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1 Introduction 1

2 Overview of Supervised Learning 9

2.1 Introduction 9

2.2 Variable Types and Terminology 9

2.3 Two Simple Approaches to Prediction：Least Squares and Nearest Neighbors 11

2.3.1 Linear Models and Least Squares 11

2.3.2 Nearest-Neighbor Methods 14

2.3.3 From Least Squares to Nearest Neighbors 16

2.4 Statistical Decision Theory 18

2.5 Local Methods in High Dimensions 22

2.6 Statistical Models,Supervised Learning and Function Approximation 28

2.6.1 A Statistical Model for the Joint Distribution Pr(X,Y) 28

2.6.2 Supervised Learning 29

2.6.3 Function Approximation 29

2.7 Structured Regression Models 32

2.7.1 Difficulty of the Problem 32

2.8 Classes of Restricted Estimators 33

2.8.1 Roughness Penalty and Bayesian Methods 34

2.8.2 Kernel Methods and Local Regression 34

2.8.3 Basis Functions and Dictionary Methods 35

2.9 Model Selection and the Bias-Variance Tradeoff 37

Bibliographic Notes 39

Exercises 39

3 Linear Methods for Regression 43

3.1 Introduction 43

3.2 Linear Regression Models and Least Squares 44

3.2.1 Example：Prostate Cancer 49

3.2.2 The Gauss-Markov Theorem 51

3.2.3 Multiple Regression from Simple Univariate Regression 52

3.2.4 Multiple Outputs 56

3.3 Subset Selection 57

3.3.1 Best-Subset Selection 57

3.3.2 Forward-and Backward-Stepwise Selection 58

3.3.3 Forward-Stagewise Regression 60

3.3.4 Prostate Cancer Data Example (Continued) 61

3.4 Shrinkage Methods 61

3.4.1 Ridge Regression 61

3.4.2 The Lasso 68

3.4.3 Discussion：Subset Selection,Ridge Regression and the Lasso 69

3.4.4 Least Angle Regression 73

3.5 Methods Using Derived Input Directions 79

3.5.1 Principal Components Regression 79

3.5.2 Partial Least Squares 80

3.6 Discussion：A Comparison of the Selection and Shrinkage Methods 82

3.7 Multiple Outcome Shrinkage and Selection 84

3.8 More on the Lasso and Related Path Algorithms 86

3.8.1 Incremental Forward Stagewise Regression 86

3.8.2 Piecewise-Linear Path Algorithms 89

3.8.3 The Dantzig Selector 89

3.8.4 The Grouped Lasso 90

3.8.5 Further Properties of the Lasso 91

3.8.6 Pathwise Coordinate Optimization 92

3.9 Computational Considerations 93

Bibliographic Notes 94

Exercises 94

4 Linear Methods for Classification 101

4.1 Introduction 101

4.2 Linear Regression of an Indicator Matrix 103

4.3 Linear Discriminant Analysis 106

4.3.1 Regularized Discriminant Analysis 112

4.3.2 Computations for LDA 113

4.3.3 Reduced-Rank Linear Discriminant Analysis 113

4.4 Logistic Regression 119

4.4.1 Fitting Logistic Regression Models 120

4.4.2 Example：South African Heart Disease 122

4.4.3 Quadratic Approximations and Inference 124

4.4.4 L1 Regularized Logistic Regression 125

4.4.5 Logistic Regression or LDA? 127

4.5 Separating Hyperplanes 129

4.5.1 Rosenblatt's Perceptron Learning Algorithm 130

4.5.2 Optimal Separating Hyperplanes 132

Bibliographic Notes 135

Exercises 135

5 Basis Expansions and Regularization 139

5.1 Introduction 139

5.2 Piecewise Polynomials and Splines 141

5.2.1 Natural Cubic Splines 144

5.2.2 Example：South African Heart Disease(Continued) 146

5.2.3 Example：Phoneme Recognition 148

5.3 Filtering and Feature Extraction 150

5.4 Smoothing Splines 151

5.4.1 Degrees of Freedom and Smoother Matrices 153

5.5 Automatic Selection of the Smoothing Parameters 156

5.5.1 Fixing the Degrees of Freedom 158

5.5.2 The Bias-Variance Tradeoff 158

5.6 Nonparametric Logistic Regression 161

5.7 Multidimensional Splines 162

5.8 Regularization and Reproducing Kernel Hilbert Spaces 167

