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Springer Texts in Statistics Series Editors: G. Casella S. Fienberg I. Olkin For further volumes: http://www.springer.com/series/417
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Gareth James • Daniela Witten • Trevor Hastie Robert Tibshirani An Introduction to Statistical Learning with Applications in R 123
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Gareth James Daniela Witten Department of Information and Department of Biostatistics Operations Management University of Washington University of Southern California Seattle, WA, USA Los Angeles, CA, USA Trevor Hastie Robert Tibshirani Department of Statistics Department of Statistics Stanford University Stanford University Stanford, CA, USA Stanford, CA, USA ISSN 1431-875X ISBN 978-1-4614-7137-0 ISBN 978-1-4614-7138-7 (eBook) DOI 10.1007/978-1-4614-7138-7 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2013936251 © Springer Science+Business Media New York 2013 (Corrected at 6th printing 2015) This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissim- ilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the pur- pose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publi- cation does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
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To our parents: Alison and Michael James Chiara Nappi and Edward Witten Valerie and Patrick Hastie Vera and Sami Tibshirani and to our families: Michael, Daniel, and Catherine Tessa and Ari Samantha, Timothy, and Lynda Charlie, Ryan, Julie, and Cheryl
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Preface Statistical learning refers to a set of tools for modeling and understanding complex datasets. It is a recently developed area in statistics and blends with parallel developments in computer science and, in particular, machine learning. The field encompasses many methods such as the lasso and sparse regression, classification and regression trees, and boosting and support vector machines. With the explosion of “Big Data” problems, statistical learning has be- come a very hot field in many scientific areas as well as marketing, finance, and other business disciplines. People with statistical learning skills are in high demand. One of the first books in this area—The Elements of Statistical Learning (ESL) (Hastie, Tibshirani, and Friedman)—was published in 2001, with a second edition in 2009. ESL has become a popular text not only in statis- tics but also in related fields. One of the reasons for ESL’s popularity is its relatively accessible style. But ESL is intended for individuals with ad- vanced training in the mathematical sciences. An Introduction to Statistical Learning (ISL) arose from the perceived need for a broader and less tech- nical treatment of these topics. In this new book, we cover many of the same topics as ESL, but we concentrate more on the applications of the methods and less on the mathematical details. We have created labs illus- trating how to implement each of the statistical learning methods using the popular statistical software package R. These labs provide the reader with valuable hands-on experience. This book is appropriate for advanced undergraduates or master’s stu- dents in statistics or related quantitative fields or for individuals in other vii
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viii Preface disciplines who wish to use statistical learning tools to analyze their data. It can be used as a textbook for a course spanning one or two semesters. We would like to thank several readers for valuable comments on prelim- inary drafts of this book: Pallavi Basu, Alexandra Chouldechova, Patrick Danaher, Will Fithian, Luella Fu, Sam Gross, Max Grazier G’Sell, Court- ney Paulson, Xinghao Qiao, Elisa Sheng, Noah Simon, Kean Ming Tan, and Xin Lu Tan. It’s tough to make predictions, especially about the future. -Yogi Berra Los Angeles, USA Gareth James Seattle, USA Daniela Witten Palo Alto, USA Trevor Hastie Palo Alto, USA Robert Tibshirani
