统计学习入门：理解复杂数据的强大工具

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《统计学习入门》(An Introduction to Statistical Learning)是一本由加雷斯·詹姆斯(Gareth James)，丹妮拉·维特恩(Daniela Witten)，特雷弗·哈斯特(Trevor Hastie)和罗伯特· Tibshirani共同编著的书籍，它为统计学习领域提供了一个深入且易于理解的概览。这本书在过去的二十年里，随着生物学、金融、市场营销和天文学等领域中大数据集的迅速增长而成为不可或缺的工具。作者们来自美国南加州大学(USC)、斯坦福大学(Stanford University)和华盛顿大学(University of Washington)，他们以其丰富的经验和专业知识，将理论与应用紧密结合起来，以R语言作为主要实践平台。本书属于Springer Texts in Statistics系列，该系列由加斯帕尔·卡塞尔(G.Casella)、萨缪尔·芬伯格(S.Fienberg)和伊万·奥尔金(I.Olkin)共同编辑，旨在为统计学研究者和从业者提供高质量的学术资源。书号为ISBN 978-1-4614-7137-0和eBook版本的ISBN 978-1-4614-7138-7，电子版可以通过DOI 10.1007/978-1-4614-7138-7获取。版权方面，此作品受到版权保护，未经许可不得复制或分发。《统计学习入门》的核心内容涵盖了统计学习的基础概念、方法和技术，包括但不限于：监督学习、无监督学习、模型选择和评估、特征工程、线性回归、决策树、随机森林、支持向量机、神经网络以及深度学习等现代机器学习技术。书中强调了如何处理高维数据、处理缺失值、正则化等关键问题，并通过实例演示如何在实际场景中应用这些统计学习方法。此外，书中还特别注重与R语言的结合，让读者能够通过实践操作熟悉和掌握统计学习工具。作者们的目标是使读者不仅了解理论，还能将其应用于解决复杂的数据分析问题，这对于任何希望在这个快速发展的领域取得进步的读者来说，都是一份宝贵的资源。《统计学习入门》是一本兼具理论深度和实践指导价值的统计学习指南，对于从事数据分析、机器学习或相关领域的专业人士和学生来说，是一本不可多得的参考书。

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 basedononeormoreinputs.Problemsof

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,onhiswage. 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,

Springer Texts in Statistics 103, DOI 10.1007/978-1-4614-7138-7

2 1. Introduction

Age

Wage

Ye a r

Wage

20 40 60 80

50 100 200 300

2003 2006 2009

12345

Education Level

Wage

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.Thereisaslow

but steady increase of approximately $10,000 in the average wage between 2003

and 2009. Right: Boxplots displaying wage as a function of education,with1

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.

Givenanemployee’sage, 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,hiseducation,andtheyear. 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

Today’s Direction

Percentage change in S&P

Two Days Previous

Percentage change in S&P

Down Up

Today’s Direction

Down Up

Today’s Direction

Down Up

−4 −2 0 2 4 6

Three Days Previous

Percentage change in S&P

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

Down Up

0.46 0.48 0.50 0.52

Today’s Direction

Predicted Probability

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, Z

and Z

.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

−40 −20 0 20 40 60

−60 −40 −20 0 20

−40 −20 0 20 40 60

FIGURE 1.4. Left: Representation of the NCI60 gene expression data set in

a two-dimensional space, Z

and Z

. 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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