5.8.1 Spaces of Functions Generated by Kernels 168

5.8.2 Examples of RKHS 170

5.9 Wavelet Smoothing 174

5.9.1 Wavelet Bases and the Wavelet Transform 176

5.9.2 Adaptive Wavelet Filtering 179

Bibliographic Notes 181

Exercises 181

Appendix：Computational Considerations for Splines 186

Appendix：B-splines 186

Appendix：Computations for Smoothing Splines 189

6 Kernel Smoothing Methods 191

6.1 One-Dimensional Kernel Smoothers 192

6.1.1 Local Linear Regression 194

6.1.2 Local Polynomial Regression 197

6.2 Selecting the Width of the Kernel 198

6.3 Local Regression in IRp 200

6.4 Structured Local Regression Models in IRp 201

6.4.1 Structured Kernels 203

6.4.2 Structured Regression Functions 203

6.5 Local Likelihood and Other Models 205

6.6 Kernel Density Estimation and Classification 208

6.6.1 Kernel Density Estimation 208

6.6.2 Kernel Density Classification 210

6.6.3 The Naive Bayes Classifier 210

6.7 Radial Basis Functions and Kernels 212

6.8 Mixture Models for Density Estimation and Classification 214

6.9 Computational Considerations 216

Bibliographic Notes 216

Exercises 216

7 Model Assessment and Selection 219

7.1 Introduction 219

7.2 Bias,Variance and Model Complexity 219

7.3 The Bias-Variance Decomposition 223

7.3.1 Example：Bias-Variance Tradeoff 226

7.4 Optimism of the Training Error Rate 228

7.5 Estimates of In-Sample Prediction Error 230

7.6 The Effective Number of Parameters 232

7.7 The Bayesian Approach and BIC 233

7.8 Minimum Description Length 235

7.9 Vapnik-Chervonenkis Dimension 237

7.9.1 Example (Continued) 239

7.10 Cross-Validation 241

7.10.1 K-Fold Cross-Validation 241

7.10.2 The Wrong and Right Way to Do Cross-validation 245

7.10.3 Does Cross-Validation Really Work? 247

7.11 Bootstrap Methods 249

7.11.1 Example(Continued) 252

7.12 Conditional or Expected Test Error? 254

Bibliographic Notes 257

Exercises 257

8 Model Inference and Averaging 261

8.1 Introduction 261

8.2 The Bootstrap and Maximum Likelihood Methods 261

8.2.1 A Smoothing Example 261

8.2.2 Maximum Likelihood Inference 265

8.2.3 Bootstrap versus Maximum Likelihood 267

8.3 Bayesian Methods 267

8.4 Relationship Between the Bootstrap and Bayesian Inference 271

8.5 The EM Algorithm 272

8.5.1 Two-Component Mixture Model 272

8.5.2 The EM Algorithm in General 276

8.5.3 EM as a Maximization-Maximization Procedure 277

8.6 MCMC for Sampling from the Posterior 279

8.7 Bagging 282

8.7.1 Example：Trees with Simulated Data 283

8.8 Model Averaging and Stacking 288

8.9 Stochastic Search：Bumping 290

Bibliographic Notes 292

Exercises 293

9 Additive Models,Trees,and Related Methods 295

9.1 Generalized Additive Models 295

9.1.1 Fitting Additive Models 297

9.1.2 Example：Additive Logistic Regression 299

9.1.3 Summary 304

9.2 Tree-Based Methods 305

9.2.1 Background 305

9.2.2 Regression Trees 307

9.2.3 Classification Trees 308

9.2.4 Other Issues 310

9.2.5 Spam Example (Continued) 313

9.3 PRIM：Bump Hunting 317

9.3.1 Spam Example (Continued) 320

9.4 MARS：Multivariate Adaptive Regression Splines 321

9.4.1 Spam Example (Continued) 326

9.4.2 Example (Simulated Data) 327

9.4.3 Other Issues 328

9.5 Hierarchical Mixtures of Experts 329

9.6 Missing Data 332

9.7 Computational Considerations 334

Bibliographic Notes 334

Exercises 335

10 Boosting and Additive Trees 337

10.1 Boosting Methods 337

10.1.1 Outline of This Chapter 340

10.2 Boosting Fits an Additive Model 341

10.3 Forward Stagewise Additive Modeling 342

10.4 Exponential Loss and AdaBoost 343

10.5 Why Exponential Loss? 345

10.6 Loss Functions and Robustness 346

