An Introduction to Statistical Learning (PDF)

Springer Texts in Statistics Gareth James Daniela Witten Trevor Hastie Robert Tibshirani An Introduction to Statistical Learning with Applications in R Springer Texts in Statistics 103 Series Editors: G. Casella S. Fienberg I. Olkin For further volumes: http://www.springer.com/series/417 Gareth James Daniela Witten Trevor Hastie Robert Tibshirani An Introduction to Statistical Learning with Applications in R 123 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 This work is subject to copyright. 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Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) 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 Ari Samantha, Timothy, and Lynda Charlie, Ryan, Julie, and Cheryl 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 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 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 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 Contents xi 4.6.4 Quadratic Discriminant Analysis........... 162 4.6.5 K-Nearest Neighbors.................. 163 4.6.6 An Application to Caravan Insurance Data..... 164 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 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.................... 320 8.2.3 Boosting......................... 321 8.3 Lab: Decision Trees....................... 324 8.3.1 Fitting Classification Trees.............. 324 8.3.2 Fitting Regression Trees................ 327 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 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 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 103, DOI 10.1007/978-1-4614-7138-7 1, © Springer Science+Business Media New York 2013 2 1. Introduction 300 300 300 200 200 200 Wage Wage Wage 50 100 50 100 50 100 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 1. Introduction 3 Yesterday Two Days Previous Three Days Previous 6 6 6 Percentage change in S&P Percentage change in S&P Percentage change in S&P 4 4 4 2 2 2 0 0 0 −2 −2 −2 −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). 4 1. Introduction 0.52 Predicted Probability 0.50 0.48 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 1. Introduction 5 20 20 0 0 Z2 Z2 −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 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 1. Introduction 7 learning was starting to explode. ESL provided one of the first accessible and comprehensive introductions to the topic. Since ESL was first published, the field of statistical learning has con- tinued to flourish. The field’s expansion has taken two forms. The most obvious growth has involved the development of new and improved statis- tical learning approaches aimed at answering a range of scientific questions across a number of fields. However, the field of statistical learning has also expanded its audience. In the 1990s, increases in computational power generated a surge of interest in the field from non-statisticians who were eager to use cutting-edge statistical tools to analyze their data. Unfortu- nately, the highly technical nature of these approaches meant that the user community remained primarily restricted to experts in statistics, computer science, and related fields with the training (and time) to understand and implement them. In recent years, new and improved software packages have significantly eased the implementation burden for many statistical learning methods. At the same time, there has been growing recognition across a number of fields, from business to health care to genetics to the social sciences and beyond, that statistical learning is a powerful tool with important practical applications. As a result, the field has moved from one of primarily academic interest to a mainstream discipline, with an enormous potential audience. This trend will surely continue with the increasing availability of enormous quantities of data and the software to analyze it. The purpose of An Introduction to Statistical Learning (ISL) is to facili- tate the transition of statistical learning from an academic to a mainstream field. ISL is not intended to replace ESL, which is a far more comprehen- sive text both in terms of the number of approaches considered and the depth to which they are explored. We consider ESL to be an important companion for professionals (with graduate degrees in statistics, machine learning, or related fields) who need to understand the technical details behind statistical learning approaches. However, the community of users of statistical learning techniques has expanded to include individuals with a wider range of interests and backgrounds. Therefore, we believe that there is now a place for a less technical and more accessible version of ESL. In teaching these topics over the years, we have discovered that they are of interest to master’s and PhD students in fields as disparate as business administration, biology, and computer science, as well as to quantitatively- oriented upper-division undergraduates. It is important for this diverse group to be able to understand the models, intuitions, and strengths and weaknesses of the various approaches. But for this audience, many of the technical details behind statistical learning methods, such as optimiza- tion algorithms and theoretical properties, are not of primary interest. We believe that these students do not need a deep understanding of these aspects in order to become informed users of the various methodologies, and 8 1. Introduction in order to contribute to their chosen fields through the use of statistical learning tools. ISLR is based on the following four premises. 1. Many statistical learning methods are relevant and useful in a wide range of academic and non-academic disciplines, beyond just the sta- tistical sciences. We believe that many contemporary statistical learn- ing procedures should, and will, become as widely available and used as is currently the case for classical methods such as linear regres- sion. As a result, rather than attempting to consider every possible approach (an impossible task), we have concentrated on presenting the methods that we believe are most widely applicable. 2. Statistical learning should not be viewed as a series of black boxes. No single approach will perform well in all possible applications. With- out understanding all of the cogs inside the box, or the interaction between those cogs, it is impossible to select the best box. Hence, we have attempted to carefully describe the model, intuition, assump- tions, and trade-oﬀs behind each of the methods that we consider. 