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Contents Preface vii 1 Introduction 1 2 Statistical Learning 15 2.1 What Is Statistical Learning? . . . . . . . . . . . . . . . . . 15 2.1.1 Why Estimate f? . . . . . . . . . . . . . . . . . . . . 17 2.1.2 How Do We Estimate f? . . . . . . . . . . . . . . . 21 2.1.3 The Trade-Off Between Prediction Accuracy and Model Interpretability . . . . . . . . . . . . . . 24 2.1.4 Supervised Versus Unsupervised Learning . . . . . . 26 2.1.5 Regression Versus Classification Problems . . . . . . 28 2.2 Assessing Model Accuracy . . . . . . . . . . . . . . . . . . . 29 2.2.1 Measuring the Quality of Fit . . . . . . . . . . . . . 29 2.2.2 The Bias-Variance Trade-Off . . . . . . . . . . . . . 33 2.2.3 The Classification Setting . . . . . . . . . . . . . . . 37 2.3 Lab: Introduction to R . . . . . . . . . . . . . . . . . . . . . 42 2.3.1 Basic Commands . . . . . . . . . . . . . . . . . . . . 42 2.3.2 Graphics . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3.3 Indexing Data . . . . . . . . . . . . . . . . . . . . . 47 2.3.4 Loading Data . . . . . . . . . . . . . . . . . . . . . . 48 2.3.5 Additional Graphical and Numerical Summaries . . 49 2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ix
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x Contents 3 Linear Regression 59 3.1 Simple Linear Regression . . . . . . . . . . . . . . . . . . . 61 3.1.1 Estimating the Coefficients . . . . . . . . . . . . . . 61 3.1.2 Assessing the Accuracy of the Coefficient Estimates . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1.3 Assessing the Accuracy of the Model . . . . . . . . . 68 3.2 Multiple Linear Regression . . . . . . . . . . . . . . . . . . 71 3.2.1 Estimating the Regression Coefficients . . . . . . . . 72 3.2.2 Some Important Questions . . . . . . . . . . . . . . 75 3.3 Other Considerations in the Regression Model . . . . . . . . 82 3.3.1 Qualitative Predictors . . . . . . . . . . . . . . . . . 82 3.3.2 Extensions of the Linear Model . . . . . . . . . . . . 86 3.3.3 Potential Problems . . . . . . . . . . . . . . . . . . . 92 3.4 The Marketing Plan . . . . . . . . . . . . . . . . . . . . . . 102 3.5 Comparison of Linear Regression with K-Nearest Neighbors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.6 Lab: Linear Regression . . . . . . . . . . . . . . . . . . . . . 109 3.6.1 Libraries . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.6.2 Simple Linear Regression . . . . . . . . . . . . . . . 110 3.6.3 Multiple Linear Regression . . . . . . . . . . . . . . 113 3.6.4 Interaction Terms . . . . . . . . . . . . . . . . . . . 115 3.6.5 Non-linear Transformations of the Predictors . . . . 115 3.6.6 Qualitative Predictors . . . . . . . . . . . . . . . . . 117 3.6.7 Writing Functions . . . . . . . . . . . . . . . . . . . 119 3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4 Classification 127 4.1 An Overview of Classification . . . . . . . . . . . . . . . . . 128 4.2 Why Not Linear Regression? . . . . . . . . . . . . . . . . . 129 4.3 Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . 130 4.3.1 The Logistic Model . . . . . . . . . . . . . . . . . . . 131 4.3.2 Estimating the Regression Coefficients . . . . . . . . 133 4.3.3 Making Predictions . . . . . . . . . . . . . . . . . . . 134 4.3.4 Multiple Logistic Regression . . . . . . . . . . . . . . 135 4.3.5 Logistic Regression for >2 Response Classes . . . . . 137 4.4 Linear Discriminant Analysis . . . . . . . . . . . . . . . . . 138 4.4.1 Using Bayes’ Theorem for Classification . . . . . . . 138 4.4.2 Linear Discriminant Analysis for p = 1 . . . . . . . . 139 4.4.3 Linear Discriminant Analysis for p >1 . . . . . . . . 142 4.4.4 Quadratic Discriminant Analysis . . . . . . . . . . . 149 4.5 A Comparison of Classification Methods . . . . . . . . . . . 151 4.6 Lab: Logistic Regression, LDA, QDA, and KNN . . . . . . 154 4.6.1 The Stock Market Data . . . . . . . . . . . . . . . . 154 4.6.2 Logistic Regression . . . . . . . . . . . . . . . . . . . 156 4.6.3 Linear Discriminant Analysis . . . . . . . . . . . . . 161
