R语言统计学习入门：数据分析应用

统计分析

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"《基于R的统计分析》是SpringerTexts in Statistics系列中的一本教材，适合统计学或相关定量领域的本科高年级学生、研究生，以及希望利用统计学习工具进行数据分析的其他学科人士阅读。本书由Gareth James、Daniela Witten、Trevor Hastie和Robert Tibshirani四位知名学者合著，提供了R语言的应用实践。" 在《统计学习导论：R语言应用》中，作者深入浅出地介绍了统计分析的基础理论和实用技巧，结合R语言这一强大的统计分析工具，帮助读者掌握统计学习的核心概念。书中涵盖了多元统计分析、预测模型、机器学习算法等多个方面，旨在提升读者的数据理解和分析能力。首先，书中讨论了基础统计概念，包括概率论、假设检验和置信区间等，这些是进行任何统计分析的基石。通过R语言，读者可以轻松实现数据的预处理、探索性数据分析和图形绘制，为后续的建模工作打下基础。其次，书中详细介绍了线性回归模型，包括简单线性回归和多元线性回归，以及如何处理缺失值和异常值。此外，非线性模型如逻辑回归和广义线性模型也在讲解之列，它们在分类和二元响应问题中扮演着重要角色。进一步，书中的重点之一是介绍了预测模型，如决策树、随机森林、支持向量机(SVM)和神经网络。这些模型在现代数据分析中广泛应用，特别是在大数据和复杂问题的解决中。书中通过实际案例和R代码，让读者能够理解和构建这些模型，并评估其性能。接着，书中探讨了聚类和降维方法，如主成分分析(PCA)和奇异值分解(SVD)，这些技术有助于发现数据中的结构和模式，简化数据的复杂性，为后续分析提供便利。最后，书中还涉及了模型选择和验证的重要概念，如交叉验证和调参策略，以确保模型的泛化能力和可靠性。此外，作者还讨论了集成学习方法，如Boosting和Bagging，这些方法可以提高模型的预测准确性和稳定性。《基于R的统计分析》是一本全面且实用的教材，不仅涵盖了统计学的基础知识，还紧跟当前数据分析的前沿趋势，结合R语言的实际操作，使读者能够将理论知识应用于实际问题中，提升数据科学技能。无论是对统计学初学者还是有一定经验的从业者，这本书都是一份宝贵的参考资料。

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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, 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,weexplore

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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