3. While it is important to know what job is performed by each cog, it is not necessary to have the skills to construct the machine inside the box! Thus, we have minimized discussion of technical details related to fitting procedures and theoretical properties. We assume that the reader is comfortable with basic mathematical concepts, but we do not assume a graduate degree in the mathematical sciences. For in- stance, we have almost completely avoided the use of matrix algebra, and it is possible to understand the entire book without a detailed knowledge of matrices and vectors. 4. We presume that the reader is interested in applying statistical learn- ing methods to real-world problems. In order to facilitate this, as well as to motivate the techniques discussed, we have devoted a section within each chapter to R computer labs. In each lab, we walk the reader through a realistic application of the methods considered in that chapter. When we have taught this material in our courses, we have allocated roughly one-third of classroom time to working through the labs, and we have found them to be extremely useful. Many of the less computationally-oriented students who were ini- tially intimidated by R’s command level interface got the hang of things over the course of the quarter or semester. We have used R because it is freely available and is powerful enough to implement all of the methods discussed in the book. It also has optional packages that can be downloaded to implement literally thousands of addi- tional methods. Most importantly, R is the language of choice for academic statisticians, and new approaches often become available in 1. Introduction 9 R years before they are implemented in commercial packages. How- ever, the labs in ISL are self-contained, and can be skipped if the reader wishes to use a diﬀerent software package or does not wish to apply the methods discussed to real-world problems. Who Should Read This Book? This book is intended for anyone who is interested in using modern statis- tical methods for modeling and prediction from data. This group includes scientists, engineers, data analysts, or quants, but also less technical indi- viduals with degrees in non-quantitative fields such as the social sciences or business. We expect that the reader will have had at least one elementary course in statistics. Background in linear regression is also useful, though not required, since we review the key concepts behind linear regression in Chapter 3. The mathematical level of this book is modest, and a detailed knowledge of matrix operations is not required. This book provides an in- troduction to the statistical programming language R. Previous exposure to a programming language, such as MATLAB or Python, is useful but not required. We have successfully taught material at this level to master’s and PhD students in business, computer science, biology, earth sciences, psychology, and many other areas of the physical and social sciences. This book could also be appropriate for advanced undergraduates who have already taken a course on linear regression. In the context of a more mathematically rigorous course in which ESL serves as the primary textbook, ISL could be used as a supplementary text for teaching computational aspects of the various approaches. Notation and Simple Matrix Algebra Choosing notation for a textbook is always a diﬃcult task. For the most part we adopt the same notational conventions as ESL. We will use n to represent the number of distinct data points, or observa- tions, in our sample. We will let p denote the number of variables that are available for use in making predictions. For example, the Wage data set con- sists of 12 variables for 3,000 people, so we have n = 3,000 observations and p = 12 variables (such as year, age, wage, and more). Note that throughout this book, we indicate variable names using colored font: Variable Name. In some examples, p might be quite large, such as on the order of thou- sands or even millions; this situation arises quite often, for example, in the analysis of modern biological data or web-based advertising data. 10 1. Introduction In general, we will let xij represent the value of the jth variable for the ith observation, where i = 1, 2,... , n and j = 1, 2,... , p. Throughout this book, i will be used to index the samples or observations (from 1 to n) and j will be used to index the variables (from 1 to p). We let X denote a n × p matrix whose (i, j)th element is xij. That is, ⎛ ⎞ x11 x12... x1p ⎜ x21 x22... x2p ⎟ ⎜ ⎟ X=⎜....... ⎟. ⎝..... ⎠ xn1 xn2... xnp For readers who are unfamiliar with matrices, it is useful to visualize X as a spreadsheet of numbers with n rows and p columns. At times we will be interested in the rows of X, which we write as x1 , x2 ,... , xn. Here xi is a vector of length p, containing the p variable measurements for the ith observation. That is, ⎛ ⎞ xi1 ⎜xi2 ⎟ ⎜ ⎟ xi = ⎜. ⎟. (1.1) ⎝.. ⎠ xip (Vectors are by default represented as columns.) For example, for the Wage data, xi is a vector of length 12, consisting of year, age, wage, and other values for the ith individual. At other times we will instead be interested in the columns of X, which we write as x1 , x2 ,... , xp. Each is a vector of length n. That is, ⎛ ⎞ x1j ⎜ x2j ⎟ ⎜ ⎟ xj = ⎜. ⎟. ⎝.. ⎠ xnj For example, for the Wage data, x1 contains the n = 3,000 values for year. Using this notation, the matrix X can be written as ' ( X = x1 x2 ··· xp , or ⎛ T⎞ x1 ⎜xT2 ⎟ ⎜ ⎟ X = ⎜. ⎟. ⎝.. ⎠ xTn 1. Introduction 11 The T notation denotes the transpose of a matrix or vector. So, for example, ⎛ ⎞ x11 x21... xn1 ⎜x12 x22... xn2 ⎟ ⎜ ⎟ XT = ⎜..... ⎟ , ⎝.... ⎠ x1p x2p... xnp while ' ( xTi = xi1 xi2 ··· xip. We use yi to denote the ith observation of the variable on which we wish to make predictions, such as wage. Hence, we write the set of all n observations in vector form as ⎛ ⎞ y1 ⎜ y2 ⎟ ⎜ ⎟ y = ⎜. ⎟. ⎝.. ⎠ yn Then our observed data consists of {(x1 , y1 ), (x2 , y2 ),... , (xn , yn )}, where each xi is a vector of length p. (If p = 1, then xi is simply a scalar.) In this text, a vector of length n will always be denoted in lower case bold ; e.g. ⎛ ⎞ a1 ⎜ a2 ⎟ ⎜ ⎟ a = ⎜. ⎟. ⎝.. ⎠ an However, vectors that are not of length n (such as feature vectors of length p, as in (1.1)) will be denoted in lower case normal font, e.g. a. Scalars will also be denoted in lower case normal font, e.g. a. In the rare cases in which these two uses for lower case normal font lead to ambiguity, we will clarify which use is intended. Matrices will be denoted using bold capitals, such as A. Random variables will be denoted using capital normal font, e.g. A, regardless of their dimensions. Occasionally we will want to indicate the dimension of a particular ob- ject. To indicate that an object is a scalar, we will use the notation a ∈ R. To indicate that it is a vector of length k, we will use a ∈ Rk (or a ∈ Rn if it is of length n). We will indicate that an object is a r × s matrix using A ∈ Rr×s. We have avoided using matrix algebra whenever possible. However, in a few instances it becomes too cumbersome to avoid it entirely. In these rare instances it is important to understand the concept of multiplying two matrices. Suppose that A ∈ Rr×d and B ∈ Rd×s. Then the product 12 1. Introduction of A and B is denoted AB. The (i, j)th element of AB is computed by multiplying each element of the ith row of A ) by the corresponding element d of the jth column of B. That is, (AB)ij = k=1 aik bkj. As an example, consider * + * + 1 2 5 6 A= and B =. 