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Contents xi 4.6.4 Quadratic Discriminant Analysis . . . . . . . . . . . 163 4.6.5 K-Nearest Neighbors . . . . . . . . . . . . . . . . . . 163 4.6.6 An Application to Caravan Insurance Data . . . . . 165 4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5 Resampling Methods 175 5.1 Cross-Validation . . . . . . . . . . . . . . . . . . . . . . . . 176 5.1.1 The Validation Set Approach . . . . . . . . . . . . . 176 5.1.2 Leave-One-Out Cross-Validation . . . . . . . . . . . 178 5.1.3 k-Fold Cross-Validation . . . . . . . . . . . . . . . . 181 5.1.4 Bias-Variance Trade-Off for k-Fold Cross-Validation . . . . . . . . . . . . . . . . . . . . 183 5.1.5 Cross-Validation on Classification Problems . . . . . 184 5.2 The Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . 187 5.3 Lab: Cross-Validation and the Bootstrap . . . . . . . . . . . 190 5.3.1 The Validation Set Approach . . . . . . . . . . . . . 191 5.3.2 Leave-One-Out Cross-Validation . . . . . . . . . . . 192 5.3.3 k-Fold Cross-Validation . . . . . . . . . . . . . . . . 193 5.3.4 The Bootstrap . . . . . . . . . . . . . . . . . . . . . 194 5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 6 Linear Model Selection and Regularization 203 6.1 Subset Selection . . . . . . . . . . . . . . . . . . . . . . . . 205 6.1.1 Best Subset Selection . . . . . . . . . . . . . . . . . 205 6.1.2 Stepwise Selection . . . . . . . . . . . . . . . . . . . 207 6.1.3 Choosing the Optimal Model . . . . . . . . . . . . . 210 6.2 Shrinkage Methods . . . . . . . . . . . . . . . . . . . . . . . 214 6.2.1 Ridge Regression . . . . . . . . . . . . . . . . . . . . 215 6.2.2 The Lasso . . . . . . . . . . . . . . . . . . . . . . . . 219 6.2.3 Selecting the Tuning Parameter . . . . . . . . . . . . 227 6.3 Dimension Reduction Methods . . . . . . . . . . . . . . . . 228 6.3.1 Principal Components Regression . . . . . . . . . . . 230 6.3.2 Partial Least Squares . . . . . . . . . . . . . . . . . 237 6.4 Considerations in High Dimensions . . . . . . . . . . . . . . 238 6.4.1 High-Dimensional Data . . . . . . . . . . . . . . . . 238 6.4.2 What Goes Wrong in High Dimensions? . . . . . . . 239 6.4.3 Regression in High Dimensions . . . . . . . . . . . . 241 6.4.4 Interpreting Results in High Dimensions . . . . . . . 243 6.5 Lab 1: Subset Selection Methods . . . . . . . . . . . . . . . 244 6.5.1 Best Subset Selection . . . . . . . . . . . . . . . . . 244 6.5.2 Forward and Backward Stepwise Selection . . . . . . 247 6.5.3 Choosing Among Models Using the Validation Set Approach and Cross-Validation . . . . . . . . . . 248
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xii Contents 6.6 Lab 2: Ridge Regression and the Lasso . . . . . . . . . . . . 251 6.6.1 Ridge Regression . . . . . . . . . . . . . . . . . . . . 251 6.6.2 The Lasso . . . . . . . . . . . . . . . . . . . . . . . . 255 6.7 Lab 3: PCR and PLS Regression . . . . . . . . . . . . . . . 256 6.7.1 Principal Components Regression . . . . . . . . . . . 256 6.7.2 Partial Least Squares . . . . . . . . . . . . . . . . . 258 6.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 7 Moving Beyond Linearity 265 7.1 Polynomial Regression . . . . . . . . . . . . . . . . . . . . . 266 7.2 Step Functions . . . . . . . . . . . . . . . . . . . . . . . . . 268 7.3 Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . 270 7.4 Regression Splines . . . . . . . . . . . . . . . . . . . . . . . 271 7.4.1 Piecewise Polynomials . . . . . . . . . . . . . . . . . 271 7.4.2 Constraints and Splines . . . . . . . . . . . . . . . . 271 7.4.3 The Spline Basis Representation . . . . . . . . . . . 273 7.4.4 Choosing the Number and Locations of the Knots . . . . . . . . . . . . . . . . . . . . . . 274 7.4.5 Comparison to Polynomial Regression . . . . . . . . 276 7.5 Smoothing Splines . . . . . . . . . . . . . . . . . . . . . . . 277 7.5.1 An Overview of Smoothing Splines . . . . . . . . . . 277 7.5.2 Choosing the Smoothing Parameter λ . . . . . . . . 278 7.6 Local Regression . . . . . . . . . . . . . . . . . . . . . . . . 280 7.7 Generalized Additive Models . . . . . . . . . . . . . . . . . 282 7.7.1 GAMs for Regression Problems . . . . . . . . . . . . 283 7.7.2 GAMs for Classification Problems . . . . . . . . . . 286 7.8 Lab: Non-linear Modeling . . . . . . . . . . . . . . . . . . . 287 7.8.1 Polynomial Regression and Step Functions . . . . . 288 7.8.2 Splines . . . . . . . . . . . . . . . . . . . . . . . . . . 293 7.8.3 GAMs . . . . . . . . . . . . . . . . . . . . . . . . . . 294 7.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 8 Tree-Based Methods 303 8.1 The Basics of Decision Trees . . . . . . . . . . . . . . . . . 303 8.1.1 Regression Trees . . . . . . . . . . . . . . . . . . . . 304 8.1.2 Classification Trees . . . . . . . . . . . . . . . . . . . 311 8.1.3 Trees Versus Linear Models . . . . . . . . . . . . . . 314 8.1.4 Advantages and Disadvantages of Trees . . . . . . . 315 8.2 Bagging, Random Forests, Boosting . . . . . . . . . . . . . 316 8.2.1 Bagging . . . . . . . . . . . . . . . . . . . . . . . . . 316 8.2.2 Random Forests . . . . . . . . . . . . . . . . . . . . 319 8.2.3 Boosting . . . . . . . . . . . . . . . . . . . . . . . . . 321 8.3 Lab: Decision Trees . . . . . . . . . . . . . . . . . . . . . . . 323 8.3.1 Fitting Classification Trees . . . . . . . . . . . . . . 323 8.3.2 Fitting Regression Trees . . . . . . . . . . . . . . . . 327