10.7 "Off-the-Shelf"Procedures for Data Mining 350

10.8 Example：Spam Data 352

10.9 Boosting Trees 353

10.10 Numerical Optimization via Gradient Boosting 358

10.10.1 Steepest Descent 358

10.10.2 Gradient Boosting 359

10.10.3 Implementations of Gradient Boosting 360

10.11 Right-Sized Trees for Boosting 361

10.12 Regularization 364

10.12.1 Shrinkage 364

10.12.2 Subsampling 365

10.13 Interpretation 367

10.13.1 Relative Importance of Predictor Variables 367

10.13.2 Partial Dependence Plots 369

10.14 Illustrations 371

10.14.1 California Housing 371

10.14.2 New Zealand Fish 375

10.14.3 Demographics Data 379

Bibliographic Notes 380

Exercises 384

11 Neural Networks 389

11.1 Introduction 389

11.2 Projection Pursuit Regression 389

11.3 Neural Networks 392

11.4 Fitting Neural Networks 395

11.5 Some Issues in Training Neural Networks 397

11.5.1 Starting Values 397

11.5.2 Overfitting 398

11.5.3 Scaling of the Inputs 398

11.5.4 Number of Hidden Units and Layers 400

11.5.5 Multiple Minima 400

11.6 Example：Simulated Data 401

11.7 Example：ZIP Code Data 404

11.8 Discussion 408

11.9 Bayesian Neural Nets and the NIPS 2003 Challenge 409

11.9.1 Bayes,Boosting and Bagging 410

11.9.2 Performance Comparisons 412

11.10 Computational Considerations 414

Bibliographic Notes 415

Exercises 415

12 Support Vector Machines and Flexible Discriminants 417

12.1 Introduction 417

12.2 The Support Vector Classifier 417

12.2.1 Computing the Support Vector Classifier 420

12.2.2 Mixture Example (Continued) 421

12.3 Support Vector Machines and Kernels 423

12.3.1 Computing the SVM for Classification 423

12.3.2 The SVM as a Penalization Method 426

12.3.3 Function Estimation and Reproducing Kernels 428

12.3.4 SVMs and the Curse of Dimensionality 431

12.3.5 A Path Algorithm for the SVM Classifier 432

12.3.6 Support Vector Machines for Regression 434

12.3.7 Regression and Kernels 436

12.3.8 Discussion 438

12.4 Generalizing Linear Discriminant Analysis 438

12.5 Flexible Discriminant Analysis 440

12.5.1 Computing the FDA Estimates 444

12.6 Penalized Discriminant Analysis 446

12.7 Mixture Discriminant Analysis 449

12.7.1 Example：Waveform Data 451

Bibliographic Notes 455

Exercises 455

13 Prototype Methods and Nearest-Neighbors 459

13.1 Introduction 459

13.2 Prototype Methods 459

13.2.1 K-means Clustering 460

13.2.2 Learning Vector Quantization 462

13.2.3 Gaussian Mixtures 463

13.3 k-Nearest-Neighbor Classifiers 463

13.3.1 Example：A Comparative Study 468

13.3.2 Example：k-Nearest-Neighbors and Image Scene Classification 470

13.3.3 Invariant Metrics and Tangent Distance 471

13.4 Adaptive Nearest-Neighbor Methods 475

13.4.1 Example 478

13.4.2 Global Dimension Reduction for Nearest-Neighbors 479

13.5 Computational Considerations 480

Bibliographic Notes 481

Exercises 481

14 Unsupervised Learning 485

14.1 Introduction 485

14.2 Association Rules 487

14.2.1 Market Basket Analysis 488

14.2.2 The Apriori Algorithm 489

14.2.3 Example：Market Basket Analysis 492

14.2.4 Unsupervised as Supervised Learning 495

14.2.5 Generalized Association Rules 497

14.2.6 Choice of Supervised Learning Method 499

14.2.7 Example：Market Basket Analysis(Continued) 499

14.3 Cluster Analysis 501

14.3.1 Proximity Matrices 503

14.3.2 Dissimilarities Based on Attributes 503

14.3.3 Object Dissimilarity 505

14.3.4 Clustering Algorithms 507

14.3.5 Combinatorial Algorithms 507

14.3.6 K-means 509

14.3.7 Gaussian Mixtures as Soft K-means Clustering 510

14.3.8 Example：Human Tumor Microarray Data 512

14.3.9 Vector Quantization 514

