Imagine a Clusteringproblem where the educational researcher

Homework answers / question archive / Imagine a Clustering problem where the educational researchers would like to find clusters of students (group of students) who have similar correlation patterns when it comes to correlation of their GPA vs Income of their parents

Imagine a Clustering problem where the educational researchers would like to find clusters of students (group of students) who have similar correlation patterns when it comes to correlation of their GPA vs Income of their parents

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Imagine a Clustering problem where the educational researchers would like to find clusters of students (group of students) who have similar correlation patterns when it comes to correlation of their GPA vs Income of their parents. And you are hired as the Data Scientist to do this job.

What kind of Objective Function would you design?

Why?

Please use your own reasoning and explain in details.

Contents

Preface vii

1 Introduction 1 1.1 What Is Data Mining? . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Motivating Challenges . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 The Origins of Data Mining . . . . . . . . . . . . . . . . . . . . 6 1.4 Data Mining Tasks . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Scope and Organization of the Book . . . . . . . . . . . . . . . 11 1.6 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Data 19 2.1 Types of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1.1 Attributes and Measurement . . . . . . . . . . . . . . . 23 2.1.2 Types of Data Sets . . . . . . . . . . . . . . . . . . . . . 29

2.2 Data Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.1 Measurement and Data Collection Issues . . . . . . . . . 37 2.2.2 Issues Related to Applications . . . . . . . . . . . . . . 43

2.3 Data Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.3.1 Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3.2 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.3.3 Dimensionality Reduction . . . . . . . . . . . . . . . . . 50 2.3.4 Feature Subset Selection . . . . . . . . . . . . . . . . . . 52 2.3.5 Feature Creation . . . . . . . . . . . . . . . . . . . . . . 55 2.3.6 Discretization and Binarization . . . . . . . . . . . . . . 57 2.3.7 Variable Transformation . . . . . . . . . . . . . . . . . . 63

2.4 Measures of Similarity and Dissimilarity . . . . . . . . . . . . . 65 2.4.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.4.2 Similarity and Dissimilarity between Simple Attributes . 67 2.4.3 Dissimilarities between Data Objects . . . . . . . . . . . 69 2.4.4 Similarities between Data Objects . . . . . . . . . . . . 72

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2.4.5 Examples of Proximity Measures . . . . . . . . . . . . . 73 2.4.6 Issues in Proximity Calculation . . . . . . . . . . . . . . 80 2.4.7 Selecting the Right Proximity Measure . . . . . . . . . . 83

2.5 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3 Exploring Data 97 3.1 The Iris Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.2 Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.2.1 Frequencies and the Mode . . . . . . . . . . . . . . . . . 99 3.2.2 Percentiles . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.2.3 Measures of Location: Mean and Median . . . . . . . . 101 3.2.4 Measures of Spread: Range and Variance . . . . . . . . 102 3.2.5 Multivariate Summary Statistics . . . . . . . . . . . . . 104 3.2.6 Other Ways to Summarize the Data . . . . . . . . . . . 105

3.3 Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.3.1 Motivations for Visualization . . . . . . . . . . . . . . . 105 3.3.2 General Concepts . . . . . . . . . . . . . . . . . . . . . . 106 3.3.3 Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.3.4 Visualizing Higher-Dimensional Data . . . . . . . . . . . 124 3.3.5 Do’s and Don’ts . . . . . . . . . . . . . . . . . . . . . . 130

3.4 OLAP and Multidimensional Data Analysis . . . . . . . . . . . 131 3.4.1 Representing Iris Data as a Multidimensional Array . . 131 3.4.2 Multidimensional Data: The General Case . . . . . . . . 133 3.4.3 Analyzing Multidimensional Data . . . . . . . . . . . . 135 3.4.4 Final Comments on Multidimensional Data Analysis . . 139

3.5 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 139 3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4 Classification: Basic Concepts, Decision Trees, and Model Evaluation 145 4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 4.2 General Approach to Solving a Classification Problem . . . . . 148 4.3 Decision Tree Induction . . . . . . . . . . . . . . . . . . . . . . 150