3 4 7 8 Then * +* + * + * + 1 2 5 6 1×5+2×7 1×6+2×8 19 22 AB = = =. 3 4 7 8 3×5+4×7 3×6+4×8 43 50 Note that this operation produces an r × s matrix. It is only possible to compute AB if the number of columns of A is the same as the number of rows of B. Organization of This Book Chapter 2 introduces the basic terminology and concepts behind statisti- cal learning. This chapter also presents the K-nearest neighbor classifier, a very simple method that works surprisingly well on many problems. Chap- ters 3 and 4 cover classical linear methods for regression and classification. In particular, Chapter 3 reviews linear regression, the fundamental start- ing point for all regression methods. In Chapter 4 we discuss two of the most important classical classification methods, logistic regression and lin- ear discriminant analysis. A central problem in all statistical learning situations involves choosing the best method for a given application. Hence, in Chapter 5 we intro- duce cross-validation and the bootstrap, which can be used to estimate the accuracy of a number of diﬀerent methods in order to choose the best one. Much of the recent research in statistical learning has concentrated on non-linear methods. However, linear methods often have advantages over their non-linear competitors in terms of interpretability and sometimes also accuracy. Hence, in Chapter 6 we consider a host of linear methods, both classical and more modern, which oﬀer potential improvements over stan- dard linear regression. These include stepwise selection, ridge regression, principal components regression, partial least squares, and the lasso. The remaining chapters move into the world of non-linear statistical learning. We first introduce in Chapter 7 a number of non-linear methods that work well for problems with a single input variable. We then show how these methods can be used to fit non-linear additive models for which there is more than one input. In Chapter 8, we investigate tree-based methods, including bagging, boosting, and random forests. Support vector machines, a set of approaches for performing both linear and non-linear classification, 1. Introduction 13 are discussed in Chapter 9. Finally, in Chapter 10, we consider a setting in which we have input variables but no output variable. In particular, we present principal components analysis, K-means clustering, and hierarchi- cal clustering. At the end of each chapter, we present one or more R lab sections in which we systematically work through applications of the various meth- ods discussed in that chapter. These labs demonstrate the strengths and weaknesses of the various approaches, and also provide a useful reference for the syntax required to implement the various methods. The reader may choose to work through the labs at his or her own pace, or the labs may be the focus of group sessions as part of a classroom environment. Within each R lab, we present the results that we obtained when we performed the lab at the time of writing this book. However, new versions of R are continuously released, and over time, the packages called in the labs will be updated. Therefore, in the future, it is possible that the results shown in the lab sections may no longer correspond precisely to the results obtained by the reader who performs the labs. As necessary, we will post updates to the labs on the book website. We use the symbol to denote sections or exercises that contain more challenging concepts. These can be easily skipped by readers who do not wish to delve as deeply into the material, or who lack the mathematical background. Data Sets Used in Labs and Exercises In this textbook, we illustrate statistical learning methods using applica- tions from marketing, finance, biology, and other areas. The ISLR package available on the book website contains a number of data sets that are required in order to perform the labs and exercises associated with this book. One other data set is contained in the MASS library, and yet another is part of the base R distribution. Table 1.1 contains a summary of the data sets required to perform the labs and exercises. A couple of these data sets are also available as text files on the book website, for use in Chapter 2. Book Website The website for this book is located at www.StatLearning.com 14 1. Introduction Name Description Auto Gas mileage, horsepower, and other information for cars. Boston Housing values and other information about Boston suburbs. Caravan Information about individuals oﬀered caravan insurance. Carseats Information about car seat sales in 400 stores. College Demographic characteristics, tuition, and more for USA colleges. Default Customer default records for a credit card company. Hitters Records and salaries for baseball players. Khan Gene expression measurements for four cancer types. NCI60 Gene expression measurements for 64 cancer cell lines. OJ Sales information for Citrus Hill and Minute Maid orange juice. Portfolio Past values of financial assets, for use in portfolio allocation. Smarket Daily percentage returns for S&P 500 over a 5-year period. USArrests Crime statistics per 100,000 residents in 50 states of USA. Wage Income survey data for males in central Atlantic region of USA. Weekly 1,089 weekly stock market returns for 21 years. TABLE 1.1. A list of data sets needed to perform the labs and exercises in this textbook. All data sets are available in the ISLR library, with the exception of Boston (part of MASS) and USArrests (part of the base R distribution). It contains a number of resources, including the R package associated with this book, and some additional data sets. Acknowledgements A few of the plots in this book were taken from ESL: Figures 6.7, 8.3, and 10.12. All other plots are new to this book. 2 Statistical Learning 2.1 What Is Statistical Learning? In order to motivate our study of statistical learning, we begin with a simple example. Suppose that we are statistical consultants hired by a client to provide advice on how to improve sales of a particular product. The Advertising data set consists of the sales of that product in 200 diﬀerent markets, along with advertising budgets for the product in each of those markets for three diﬀerent media: TV, radio, and newspaper. The data are displayed