14.3.10 K-medoids 515

14.3.11 Practical Issues 518

14.3.12 Hierarchical Clustering 520

14.4 Self-Organizing Maps 528

14.5 Principal Components,Curves and Surfaces 534

14.5.1 Principal Components 534

14.5.2 Principal Curves and Surfaces 541

14.5.3 Spectral Clustering 544

14.5.4 Kernel Principal Components 547

14.5.5 Sparse Principal Components 550

14.6 Non-negative Matrix Factorization 553

14.6.1 Archetypal Analysis 554

14.7 Independent Component Analysis and Exploratory Projection Pursuit 557

14.7.1 Latent Variables and Factor Analysis 558

14.7.2 Independent Component Analysis 560

14.7.3 Exploratory Projection Pursuit 565

14.7.4 A Direct Approach to ICA 565

14.8 Multidimensional Scaling 570

14.9 Nonlinear Dimension Reduction and Local Multidimensional Scaling 572

14.10 The Google PageRank Algorithm 576

Bibliographic Notes 578

Exercises 579

15 Random Forests 587

15.1 Introduction 587

15.2 Definition of Random Forests 587

15.3 Details of Random Forests 592

15.3.1 Out of Bag Samples 592

15.3.2 Variable Importance 593

15.3.3 Proximity Plots 595

15.3.4 Random Forests and Overfitting 596

15.4 Analysis of Random Forests 597

15.4.1 Variance and the De-Correlation Effect 597

15.4.2 Bias 600

15.4.3 Adaptive Nearest Neighbors 601

Bibliographic Notes 602

Exercises 603

16 Ensemble Learning 605

16.1 Introduction 605

16.2 Boosting and Regularization Paths 607

16.2.1 Penalized Regression 607

16.2.2 The"Bet on Sparsity"Principle 610

16.2.3 Regularization Paths,Over-fitting and Margins 613

16.3 Learning Ensembles 616

16.3.1 Learning a Good Ensemble 617

16.3.2 Rule Ensembles 622

Bibliographic Notes 623

Exercises 624

17 Undirected Graphical Models 625

17.1 Introduction 625

17.2 Markov Graphs and Their Properties 627

17.3 Undirected Graphical Models for Continuous Variables 630

17.3.1 Estimation of the Parameters when the Graph Structure is Known 631

17.3.2 Estimation of the Graph Structure 635

17.4 Undirected Graphical Models for Discrete Variables 638

17.4.1 Estimation of the Parameters when the Graph Structure is Known 639

17.4.2 Hidden Nodes 641

17.4.3 Estimation of the Graph Structure 642

17.4.4 Restricted Boltzmann Machines 643

Exercises 645

18 High-Dimensional Problems：p＞＞N 649

18.1 When p is Much Bigger than N 649

18.2 Diagonal Linear Discriminant Analysis and Nearest Shrunken Centroids 651

18.3 Linear Classifiers with Quadratic Regularization 654

18.3.1 Regularized Discriminant Analysis 656

18.3.2 Logistic Regression with Quadratic Regularization 657

18.3.3 The Support Vector Classifier 657

18.3.4 Feature Selection 658

18.3.5 Computational Shortcuts When p＞＞N 659

18.4 Linear Classifiers with L1 Regularization 661

18.4.1 Application of Lasso to Protein Mass Spectroscopy 664

18.4.2 The Fused Lasso for Functional Data 666

18.5 Classification When Features are Unavailable 668

18.5.1 Example：String Kernels and Protein Classification 668

18.5.2 Classification and Other Models Using Inner-Product Kernels and Pairwise Distances 670

18.5.3 Example：Abstracts Classification 672

18.6 High-Dimensional Regression：Supervised Principal Components 674

18.6.1 Connection to Latent-Variable Modeling 678

18.6.2 Relationship with Partial Least Squares 680

18.6.3 Pre-Conditioning for Feature Selection 681

18.7 Feature Assessment and the Multiple-Testing Problem 683

18.7.1 The False Discovery Rate 687

18.7.2 Asymmetric Cutpoints and the SAM Procedure 690

18.7.3 A Bayesian Interpretation of the FDR 692

18.8 Bibliographic Notes 693

Exercises 694

References 699

Author Index 729

Index 737