4.3.1 How a Decision Tree Works . . . . . . . . . . . . . . . . 150 4.3.2 How to Build a Decision Tree . . . . . . . . . . . . . . . 151 4.3.3 Methods for Expressing Attribute Test Conditions . . . 155 4.3.4 Measures for Selecting the Best Split . . . . . . . . . . . 158 4.3.5 Algorithm for Decision Tree Induction . . . . . . . . . . 164 4.3.6 An Example: Web Robot Detection . . . . . . . . . . . 166

Contents

Preface vii

2 Data 19 2.1 Types of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1.1 Attributes and Measurement . . . . . . . . . . . . . . . 23 2.1.2 Types of Data Sets . . . . . . . . . . . . . . . . . . . . . 29

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2.5 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3 Exploring Data 97 3.1 The Iris Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.2 Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.5 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 139 3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

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4.3.7 Characteristics of Decision Tree Induction . . . . . . . . 168 4.4 Model Overfitting . . . . . . . . . . . . . . . . . . . . . . . . . . 172

4.4.1 Overfitting Due to Presence of Noise . . . . . . . . . . . 175 4.4.2 Overfitting Due to Lack of Representative Samples . . . 177 4.4.3 Overfitting and the Multiple Comparison Procedure . . 178 4.4.4 Estimation of Generalization Errors . . . . . . . . . . . 179 4.4.5 Handling Overfitting in Decision Tree Induction . . . . 184

4.5 Evaluating the Performance of a Classifier . . . . . . . . . . . . 186 4.5.1 Holdout Method . . . . . . . . . . . . . . . . . . . . . . 186 4.5.2 Random Subsampling . . . . . . . . . . . . . . . . . . . 187 4.5.3 Cross-Validation . . . . . . . . . . . . . . . . . . . . . . 187 4.5.4 Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . 188

4.6 Methods for Comparing Classifiers . . . . . . . . . . . . . . . . 188 4.6.1 Estimating a Confidence Interval for Accuracy . . . . . 189 4.6.2 Comparing the Performance of Two Models . . . . . . . 191 4.6.3 Comparing the Performance of Two Classifiers . . . . . 192

4.7 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 193 4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

5 Classification: Alternative Techniques 207 5.1 Rule-Based Classifier . . . . . . . . . . . . . . . . . . . . . . . . 207

5.1.1 How a Rule-Based Classifier Works . . . . . . . . . . . . 209 5.1.2 Rule-Ordering Schemes . . . . . . . . . . . . . . . . . . 211 5.1.3 How to Build a Rule-Based Classifier . . . . . . . . . . . 212 5.1.4 Direct Methods for Rule Extraction . . . . . . . . . . . 213 5.1.5 Indirect Methods for Rule Extraction . . . . . . . . . . 221 5.1.6 Characteristics of Rule-Based Classifiers . . . . . . . . . 223

5.2 Nearest-Neighbor classifiers . . . . . . . . . . . . . . . . . . . . 223 5.2.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 225 5.2.2 Characteristics of Nearest-Neighbor Classifiers . . . . . 226

5.3 Bayesian Classifiers . . . . . . . . . . . . . . . . . . . . . . . . . 227 5.3.1 Bayes Theorem . . . . . . . . . . . . . . . . . . . . . . . 228 5.3.2 Using the Bayes Theorem for Classification . . . . . . . 229 5.3.3 Na??ve Bayes Classifier . . . . . . . . . . . . . . . . . . . 231 5.3.4 Bayes Error Rate . . . . . . . . . . . . . . . . . . . . . . 238 5.3.5 Bayesian Belief Networks . . . . . . . . . . . . . . . . . 240

5.4 Artificial Neural Network (ANN) . . . . . . . . . . . . . . . . . 246 5.4.1 Perceptron . . . . . . . . . . . . . . . . . . . . . . . . . 247 5.4.2 Multilayer Artificial Neural Network . . . . . . . . . . . 251 5.4.3 Characteristics of ANN . . . . . . . . . . . . . . . . . . 255