in Figure 2.1. It is not possible for our client to directly increase sales of the product. On the other hand, they can control the advertising expenditure in each of the three media. Therefore, if we determine that there is an association between advertising and sales, then we can instruct our client to adjust advertising budgets, thereby indirectly increasing sales. In other words, our goal is to develop an accurate model that can be used to predict sales on the basis of the three media budgets. In this setting, the advertising budgets are input variables while sales input is an output variable. The input variables are typically denoted using the variable symbol X, with a subscript to distinguish them. So X1 might be the TV output variable budget, X2 the radio budget, and X3 the newspaper budget. The inputs go by diﬀerent names, such as predictors, independent variables, features, predictor or sometimes just variables. The output variable—in this case, sales—is independent often called the response or dependent variable, and is typically denoted variable using the symbol Y. Throughout this book, we will use all of these terms feature variable interchangeably. response dependent variable G. James et al., An Introduction to Statistical Learning: with Applications in R, 15 Springer Texts in Statistics 103, DOI 10.1007/978-1-4614-7138-7 2, © Springer Science+Business Media New York 2013 16 2. Statistical Learning 25 25 25 20 20 20 Sales Sales Sales 15 15 15 10 10 10 5 5 5 0 50 100 200 300 0 10 20 30 40 50 0 20 40 60 80 100 TV Radio Newspaper FIGURE 2.1. The Advertising data set. The plot displays sales, in thousands of units, as a function of TV, radio, and newspaper budgets, in thousands of dollars, for 200 diﬀerent markets. In each plot we show the simple least squares fit of sales to that variable, as described in Chapter 3. In other words, each blue line represents a simple model that can be used to predict sales using TV, radio, and newspaper, respectively. More generally, suppose that we observe a quantitative response Y and p diﬀerent predictors, X1 , X2 ,... , Xp. We assume that there is some relationship between Y and X = (X1 , X2 ,... , Xp ), which can be written in the very general form Y = f (X) + ϵ. (2.1) Here f is some fixed but unknown function of X1 ,... , Xp , and ϵ is a random error term, which is independent of X and has mean zero. In this formula- error term tion, f represents the systematic information that X provides about Y. systematic As another example, consider the left-hand panel of Figure 2.2, a plot of income versus years of education for 30 individuals in the Income data set. The plot suggests that one might be able to predict income using years of education. However, the function f that connects the input variable to the output variable is in general unknown. In this situation one must estimate f based on the observed points. Since Income is a simulated data set, f is known and is shown by the blue curve in the right-hand panel of Figure 2.2. The vertical lines represent the error terms ϵ. We note that some of the 30 observations lie above the blue curve and some lie below it; overall, the errors have approximately mean zero. In general, the function f may involve more than one input variable. In Figure 2.3 we plot income as a function of years of education and seniority. Here f is a two-dimensional surface that must be estimated based on the observed data. 2.1 What Is Statistical Learning? 17 80 80 70 70 60 60 Income Income 50 50 40 40 30 30 20 20 10 12 14 16 18 20 22 10 12 14 16 18 20 22 Years of Education Years of Education FIGURE 2.2. The Income data set. Left: The red dots are the observed values of income (in tens of thousands of dollars) and years of education for 30 indi- viduals. Right: The blue curve represents the true underlying relationship between income and years of education, which is generally unknown (but is known in this case because the data were simulated). The black lines represent the error associated with each observation. Note that some errors are positive (if an ob- servation lies above the blue curve) and some are negative (if an observation lies below the curve). Overall, these errors have approximately mean zero. In essence, statistical learning refers to a set of approaches for estimating f. In this chapter we outline some of the key theoretical concepts that arise in estimating f , as well as tools for evaluating the estimates obtained. 2.1.1 Why Estimate f ? There are two main reasons that we may wish to estimate f : prediction and inference. We discuss each in turn. Prediction In many situations, a set of inputs X are readily available, but the output Y cannot be easily obtained. In this setting, since the error term averages to zero, we can predict Y using Ŷ = fˆ(X), (2.2) where fˆ represents our estimate for f , and Ŷ represents the resulting pre- diction for Y. In this setting, fˆ is often treated as a black box, in the sense that one is not typically concerned with the exact form of fˆ, provided that it yields accurate predictions for Y. 18 2. Statistical Learning Incom e yit Ye or a ni rs Se of Ed uc ati on FIGURE 2.3. The plot displays income as a function of years of education and seniority in the Income data set. The blue surface represents the true un- derlying relationship between income and years of education and seniority, which is known since the data are simulated. The red dots indicate the observed values of these quantities for 30 individuals. As an example, suppose that X1 ,... , Xp are characteristics of a patient’s blood sample that can be easily measured in a lab, and Y is a variable encoding the patient’s risk for a severe adverse reaction to a particular drug. It is natural to seek to predict Y using X, since we can then avoid giving the drug in question to patients who are at high risk of an adverse reaction—that is, patients for whom the estimate of Y is high. The accuracy of Ŷ as a prediction for Y depends on two quantities, which we will call the reducible error and the irreducible error. In general, reducible fˆ will not be a perfect estimate for f , and this inaccuracy will introduce error some error. This error is reducible because we can potentially improve the irreducible error accuracy of fˆ by using the most appropriate statistical learning technique to estimate f. However, even if it were possible to form a perfect estimate for f , so that our estimated response took the form Ŷ = f (X), our prediction would still have some error in it! This is because Y is also a function of ϵ, which, by definition, cannot be predicted using X. Therefore, variability associated with ϵ also aﬀects the accuracy of our predictions. This is known as the irreducible error, because no matter how well we estimate f , we cannot reduce the error introduced by ϵ. Why is the irreducible error larger than zero? The quantity ϵ may con- tain unmeasured variables that are useful in predicting Y : since we don’t measure them, f cannot use them for its prediction. The quantity ϵ may also contain unmeasurable variation. For example, the risk of an adverse reaction might vary for a given patient on a given day, depending on 2.1 What Is Statistical Learning? 