example, in the e-commerce sector, data about our online viewing or
shopping preferences can be used to provide personalized recommendations of
products. Data mining also plays a prominent role in supporting several other
Internet-based services, such as filtering spam messages, answering search
queries, and suggesting social updates and connections. The large corpus of text,
images, and videos available on the Internet has enabled a number of
advancements in data mining methods, including deep learning, which is discussed
in Chapter 4 . These developments have led to great advances in a number of
applications, such as object recognition, natural language translation, and
autonomous driving.
Another domain that has undergone a rapid big data transformation is the use of
mobile sensors and devices, such as smart phones and wearable computing
devices. With better sensor technologies, it has become possible to collect a variety
of information about our physical world using low-cost sensors embedded on
everyday objects that are connected to each other, termed the Internet of Things
(IOT). This deep integration of physical sensors in digital systems is beginning to
generate large amounts of diverse and distributed data about our environment,
which can be used for designing convenient, safe, and energy-efficient home
systems, as well as for urban planning of smart cities.
Medicine, Science, and Engineering Researchers in medicine, science, and
engineering are rapidly accumulating data that is key to significant new discoveries.
For example, as an important step toward improving our understanding of the
Earth’s climate system, NASA has deployed a series of Earth-orbiting satellites that
continuously generate global observations of the land surface, oceans, and
atmosphere. However, because of the size and spatio-temporal nature of the data,
traditional methods are often not suitable for analyzing these data sets. Techniques
developed in data mining can aid Earth scientists in answering questions such as
the following: “What is the relationship between the frequency and intensity of
ecosystem disturbances such as droughts and hurricanes to global warming?”
“How is land surface precipitation and temperature affected by ocean surface
temperature?”; and “How well can we predict the beginning and end of the growing
season for a region?”
As another example, researchers in molecular biology hope to use the large
amounts of genomic data to better understand the structure and function of genes.
In the past, traditional methods in molecular biology allowed scientists to study only
a few genes at a time in a given experiment. Recent breakthroughs in microarray
technology have enabled scientists to compare the behavior of thousands of genes
under various situations. Such comparisons can help determine the function of
each gene, and perhaps isolate the genes responsible for certain diseases.
However, the noisy, high-dimensional nature of data requires new data analysis
methods. In addition to analyzing gene expression data, data mining can also be
used to address other important biological challenges such as protein structure
prediction, multiple sequence alignment, the modeling of biochemical pathways,
and phylogenetics.
Another example is the use of data mining techniques to analyze electronic health
record (EHR) data, which has become increasingly available. Not very long ago,
studies of patients required manually examining the physical records of individual
patients and extracting very specific pieces of information pertinent to the particular
question being investigated. EHRs allow for a faster and broader exploration of
such data. However, there are significant challenges since the observations on any
one patient typically occur during their visits to a doctor or hospital and only a small
number of details about the health of the patient are measured during any particular
visit.
Currently, EHR analysis focuses on simple types of data, e.g., a patient’s blood
pressure or the diagnosis code of a disease. However, large amounts of more
complex types of medical data are also being collected, such as
electrocardiograms (ECGs) and neuroimages from magnetic resonance imaging
(MRI) or functional Magnetic Resonance Imaging (fMRI). Although challenging to
analyze, this data also provides vital information about patients. Integrating and
analyzing such data, with traditional EHR and genomic data is one of the
capabilities needed to enable precision medicine, which aims to provide more
personalized patient care.
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).
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 timeconsuming 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-ofcore 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 parentchild 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
shows the relationship of data mining to
gathered in one location. Figure 1.2
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.
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.
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).
illustrate point-of-sale data collected at
The transactions shown in Table 1.1
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
, 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.
{Diapers} → {Milk}
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
and 4
cover classification. Chapter 3
provides a foundation by
3
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 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, Self-Organizing 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, distance-based,
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 Pacific-Asia
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 privacypreserving 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 theoryguided 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. Discuss whether or not each of the following activities is a data mining task.
a. Dividing the customers of a company according to their gender.
b. Dividing the customers of a company according to their profitability.
c. Computing the total sales of a company.
d. Sorting a student database based on student identification numbers.
e. Predicting the outcomes of tossing a (fair) pair of dice.
f. Predicting the future stock price of a company using historical records.
g. Monitoring the heart rate of a patient for abnormalities.
h. Monitoring seismic waves for earthquake activities.
i. Extracting the frequencies of a sound wave.
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. For each of the following data sets, explain whether or not data privacy is an
important issue.
a. Census data collected from 1900–1950.
b. IP addresses and visit times of web users who visit your website.
c. Images from Earth-orbiting satellites.
d. Names and addresses of people from the telephone book.
e. 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
â‹®
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
â‹®
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.
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
Categorical
Nominal
(Qualitative)
Description
Examples
Operations
The values of a nominal
zip codes,
mode,
attribute are just different
employee ID
entropy,
names; i.e., nominal
numbers, eye
contingency
values provide only
color, gender
correlation,
The values of an ordinal
hardness of
median,
attribute provide enough
minerals,
percentiles,
information to order
{good, better,
rank
objects. ()
best}, grades,
correlation,
street
run tests,
numbers
sign tests
For interval attributes, the
calendar
mean,
differences between
dates,
standard
values are meaningful, i.e.,
temperature
deviation,
a unit of measurement
in Celsius or
Pearson’s
exists. (+, −)
Fahrenheit
correlation,
enough information to
χ2 test
distinguish one object from
another. (=, ≠ )
Ordinal
Numeric
(Quantitative)
Interval
t and F
tests
Attribute Type
Ratio
Description
Examples
Operations
For ratio variables, both
temperature
geometric
differences and ratios are
in Kelvin,
mean,
meaningful. (×, /)
monetary
harmonic
quantities,
mean,
counts, age,
percent
mass, length,
variation
electrical
current
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 integervalued 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 meaningpreserving transformations for the four attribute types of Table 2.2 .
Table 2.3. Transformations that define attribute levels.
Attribute Type
Transformation
Comment
Attribute Type
Categorical
Nominal
(Qualitative)
Transformation
Comment
Any one-to-one mapping, e.g., a
If all employee ID
permutation of values
numbers are
reassigned, it will
not make any
difference.
Ordinal
An order-preserving change of
An attribute
values, i.e.,
encompassing the
new_value = f (old_value),
notion of good,
where f is a monotonic function.
better, best can be
represented equally
well by the values
{1, 2, 3} or by {0.5,
1, 10}.
Numeric
Interval
(Quantitative)
new_value = a × old_value + b,
The Fahrenheit and
a and b constants.
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
â—¦
of 1 . This is not true when temperature is measured on either the Celsius or
â—¦
Fahrenheit scales, because, physically, a temperature of 1 Fahrenheit (Celsius)
â—¦
is not much different than a temperature of 2 Fahrenheit (Celsius). The
â—¦
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 nonzero 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…
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