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5.5 Support Vector Machine (SVM) . . . . . . . . . . . . . . . . . . 256 5.5.1 Maximum Margin Hyperplanes . . . . . . . . . . . . . . 256 5.5.2 Linear SVM: Separable Case . . . . . . . . . . . . . . . 259 5.5.3 Linear SVM: Nonseparable Case . . . . . . . . . . . . . 266 5.5.4 Nonlinear SVM . . . . . . . . . . . . . . . . . . . . . . . 270 5.5.5 Characteristics of SVM . . . . . . . . . . . . . . . . . . 276

5.6 Ensemble Methods . . . . . . . . . . . . . . . . . . . . . . . . . 276 5.6.1 Rationale for Ensemble Method . . . . . . . . . . . . . . 277 5.6.2 Methods for Constructing an Ensemble Classifier . . . . 278 5.6.3 Bias-Variance Decomposition . . . . . . . . . . . . . . . 281 5.6.4 Bagging . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 5.6.5 Boosting . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 5.6.6 Random Forests . . . . . . . . . . . . . . . . . . . . . . 290 5.6.7 Empirical Comparison among Ensemble Methods . . . . 294

5.7 Class Imbalance Problem . . . . . . . . . . . . . . . . . . . . . 294 5.7.1 Alternative Metrics . . . . . . . . . . . . . . . . . . . . . 295 5.7.2 The Receiver Operating Characteristic Curve . . . . . . 298 5.7.3 Cost-Sensitive Learning . . . . . . . . . . . . . . . . . . 302 5.7.4 Sampling-Based Approaches . . . . . . . . . . . . . . . . 305

5.8 Multiclass Problem . . . . . . . . . . . . . . . . . . . . . . . . . 306 5.9 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 309 5.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

6 Association Analysis: Basic Concepts and Algorithms 327 6.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . 328 6.2 Frequent Itemset Generation . . . . . . . . . . . . . . . . . . . 332

6.2.1 The Apriori Principle . . . . . . . . . . . . . . . . . . . 333 6.2.2 Frequent Itemset Generation in the Apriori Algorithm . 335 6.2.3 Candidate Generation and Pruning . . . . . . . . . . . . 338 6.2.4 Support Counting . . . . . . . . . . . . . . . . . . . . . 342 6.2.5 Computational Complexity . . . . . . . . . . . . . . . . 345

6.3 Rule Generation . . . . . . . . . . . . . . . . . . . . . . . . . . 349 6.3.1 Confidence-Based Pruning . . . . . . . . . . . . . . . . . 350 6.3.2 Rule Generation in Apriori Algorithm . . . . . . . . . . 350 6.3.3 An Example: Congressional Voting Records . . . . . . . 352

6.4 Compact Representation of Frequent Itemsets . . . . . . . . . . 353 6.4.1 Maximal Frequent Itemsets . . . . . . . . . . . . . . . . 354 6.4.2 Closed Frequent Itemsets . . . . . . . . . . . . . . . . . 355

6.5 Alternative Methods for Generating Frequent Itemsets . . . . . 359 6.6 FP-Growth Algorithm . . . . . . . . . . . . . . . . . . . . . . . 363

Contents xv

4.3.7 Characteristics of Decision Tree Induction . . . . . . . . 168 4.4 Model Overfitting . . . . . . . . . . . . . . . . . . . . . . . . . . 172

4.7 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 193 4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

5 Classification: Alternative Techniques 207 5.1 Rule-Based Classifier . . . . . . . . . . . . . . . . . . . . . . . . 207

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6.5 Alternative Methods for Generating Frequent Itemsets . . . . . 359 6.6 FP-Growth Algorithm . . . . . . . . . . . . . . . . . . . . . . . 363

Contents xvii

6.6.1 FP-Tree Representation . . . . . . . . . . . . . . . . . . 363 6.6.2 Frequent Itemset Generation in FP-Growth Algorithm . 366