19 manufacturing variation in the drug itself or the patient’s general feeling of well-being on that day. Consider a given estimate fˆ and a set of predictors X, which yields the prediction Ŷ = fˆ(X). Assume for a moment that both fˆ and X are fixed. Then, it is easy to show that E(Y − Ŷ )2 = E[f (X) + ϵ − fˆ(X)]2 = [f (X) − fˆ(X)]2 + Var(ϵ) , (2.3) , -. / , -. / Reducible Irreducible 2 where E(Y − Ŷ ) represents the average, or expected value, of the squared expected diﬀerence between the predicted and actual value of Y , and Var(ϵ) repre- value sents the variance associated with the error term ϵ. variance The focus of this book is on techniques for estimating f with the aim of minimizing the reducible error. It is important to keep in mind that the irreducible error will always provide an upper bound on the accuracy of our prediction for Y. This bound is almost always unknown in practice. Inference We are often interested in understanding the way that Y is aﬀected as X1 ,... , Xp change. In this situation we wish to estimate f , but our goal is not necessarily to make predictions for Y. We instead want to understand the relationship between X and Y , or more specifically, to understand how Y changes as a function of X1 ,... , Xp. Now fˆ cannot be treated as a black box, because we need to know its exact form. In this setting, one may be interested in answering the following questions: Which predictors are associated with the response? It is often the case that only a small fraction of the available predictors are substantially associated with Y. Identifying the few important predictors among a large set of possible variables can be extremely useful, depending on the application. What is the relationship between the response and each predictor? Some predictors may have a positive relationship with Y , in the sense that increasing the predictor is associated with increasing values of Y. Other predictors may have the opposite relationship. Depending on the complexity of f , the relationship between the response and a given predictor may also depend on the values of the other predictors. Can the relationship between Y and each predictor be adequately sum- marized using a linear equation, or is the relationship more compli- cated? Historically, most methods for estimating f have taken a linear form. In some situations, such an assumption is reasonable or even de- sirable. But often the true relationship is more complicated, in which case a linear model may not provide an accurate representation of the relationship between the input and output variables. 20 2. Statistical Learning In this book, we will see a number of examples that fall into the prediction setting, the inference setting, or a combination of the two. For instance, consider a company that is interested in conducting a direct-marketing campaign. The goal is to identify individuals who will respond positively to a mailing, based on observations of demographic vari- ables measured on each individual. In this case, the demographic variables serve as predictors, and response to the marketing campaign (either pos- itive or negative) serves as the outcome. The company is not interested in obtaining a deep understanding of the relationships between each in- dividual predictor and the response; instead, the company simply wants an accurate model to predict the response using the predictors. This is an example of modeling for prediction. In contrast, consider the Advertising data illustrated in Figure 2.1. One may be interested in answering questions such as: – Which media contribute to sales? – Which media generate the biggest boost in sales? or – How much increase in sales is associated with a given increase in TV advertising? This situation falls into the inference paradigm. Another example involves modeling the brand of a product that a customer might purchase based on variables such as price, store location, discount levels, competition price, and so forth. In this situation one might really be most interested in how each of the individual variables aﬀects the probability of purchase. For instance, what eﬀect will changing the price of a product have on sales? This is an example of modeling for inference. Finally, some modeling could be conducted both for prediction and infer- ence. For example, in a real estate setting, one may seek to relate values of homes to inputs such as crime rate, zoning, distance from a river, air qual- ity, schools, income level of community, size of houses, and so forth. In this case one might be interested in how the individual input variables aﬀect the prices—that is, how much extra will a house be worth if it has a view of the river? This is an inference problem. Alternatively, one may simply be interested in predicting the value of a home given its characteristics: is this house under- or over-valued? This is a prediction problem. Depending on whether our ultimate goal is prediction, inference, or a combination of the two, diﬀerent methods for estimating f may be appro- priate. For example, linear models allow for relatively simple and inter- linear model pretable inference, but may not yield as accurate predictions as some other approaches. In contrast, some of the highly non-linear approaches that we discuss in the later chapters of this book can potentially provide quite accu- rate predictions for Y , but this comes at the expense of a less interpretable model for which inference is more challenging. 2.1 What Is Statistical Learning? 21 2.1.2 How Do We Estimate f ? Throughout this book, we explore many linear and non-linear approaches for estimating f. However, these methods generally share certain charac- teristics. We provide an overview of these shared characteristics in this section. We will always assume that we have observed a set of n diﬀerent data points. For example in Figure 2.2 we observed n = 30 data points. These observations are called the training data because we will use these training data observations to train, or teach, our method how to estimate f. Let xij represent the value of the jth predictor, or input, for observation i, where i = 1, 2,... , n and j = 1, 2,... , p. Correspondingly, let yi represent the response variable for the ith observation. Then our training data consist of {(x1 , y1 ), (x2 , y2 ),... , (xn , yn )} where xi = (xi1 , xi2 ,... , xip )T. Our goal is to apply a statistical learning method to the training data in order to estimate the unknown function f. In other words, we want to find a function fˆ such that Y ≈ fˆ(X) for any observation (X, Y ). Broadly speaking, most statistical learning methods for this task can be character- ized as either parametric or non-parametric. We now briefly discuss these parametric two types of approaches. non- parametric Parametric Methods Parametric methods involve a two-step model-based approach. 