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Contents xiii 8.3.3 Bagging and Random Forests . . . . . . . . . . . . . 328 8.3.4 Boosting . . . . . . . . . . . . . . . . . . . . . . . . . 330 8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 9 Support Vector Machines 337 9.1 Maximal Margin Classifier . . . . . . . . . . . . . . . . . . . 338 9.1.1 What Is a Hyperplane? . . . . . . . . . . . . . . . . 338 9.1.2 Classification Using a Separating Hyperplane . . . . 339 9.1.3 The Maximal Margin Classifier . . . . . . . . . . . . 341 9.1.4 Construction of the Maximal Margin Classifier . . . 342 9.1.5 The Non-separable Case . . . . . . . . . . . . . . . . 343 9.2 Support Vector Classifiers . . . . . . . . . . . . . . . . . . . 344 9.2.1 Overview of the Support Vector Classifier . . . . . . 344 9.2.2 Details of the Support Vector Classifier . . . . . . . 345 9.3 Support Vector Machines . . . . . . . . . . . . . . . . . . . 349 9.3.1 Classification with Non-linear Decision Boundaries . . . . . . . . . . . . . . . . . . . . . . . 349 9.3.2 The Support Vector Machine . . . . . . . . . . . . . 350 9.3.3 An Application to the Heart Disease Data . . . . . . 354 9.4 SVMs with More than Two Classes . . . . . . . . . . . . . . 355 9.4.1 One-Versus-One Classification . . . . . . . . . . . . . 355 9.4.2 One-Versus-All Classification . . . . . . . . . . . . . 356 9.5 Relationship to Logistic Regression . . . . . . . . . . . . . . 356 9.6 Lab: Support Vector Machines . . . . . . . . . . . . . . . . 359 9.6.1 Support Vector Classifier . . . . . . . . . . . . . . . 359 9.6.2 Support Vector Machine . . . . . . . . . . . . . . . . 363 9.6.3 ROC Curves . . . . . . . . . . . . . . . . . . . . . . 365 9.6.4 SVM with Multiple Classes . . . . . . . . . . . . . . 366 9.6.5 Application to Gene Expression Data . . . . . . . . 366 9.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 10 Unsupervised Learning 373 10.1 The Challenge of Unsupervised Learning . . . . . . . . . . . 373 10.2 Principal Components Analysis . . . . . . . . . . . . . . . . 374 10.2.1 What Are Principal Components? . . . . . . . . . . 375 10.2.2 Another Interpretation of Principal Components . . 379 10.2.3 More on PCA . . . . . . . . . . . . . . . . . . . . . . 380 10.2.4 Other Uses for Principal Components . . . . . . . . 385 10.3 Clustering Methods . . . . . . . . . . . . . . . . . . . . . . . 385 10.3.1 K-Means Clustering . . . . . . . . . . . . . . . . . . 386 10.3.2 Hierarchical Clustering . . . . . . . . . . . . . . . . . 390 10.3.3 Practical Issues in Clustering . . . . . . . . . . . . . 399 10.4 Lab 1: Principal Components Analysis . . . . . . . . . . . . 401
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xiv Contents 10.5 Lab 2: Clustering . . . . . . . . . . . . . . . . . . . . . . . . 404 10.5.1 K-Means Clustering . . . . . . . . . . . . . . . . . . 404 10.5.2 Hierarchical Clustering . . . . . . . . . . . . . . . . . 406 10.6 Lab 3: NCI60 Data Example . . . . . . . . . . . . . . . . . 407 10.6.1 PCA on the NCI60 Data . . . . . . . . . . . . . . . 408 10.6.2 Clustering the Observations of the NCI60 Data . . . 410 10.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 Index 419
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1 Introduction An Overview of Statistical Learning Statistical learning refers to a vast set of tools for understanding data. These tools can be classified as supervised or unsupervised. Broadly speaking, supervised statistical learning involves building a statistical model for pre- dicting, or estimating, an output based on one or more inputs. Problems of this nature occur in fields as diverse as business, medicine, astrophysics, and public policy. With unsupervised statistical learning, there are inputs but no supervising output; nevertheless we can learn relationships and struc- ture from such data. To provide an illustration of some applications of