6.7 Evaluation of Association Patterns . . . . . . . . . . . . . . . . 370 6.7.1 Objective Measures of Interestingness . . . . . . . . . . 371 6.7.2 Measures beyond Pairs of Binary Variables . . . . . . . 382 6.7.3 Simpson’s Paradox . . . . . . . . . . . . . . . . . . . . . 384

6.8 Effect of Skewed Support Distribution . . . . . . . . . . . . . . 386 6.9 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 390 6.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

7 Association Analysis: Advanced Concepts 415 7.1 Handling Categorical Attributes . . . . . . . . . . . . . . . . . 415 7.2 Handling Continuous Attributes . . . . . . . . . . . . . . . . . 418

7.2.1 Discretization-Based Methods . . . . . . . . . . . . . . . 418 7.2.2 Statistics-Based Methods . . . . . . . . . . . . . . . . . 422 7.2.3 Non-discretization Methods . . . . . . . . . . . . . . . . 424

7.3 Handling a Concept Hierarchy . . . . . . . . . . . . . . . . . . 426 7.4 Sequential Patterns . . . . . . . . . . . . . . . . . . . . . . . . . 429

7.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . 429 7.4.2 Sequential Pattern Discovery . . . . . . . . . . . . . . . 431 7.4.3 Timing Constraints . . . . . . . . . . . . . . . . . . . . . 436 7.4.4 Alternative Counting Schemes . . . . . . . . . . . . . . 439

7.5 Subgraph Patterns . . . . . . . . . . . . . . . . . . . . . . . . . 442 7.5.1 Graphs and Subgraphs . . . . . . . . . . . . . . . . . . . 443 7.5.2 Frequent Subgraph Mining . . . . . . . . . . . . . . . . 444 7.5.3 Apriori -like Method . . . . . . . . . . . . . . . . . . . . 447 7.5.4 Candidate Generation . . . . . . . . . . . . . . . . . . . 448 7.5.5 Candidate Pruning . . . . . . . . . . . . . . . . . . . . . 453 7.5.6 Support Counting . . . . . . . . . . . . . . . . . . . . . 457

7.6 Infrequent Patterns . . . . . . . . . . . . . . . . . . . . . . . . . 457 7.6.1 Negative Patterns . . . . . . . . . . . . . . . . . . . . . 458 7.6.2 Negatively Correlated Patterns . . . . . . . . . . . . . . 458 7.6.3 Comparisons among Infrequent Patterns, Negative Pat-

terns, and Negatively Correlated Patterns . . . . . . . . 460 7.6.4 Techniques for Mining Interesting Infrequent Patterns . 461 7.6.5 Techniques Based on Mining Negative Patterns . . . . . 463 7.6.6 Techniques Based on Support Expectation . . . . . . . . 465

7.7 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 469 7.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

xviii Contents

8 Cluster Analysis: Basic Concepts and Algorithms 487 8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

8.1.1 What Is Cluster Analysis? . . . . . . . . . . . . . . . . . 490 8.1.2 Different Types of Clusterings . . . . . . . . . . . . . . . 491 8.1.3 Different Types of Clusters . . . . . . . . . . . . . . . . 493

8.2 K-means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 8.2.1 The Basic K-means Algorithm . . . . . . . . . . . . . . 497 8.2.2 K-means: Additional Issues . . . . . . . . . . . . . . . . 506 8.2.3 Bisecting K-means . . . . . . . . . . . . . . . . . . . . . 508 8.2.4 K-means and Different Types of Clusters . . . . . . . . 510 8.2.5 Strengths and Weaknesses . . . . . . . . . . . . . . . . . 510 8.2.6 K-means as an Optimization Problem . . . . . . . . . . 513

8.3 Agglomerative Hierarchical Clustering . . . . . . . . . . . . . . 515 8.3.1 Basic Agglomerative Hierarchical Clustering Algorithm 516 8.3.2 Specific Techniques . . . . . . . . . . . . . . . . . . . . . 518 8.3.3 The Lance-Williams Formula for Cluster Proximity . . . 524 8.3.4 Key Issues in Hierarchical Clustering . . . . . . . . . . . 524 8.3.5 Strengths and Weaknesses . . . . . . . . . . . . . . . . . 526