1. First, we make an assumption about the functional form, or shape, of f. For example, one very simple assumption is that f is linear in X: f (X) = β0 + β1 X1 + β2 X2 +... + βp Xp. (2.4) This is a linear model, which will be discussed extensively in Chap- ter 3. Once we have assumed that f is linear, the problem of estimat- ing f is greatly simplified. Instead of having to estimate an entirely arbitrary p-dimensional function f (X), one only needs to estimate the p + 1 coeﬃcients β0 , β1 ,... , βp. 2. After a model has been selected, we need a procedure that uses the training data to fit or train the model. In the case of the linear model fit (2.4), we need to estimate the parameters β0 , β1 ,... , βp. That is, we train want to find values of these parameters such that Y ≈ β0 + β1 X 1 + β2 X 2 +... + βp X p. The most common approach to fitting the model (2.4) is referred to as (ordinary) least squares, which we discuss in Chapter 3. How- least squares ever, least squares is one of many possible ways way to fit the linear model. In Chapter 6, we discuss other approaches for estimating the parameters in (2.4). The model-based approach just described is referred to as parametric; it reduces the problem of estimating f down to one of estimating a set of 22 2. Statistical Learning Incom e ity Ye r io ar n so Se fE du ca tio n FIGURE 2.4. A linear model fit by least squares to the Income data from Fig- ure 2.3. The observations are shown in red, and the yellow plane indicates the least squares fit to the data. parameters. Assuming a parametric form for f simplifies the problem of estimating f because it is generally much easier to estimate a set of pa- rameters, such as β0 , β1 ,... , βp in the linear model (2.4), than it is to fit an entirely arbitrary function f. The potential disadvantage of a paramet- ric approach is that the model we choose will usually not match the true unknown form of f. If the chosen model is too far from the true f , then our estimate will be poor. We can try to address this problem by choos- ing flexible models that can fit many diﬀerent possible functional forms flexible for f. But in general, fitting a more flexible model requires estimating a greater number of parameters. These more complex models can lead to a phenomenon known as overfitting the data, which essentially means they overfitting follow the errors, or noise, too closely. These issues are discussed through- noise out this book. Figure 2.4 shows an example of the parametric approach applied to the Income data from Figure 2.3. We have fit a linear model of the form income ≈ β0 + β1 × education + β2 × seniority. Since we have assumed a linear relationship between the response and the two predictors, the entire fitting problem reduces to estimating β0 , β1 , and β2 , which we do using least squares linear regression. Comparing Figure 2.3 to Figure 2.4, we can see that the linear fit given in Figure 2.4 is not quite right: the true f has some curvature that is not captured in the linear fit. However, the linear fit still appears to do a reasonable job of capturing the positive relationship between years of education and income, as well as the 2.1 What Is Statistical Learning? 23 Incom e ity Ye or a ni rs Se of Ed uc ati on FIGURE 2.5. A smooth thin-plate spline fit to the Income data from Figure 2.3 is shown in yellow; the observations are displayed in red. Splines are discussed in Chapter 7. slightly less positive relationship between seniority and income. It may be that with such a small number of observations, this is the best we can do. Non-parametric Methods Non-parametric methods do not make explicit assumptions about the func- tional form of f. Instead they seek an estimate of f that gets as close to the data points as possible without being too rough or wiggly. Such approaches can have a major advantage over parametric approaches: by avoiding the assumption of a particular functional form for f , they have the potential to accurately fit a wider range of possible shapes for f. Any parametric approach brings with it the possibility that the functional form used to estimate f is very diﬀerent from the true f , in which case the resulting model will not fit the data well. In contrast, non-parametric approaches completely avoid this danger, since essentially no assumption about the form of f is made. But non-parametric approaches do suﬀer from a major disadvantage: since they do not reduce the problem of estimating f to a small number of parameters, a very large number of observations (far more than is typically needed for a parametric approach) is required in order to obtain an accurate estimate for f. An example of a non-parametric approach to fitting the Income data is shown in Figure 2.5. A thin-plate spline is used to estimate f. This ap- thin-plate proach does not impose any pre-specified model on f. It instead attempts spline to produce an estimate for f that is as close as possible to the observed data, subject to the fit—that is, the yellow surface in Figure 2.5—being 24 2. Statistical Learning Incom e yit Ye or ar ni so Se fE du ca tio n FIGURE 2.6. A rough thin-plate spline fit to the Income data from Figure 2.3. This fit makes zero errors on the training data. smooth. In this case, the non-parametric fit has produced a remarkably ac- curate estimate of the true f shown in Figure 2.3. In order to fit a thin-plate spline, the data analyst must select a level of smoothness. Figure 2.6 shows the same thin-plate spline fit using a lower level of smoothness, allowing for a rougher fit. The resulting estimate fits the observed data perfectly! However, the spline fit shown in Figure 2.6 is far more variable than the true function f , from Figure 2.3. This is an example of overfitting the data, which we discussed previously. It is an undesirable situation because the fit obtained will not yield accurate estimates of the response on new observations that were not part of the original training data set. We dis- cuss methods for choosing the correct amount of smoothness in Chapter 5. Splines are discussed in Chapter 7. As we have seen, there are advantages and disadvantages to parametric and non-parametric methods for statistical learning. We explore both types of methods throughout this book. 2.1.3 The Trade-Oﬀ Between Prediction Accuracy and Model Interpretability Of the many methods that we examine in this book, some are less flexible, or more restrictive, in the sense that they can produce just a relatively small range of shapes to estimate f. For example, linear regression is a relatively inflexible approach, because it can only generate linear functions such as the lines shown in Figure 2.1 or the plane shown in Figure 2.3. 