statistical learning, we briefly discuss three real-world data sets that are considered in this book. Wage Data In this application (which we refer to as the Wage data set throughout this book), we examine a number of factors that relate to wages for a group of males from the Atlantic region of the United States. In particular, we wish to understand the association between an employee’s age and education, as well as the calendar year, on his wage. Consider, for example, the left-hand panel of Figure 1.1, which displays wage versus age for each of the individu- als in the data set. There is evidence that wage increases with age but then decreases again after approximately age 60. The blue line, which provides an estimate of the average wage for a given age, makes this trend clearer. G. James et al., An Introduction to Statistical Learning: with Applications in R, 1 Springer Texts in Statistics, DOI 10.1007/978-1-4614-7138-7 1, © Springer Science+Business Media New York 2013
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2 1. Introduction 300 300 300 200 200 200 Wage Wage Wage 100 100 100 50 50 50 20 40 60 80 2003 2006 2009 1 2 3 4 5 Age Year Education Level FIGURE 1.1. Wage data, which contains income survey information for males from the central Atlantic region of the United States. Left: wage as a function of age. On average, wage increases with age until about 60 years of age, at which point it begins to decline. Center: wage as a function of year. There is a slow but steady increase of approximately $10,000 in the average wage between 2003 and 2009. Right: Boxplots displaying wage as a function of education, with 1 indicating the lowest level (no high school diploma) and 5 the highest level (an advanced graduate degree). On average, wage increases with the level of education. Given an employee’s age, we can use this curve to predict his wage. However, it is also clear from Figure 1.1 that there is a significant amount of vari- ability associated with this average value, and so age alone is unlikely to provide an accurate prediction of a particular man’s wage. We also have information regarding each employee’s education level and the year in which the wage was earned. The center and right-hand panels of Figure 1.1, which display wage as a function of both year and education, in- dicate that both of these factors are associated with wage. Wages increase by approximately $10,000, in a roughly linear (or straight-line) fashion, between 2003 and 2009, though this rise is very slight relative to the vari- ability in the data. Wages are also typically greater for individuals with higher education levels: men with the lowest education level (1) tend to have substantially lower wages than those with the highest education level (5). Clearly, the most accurate prediction of a given man’s wage will be obtained by combining his age, his education, and the year. In Chapter 3, we discuss linear regression, which can be used to predict wage from this data set. Ideally, we should predict wage in a way that accounts for the non-linear relationship between wage and age. In Chapter 7, we discuss a class of approaches for addressing this problem. Stock Market Data The Wage data involves predicting a continuous or quantitative output value. This is often referred to as a regression problem. However, in certain cases we may instead wish to predict a non-numerical value—that is, a categorical
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1. Introduction 3 Yesterday Two Days Previous Three Days Previous 6 6 6 4 4 4 2 2 2 0 0 0 −2 −2 −2 Percentage change in S&P Percentage change in S&P Percentage change in S&P −4 −4 −4 Down Up Down Up Down Up Today’s Direction Today’s Direction Today’s Direction FIGURE 1.2. Left: Boxplots of the previous day’s percentage change in the S&P index for the days for which the market increased or decreased, obtained from the Smarket data. Center and Right: Same as left panel, but the percentage changes for 2 and 3 days previous are shown. or qualitative output. For example, in Chapter 4 we examine a stock mar- ket data set that contains the daily movements in the Standard & Poor’s 500 (S&P) stock index over a 5-year period between 2001 and 2005. We refer to this as the Smarket data. The goal is to predict whether the index will increase or decrease on a given day using the past 5 days’ percentage changes in the index. Here the statistical learning problem does not in- volve predicting a numerical value. Instead it involves predicting whether a given day’s stock market performance will fall into the Up bucket or the Down bucket. This is known as a classification problem. A model that could accurately predict the direction in which the market will move would be very useful! The left-hand panel of Figure 1.2 displays two boxplots of the previous day’s percentage changes in the stock index: one for the 648 days for which the market increased on the subsequent day, and one for the 602 days for which the market decreased. The two plots look almost identical, suggest- ing that there is no simple strategy for using yesterday’s movement in the S&P to predict today’s returns. The remaining panels, which display box- plots for the percentage changes 2 and 3 days previous to today, similarly indicate little association between past and present returns. Of course, this lack of pattern is to be expected: in the presence of strong correlations be- tween successive days’ returns, one could adopt a simple trading strategy to generate profits from the market. Nevertheless, in Chapter 4, we explore these data using several different statistical learning methods. Interestingly, there are hints of some weak trends in the data that suggest that, at least for this 5-year period, it is possible to correctly predict the direction of movement in the market approximately 60% of the time (Figure 1.3).
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4 1. Introduction 0.52 0.50 0.48 Predicted Probability 0.46 Down Up Today’s Direction FIGURE 1.3. We fit a quadratic discriminant analysis model to the subset of the Smarket data corresponding to the 2001–2004 time period, and predicted the probability of a stock market decrease using the 2005 data. On average, the predicted probability of decrease is higher for the days in which the market does decrease. Based on these results, we are able to correctly predict the direction of movement in the market 60% of the time. Gene Expression Data The previous two applications illustrate data sets with both input and output variables. However, another important class of problems involves situations in which we only observe input variables, with no corresponding output. For example, in a marketing setting, we might have demographic information for a number of current or potential customers. We may wish to understand which types of customers are similar to each other by grouping individuals according to their observed characteristics. This is known as a clustering problem. Unlike in the previous examples, here we are not trying to predict an output variable. We devote Chapter 10 to a discussion of statistical learning methods for problems in which no natural output variable is available. We consider the NCI60 data set, which consists of 6,830 gene expression measurements for each of 64 cancer cell lines. Instead of predicting a particular output variable, we are interested in determining whether there are groups, or clusters, among the cell lines based on their gene expression measurements. This is a difficult question to address, in part because there are thousands of gene expression measurements per cell line, making it hard to visualize the data. The left-hand panel of Figure 1.4 addresses this problem by represent- ing each of the 64 cell lines using just two numbers, Z1 and Z2. These are the first two principal components of the data, which summarize the 6,830 expression measurements for each cell line down to two numbers or dimensions. While it is likely that this dimension reduction has resulted in
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1. Introduction 5 20 20 0 0 2 2 Z Z −20 −20 −40 −40 −60 −60 −40 −20 0 20 40 60 −40 −20 0 20 40 60 Z1 Z1 FIGURE 1.4. Left: Representation of the NCI60 gene expression data set in a two-dimensional space, Z1 and Z2. Each point corresponds to one of the 64 cell lines. There appear to be four groups of cell lines, which we have represented using different colors. Right: Same as left panel except that we have represented each of the 14 different types of cancer using a different colored symbol. Cell lines corresponding to the same cancer type tend to be nearby in the two-dimensional space. some loss of information, it is now possible