8.4 DBSCAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 8.4.1 Traditional Density: Center-Based Approach . . . . . . 527 8.4.2 The DBSCAN Algorithm . . . . . . . . . . . . . . . . . 528 8.4.3 Strengths and Weaknesses . . . . . . . . . . . . . . . . . 530

8.5 Cluster Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 532 8.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 533 8.5.2 Unsupervised Cluster Evaluation Using Cohesion and

Separation . . . . . . . . . . . . . . . . . . . . . . . . . 536 8.5.3 Unsupervised Cluster Evaluation Using the Proximity

Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 8.5.4 Unsupervised Evaluation of Hierarchical Clustering . . . 544 8.5.5 Determining the Correct Number of Clusters . . . . . . 546 8.5.6 Clustering Tendency . . . . . . . . . . . . . . . . . . . . 547 8.5.7 Supervised Measures of Cluster Validity . . . . . . . . . 548 8.5.8 Assessing the Significance of Cluster Validity Measures . 553

8.6 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 555 8.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

9 Cluster Analysis: Additional Issues and Algorithms 569 9.1 Characteristics of Data, Clusters, and Clustering Algorithms . 570

9.1.1 Example: Comparing K-means and DBSCAN . . . . . . 570 9.1.2 Data Characteristics . . . . . . . . . . . . . . . . . . . . 571

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6.6.1 FP-Tree Representation . . . . . . . . . . . . . . . . . . 363 6.6.2 Frequent Itemset Generation in FP-Growth Algorithm . 366

7 Association Analysis: Advanced Concepts 415 7.1 Handling Categorical Attributes . . . . . . . . . . . . . . . . . 415 7.2 Handling Continuous Attributes . . . . . . . . . . . . . . . . . 418

7.3 Handling a Concept Hierarchy . . . . . . . . . . . . . . . . . . 426 7.4 Sequential Patterns . . . . . . . . . . . . . . . . . . . . . . . . . 429

7.7 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 469 7.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

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8 Cluster Analysis: Basic Concepts and Algorithms 487 8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

Separation . . . . . . . . . . . . . . . . . . . . . . . . . 536 8.5.3 Unsupervised Cluster Evaluation Using the Proximity

8.6 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 555 8.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

9 Cluster Analysis: Additional Issues and Algorithms 569 9.1 Characteristics of Data, Clusters, and Clustering Algorithms . 570

9.1.1 Example: Comparing K-means and DBSCAN . . . . . . 570 9.1.2 Data Characteristics . . . . . . . . . . . . . . . . . . . . 571

Contents xix

9.1.3 Cluster Characteristics . . . . . . . . . . . . . . . . . . . 573 9.1.4 General Characteristics of Clustering Algorithms . . . . 575

9.2 Prototype-Based Clustering . . . . . . . . . . . . . . . . . . . . 577 9.2.1 Fuzzy Clustering . . . . . . . . . . . . . . . . . . . . . . 577 9.2.2 Clustering Using Mixture Models . . . . . . . . . . . . . 583 9.2.3 Self-Organizing Maps (SOM) . . . . . . . . . . . . . . . 594

9.3 Density-Based Clustering . . . . . . . . . . . . . . . . . . . . . 600 9.3.1 Grid-Based Clustering . . . . . . . . . . . . . . . . . . . 601 9.3.2 Subspace Clustering . . . . . . . . . . . . . . . . . . . . 604 9.3.3 DENCLUE: A Kernel-Based Scheme for Density-Based

Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . 608 9.4 Graph-Based Clustering . . . . . . . . . . . . . . . . . . . . . . 612

9.4.1 Sparsification . . . . . . . . . . . . . . . . . . . . . . . . 613 9.4.2 Minimum Spanning Tree (MST) Clustering . . . . . . . 614 9.4.3 OPOSSUM: Optimal Partitioning of Sparse Similarities

Using METIS . . . . . . . . . . . . . . . . . . . . . . . . 616 9.4.4 Chameleon: Hierarchical Clustering with Dynamic

Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 616 9.4.5 Shared Nearest Neighbor Similarity . . . . . . . . . . . 622 9.4.6 The Jarvis-Patrick Clustering Algorithm . . . . . . . . . 625 9.4.7 SNN Density . . . . . . . . . . . . . . . . . . . . . . . . 627 9.4.8 SNN Density-Based Clustering . . . . . . . . . . . . . . 629

9.5 Scalable Clustering Algorithms . . . . . . . . . . . . . . . . . . 630 9.5.1 Scalability: General Issues and Approaches . . . . . . . 630 9.5.2 BIRCH . . . . . . . . . . . . . . . . . . . . . . . . . . . 633 9.5.3 CURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

9.6 Which Clustering Algorithm? . . . . . . . . . . . . . . . . . . . 639 9.7 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 643 9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647

10 Anomaly Detection 651 10.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653

10.1.1 Causes of Anomalies . . . . . . . . . . . . . . . . . . . . 653 10.1.2 Approaches to Anomaly Detection . . . . . . . . . . . . 654 10.1.3 The Use of Class Labels . . . . . . . . . . . . . . . . . . 655 10.1.4 Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656

10.2 Statistical Approaches . . . . . . . . . . . . . . . . . . . . . . . 658 10.2.1 Detecting Outliers in a Univariate Normal Distribution 659 10.2.2 Outliers in a Multivariate Normal Distribution . . . . . 661 10.2.3 A Mixture Model Approach for Anomaly Detection . . . 662

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10.2.4 Strengths and Weaknesses . . . . . . . . . . . . . . . . . 665 10.3 Proximity-Based Outlier Detection . . . . . . . . . . . . . . . . 666

10.3.1 Strengths and Weaknesses . . . . . . . . . . . . . . . . . 666 10.4 Density-Based Outlier Detection . . . . . . . . . . . . . . . . . 668

10.4.1 Detection of Outliers Using Relative Density . . . . . . 669 10.4.2 Strengths and Weaknesses . . . . . . . . . . . . . . . . . 670

10.5 Clustering-Based Techniques . . . . . . . . . . . . . . . . . . . 671 10.5.1 Assessing the Extent to Which an Object Belongs to a

Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 10.5.2 Impact of Outliers on the Initial Clustering . . . . . . . 674 10.5.3 The Number of Clusters to Use . . . . . . . . . . . . . . 674 10.5.4 Strengths and Weaknesses . . . . . . . . . . . . . . . . . 674

10.6 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 675 10.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680

Appendix A Linear Algebra 685 A.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685

A.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 685 A.1.2 Vector Addition and Multiplication by a Scalar . . . . . 685 A.1.3 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . 687 A.1.4 The Dot Product, Orthogonality, and Orthogonal

Projections . . . . . . . . . . . . . . . . . . . . . . . . . 688 A.1.5 Vectors and Data Analysis . . . . . . . . . . . . . . . . 690

A.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691 A.2.1 Matrices: Definitions . . . . . . . . . . . . . . . . . . . . 691 A.2.2 Matrices: Addition and Multiplication by a Scalar . . . 692 A.2.3 Matrices: Multiplication . . . . . . . . . . . . . . . . . . 693 A.2.4 Linear Transformations and Inverse Matrices . . . . . . 695 A.2.5 Eigenvalue and Singular Value Decomposition . . . . . . 697 A.2.6 Matrices and Data Analysis . . . . . . . . . . . . . . . . 699

A.3 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . 700

Appendix B Dimensionality Reduction 701 B.1 PCA and SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . 701

B.1.1 Principal Components Analysis (PCA) . . . . . . . . . . 701 B.1.2 SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706

B.2 Other Dimensionality Reduction Techniques . . . . . . . . . . . 708 B.2.1 Factor Analysis . . . . . . . . . . . . . . . . . . . . . . . 708 B.2.2 Locally Linear Embedding (LLE) . . . . . . . . . . . . . 710 B.2.3 Multidimensional Scaling, FastMap, and ISOMAP . . . 712