2.1 What Is Statistical Learning? 25 High Subset Selection Lasso Least Squares Interpretability Generalized Additive Models Trees Bagging, Boosting Support Vector Machines Low Low High Flexibility FIGURE 2.7. A representation of the tradeoﬀ between flexibility and inter- pretability, using diﬀerent statistical learning methods. In general, as the flexibil- ity of a method increases, its interpretability decreases. Other methods, such as the thin plate splines shown in Figures 2.5 and 2.6, are considerably more flexible because they can generate a much wider range of possible shapes to estimate f. One might reasonably ask the following question: why would we ever choose to use a more restrictive method instead of a very flexible approach? There are several reasons that we might prefer a more restrictive model. If we are mainly interested in inference, then restrictive models are much more interpretable. For instance, when inference is the goal, the linear model may be a good choice since it will be quite easy to understand the relationship between Y and X1 , X2 ,... , Xp. In contrast, very flexible approaches, such as the splines discussed in Chapter 7 and displayed in Figures 2.5 and 2.6, and the boosting methods discussed in Chapter 8, can lead to such complicated estimates of f that it is diﬃcult to understand how any individual predictor is associated with the response. Figure 2.7 provides an illustration of the trade-oﬀ between flexibility and interpretability for some of the methods that we cover in this book. Least squares linear regression, discussed in Chapter 3, is relatively inflexible but is quite interpretable. The lasso, discussed in Chapter 6, relies upon the lasso linear model (2.4) but uses an alternative fitting procedure for estimating the coeﬃcients β0 , β1 ,... , βp. The new procedure is more restrictive in es- timating the coeﬃcients, and sets a number of them to exactly zero. Hence in this sense the lasso is a less flexible approach than linear regression. It is also more interpretable than linear regression, because in the final model the response variable will only be related to a small subset of the predictors—namely, those with nonzero coeﬃcient estimates. Generalized 26 2. Statistical Learning additive models (GAMs), discussed in Chapter 7, instead extend the lin- generalized ear model (2.4) to allow for certain non-linear relationships. Consequently, additive GAMs are more flexible than linear regression. They are also somewhat model less interpretable than linear regression, because the relationship between each predictor and the response is now modeled using a curve. Finally, fully non-linear methods such as bagging, boosting, and support vector machines bagging with non-linear kernels, discussed in Chapters 8 and 9, are highly flexible boosting approaches that are harder to interpret. support We have established that when inference is the goal, there are clear ad- vector machine vantages to using simple and relatively inflexible statistical learning meth- ods. In some settings, however, we are only interested in prediction, and the interpretability of the predictive model is simply not of interest. For instance, if we seek to develop an algorithm to predict the price of a stock, our sole requirement for the algorithm is that it predict accurately— interpretability is not a concern. In this setting, we might expect that it will be best to use the most flexible model available. Surprisingly, this is not always the case! We will often obtain more accurate predictions using a less flexible method. This phenomenon, which may seem counterintuitive at first glance, has to do with the potential for overfitting in highly flexible methods. We saw an example of overfitting in Figure 2.6. We will discuss this very important concept further in Section 2.2 and throughout this book. 2.1.4 Supervised Versus Unsupervised Learning Most statistical learning problems fall into one of two categories: supervised supervised or unsupervised. The examples that we have discussed so far in this chap- unsupervised ter all fall into the supervised learning domain. For each observation of the predictor measurement(s) xi , i = 1,... , n there is an associated response measurement yi. We wish to fit a model that relates the response to the predictors, with the aim of accurately predicting the response for future observations (prediction) or better understanding the relationship between the response and the predictors (inference). Many classical statistical learn- ing methods such as linear regression and logistic regression (Chapter 4), as logistic well as more modern approaches such as GAM, boosting, and support vec- regression tor machines, operate in the supervised learning domain. The vast majority of this book is devoted to this setting. In contrast, unsupervised learning describes the somewhat more chal- lenging situation in which for every observation i = 1,... , n, we observe a vector of measurements xi but no associated response yi. It is not pos- sible to fit a linear regression model, since there is no response variable to predict. In this setting, we are in some sense working blind; the sit- uation is referred to as unsupervised because we lack a response vari- able that can supervise our analysis. What sort of statistical analysis is 2.1 What Is Statistical Learning? 27 12 8 10 8 6 6 4 4 2 2 0 2 4 6 8 10 12 0 2 4 6 FIGURE 2.8. A clustering data set involving three groups. Each group is shown using a diﬀerent colored symbol. Left: The three groups are well-separated. In this setting, a clustering approach should successfully identify the three groups. Right: There is some overlap among the groups. Now the clustering task is more challenging. possible? We can seek to understand the relationships between the variables or between the observations. One statistical learning tool that we may use in this setting is cluster analysis, or clustering. The goal of cluster analysis cluster is to ascertain, on the basis of x1 ,... , xn , whether the observations fall into analysis relatively distinct groups. For example, in a market segmentation study we might observe multiple characteristics (variables) for potential customers, such as zip code, family income, and shopping habits. We might believe that the customers fall into diﬀerent groups, such as big spenders versus low spenders. If the information about each customer’s spending patterns were available, then a supervised analysis would