to visually examine the data for evidence of clustering. Deciding on the number of clusters is often a diffi- cult problem. But the left-hand panel of Figure 1.4 suggests at least four groups of cell lines, which we have represented using separate colors. We can now examine the cell lines within each cluster for similarities in their types of cancer, in order to better understand the relationship between gene expression levels and cancer. In this particular data set, it turns out that the cell lines correspond to 14 different types of cancer. (However, this information was not used to create the left-hand panel of Figure 1.4.) The right-hand panel of Fig- ure 1.4 is identical to the left-hand panel, except that the 14 cancer types are shown using distinct colored symbols. There is clear evidence that cell lines with the same cancer type tend to be located near each other in this two-dimensional representation. In addition, even though the cancer infor- mation was not used to produce the left-hand panel, the clustering obtained does bear some resemblance to some of the actual cancer types observed in the right-hand panel. This provides some independent verification of the accuracy of our clustering analysis. A Brief History of Statistical Learning Though the term statistical learning is fairly new, many of the concepts that underlie the field were developed long ago. At the beginning of the nineteenth century, Legendre and Gauss published papers on the method
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6 1. Introduction of least squares, which implemented the earliest form of what is now known as linear regression. The approach was first successfully applied to problems in astronomy. Linear regression is used for predicting quantitative values, such as an individual’s salary. In order to predict qualitative values, such as whether a patient survives or dies, or whether the stock market increases or decreases, Fisher proposed linear discriminant analysis in 1936. In the 1940s, various authors put forth an alternative approach, logistic regression. In the early 1970s, Nelder and Wedderburn coined the term generalized linear models for an entire class of statistical learning methods that include both linear and logistic regression as special cases. By the end of the 1970s, many more techniques for learning from data were available. However, they were almost exclusively linear methods, be- cause fitting non-linear relationships was computationally infeasible at the time. By the 1980s, computing technology had finally improved sufficiently that non-linear methods were no longer computationally prohibitive. In mid 1980s Breiman, Friedman, Olshen and Stone introduced classification and regression trees, and were among the first to demonstrate the power of a detailed practical implementation of a method, including cross-validation for model selection. Hastie and Tibshirani coined the term generalized addi- tive models in 1986 for a class of non-linear extensions to generalized linear models, and also provided a practical software implementation. Since that time, inspired by the advent of machine learning and other disciplines, statistical learning has emerged as a new subfield in statistics, focused on supervised and unsupervised modeling and prediction. In recent years, progress in statistical learning has been marked by the increasing availability of powerful and relatively user-friendly software, such as the popular and freely available R system. This has the potential to continue the transformation of the field from a set of techniques used and developed by statisticians and computer scientists to an essential toolkit for a much broader community. This Book The Elements of Statistical Learning (ESL) by Hastie, Tibshirani, and Friedman was first published in 2001. Since that time, it has become an important reference on the fundamentals of statistical machine learning. Its success derives from its comprehensive and detailed treatment of many important topics in statistical learning, as well as the fact that (relative to many upper-level statistics textbooks) it is accessible to a wide audience. However, the greatest factor behind the success of ESL has been its topical nature. At the time of its publication, interest in the field of statistical
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