be possible. However, this information is not available—that is, we do not know whether each poten- tial customer is a big spender or not. In this setting, we can try to cluster the customers on the basis of the variables measured, in order to identify distinct groups of potential customers. Identifying such groups can be of interest because it might be that the groups diﬀer with respect to some property of interest, such as spending habits. Figure 2.8 provides a simple illustration of the clustering problem. We have plotted 150 observations with measurements on two variables, X1 and X2. Each observation corresponds to one of three distinct groups. For illustrative purposes, we have plotted the members of each group using diﬀerent colors and symbols. However, in practice the group memberships are unknown, and the goal is to determine the group to which each ob- servation belongs. In the left-hand panel of Figure 2.8, this is a relatively easy task because the groups are well-separated. In contrast, the right-hand panel illustrates a more challenging problem in which there is some overlap 28 2. Statistical Learning between the groups. A clustering method could not be expected to assign all of the overlapping points to their correct group (blue, green, or orange). In the examples shown in Figure 2.8, there are only two variables, and so one can simply visually inspect the scatterplots of the observations in order to identify clusters. However, in practice, we often encounter data sets that contain many more than two variables. In this case, we cannot easily plot the observations. For instance, if there are p variables in our data set, then p(p − 1)/2 distinct scatterplots can be made, and visual inspection is simply not a viable way to identify clusters. For this reason, automated clustering methods are important. We discuss clustering and other unsupervised learning approaches in Chapter 10. Many problems fall naturally into the supervised or unsupervised learn- ing paradigms. However, sometimes the question of whether an analysis should be considered supervised or unsupervised is less clear-cut. For in- stance, suppose that we have a set of n observations. For m of the observa- tions, where m < n, we have both predictor measurements and a response measurement. For the remaining n − m observations, we have predictor measurements but no response measurement. Such a scenario can arise if the predictors can be measured relatively cheaply but the corresponding responses are much more expensive to collect. We refer to this setting as a semi-supervised learning problem. In this setting, we wish to use a sta- semi- tistical learning method that can incorporate the m observations for which supervised response measurements are available as well as the n − m observations for learning which they are not. Although this is an interesting topic, it is beyond the scope of this book. 2.1.5 Regression Versus Classification Problems Variables can be characterized as either quantitative or qualitative (also quantitative known as categorical). Quantitative variables take on numerical values. qualitative Examples include a person’s age, height, or income, the value of a house, categorical and the price of a stock. In contrast, qualitative variables take on val- ues in one of K diﬀerent classes, or categories. Examples of qualitative class variables include a person’s gender (male or female), the brand of prod- uct purchased (brand A, B, or C), whether a person defaults on a debt (yes or no), or a cancer diagnosis (Acute Myelogenous Leukemia, Acute Lymphoblastic Leukemia, or No Leukemia). We tend to refer to problems with a quantitative response as regression problems, while those involv- regression ing a qualitative response are often referred to as classification problems. classification However, the distinction is not always that crisp. Least squares linear re- gression (Chapter 3) is used with a quantitative response, whereas logistic regression (Chapter 4) is typically used with a qualitative (two-class, or binary) response. As such it is often used as a classification method. But binary since it estimates class probabilities, it can be thought of as a regression 2.2 Assessing Model Accuracy 29 method as well. Some statistical methods, such as K-nearest neighbors (Chapters 2 and 4) and boosting (Chapter 8), can be used in the case of either quantitative or qualitative responses. We tend to select statistical learning methods on the basis of whether the response is quantitative or qualitative; i.e. we might use linear regres- sion when quantitative and logistic regression when qualitative. However, whether the predictors are qualitative or quantitative is generally consid- ered less important. Most of the statistical learning methods discussed in this book can be applied regardless of the predictor variable type, provided that any qualitative predictors are properly coded before the analysis is performed. This is discussed in Chapter 3. 2.2 Assessing Model Accuracy One of the key aims of this book is to introduce the reader to a wide range of statistical learning methods that extend far beyond the standard linear regression approach. Why is it necessary to introduce so many diﬀerent statistical learning approaches, rather than just a single best method? There is no free lunch in statistics: no one method dominates all others over all possible data sets. On a particular data set, one specific method may work best, but some other method may work better on a similar but diﬀerent data set. Hence it is an important task to decide for any given set of data which method produces the best results. Selecting the best approach can be one of the most challenging parts of performing statistical learning in practice. In this section, we discuss some of the most important concepts that arise in selecting a statistical learning procedure for a specific data set. As the book progresses, we will explain how the concepts presented here can be applied in practice. 2.2.1 Measuring the Quality of Fit In order to evaluate the performance of a statistical learning method on a given data set, we need some way to measure how well its predictions actually match the observed data. That is, we need to quantify the extent to which the predicted response value for a given observation is close to the true response value for that observation. In the regression setting, the most commonly-used measure is the mean squared error (MSE), given by mean squared n error 10 M SE = (yi − fˆ(xi ))2 , (2.5) n i=1 30 2. Statistical Learning where fˆ(xi ) is the prediction that fˆ gives for the ith observation. The MSE will be small if the predicted responses are very close to the true responses, and will be large if for some of the

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