R语言统计计算实践指南

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"Statistical Computing with R" 本书《Statistical Computing with R》专注于实践而非理论，为读者提供了一个学习统计计算的实用教程。作者通过实际的计算问题实例来解释编程概念，帮助读者理解R语言计算环境的运用。书中首先回顾了概率论和经典统计推断的基本概念，然后通过完整实现的R代码示例展示每一种算法。书中的章节涵盖了多个统计计算的重要主题，包括但不限于： 1. **Monte Carlo方法**：这是一种基于随机抽样或重复试验的计算技术，常用于模拟复杂系统或求解数学问题。在R中，有丰富的包支持Monte Carlo模拟，如`rstan`用于贝叶斯分析，`gtools`提供随机抽样的功能。 2. **聚类分析（Clustering）**：这是一种无监督学习方法，用于将数据集中的对象分组到相似的类别。R中常用的聚类库有`cluster`，包含多种聚类算法如K-means、层次聚类等。 3. **Bootstrap方法**：Bootstrap是一种统计推断方法，通过从原始样本中多次重抽样以估计统计量的分布。在R中，`boot`包提供了实现Bootstrap的工具。 4. **非参数回归（Nonparametric Regression）**：非参数回归不假设数据分布的具体形式，如核平滑、局部多项式回归等。R中的`np`包提供了进行非参数回归分析的功能。 5. **密度估计（Density Estimation）**：用于估计数据的未知概率密度函数。R中的`density`函数是基本的密度估计工具，而`kde2d`等则可用于二维数据的密度估计。 6. **拟合优度检验（Goodness-of-Fit）**：评估模型对数据的拟合程度。R提供了多种检验，如`chisq.test`用于卡方检验，`ks.test`进行 Kolmogorov-Smirnov 检验。书中还包含大量练习题供学生练习，同时提供教师用的解答手册，这使得本书不仅适合自学，也适合作为大学课程的教学材料。通过这本书，读者可以深入理解如何利用R进行高效且灵活的统计计算，并解决实际问题。

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Preface

This book is an introduction to statistical computing and computational sta-

tistics. Computational statistics is a rapidly expanding area in statistical

research and applications. It includes computationally intensive methods in

statistics, such as Monte Carlo methods, bootstrap, MCMC, density estima-

tion, nonparametric regression, classiﬁcation and clustering, and visualization

of multivariate data. Gentle [113] and Wegman [295] describe computational

statistics as computationally intensive methods in statistics. Statistical com-

puting, at least traditionally, focused on numerical algorithms for statistics

(see e.g. Thisted [269]). Generally a book has only one of these terms in

the title; for example, Givens and Hoeting’s “Computational Statistics” [121]

includes classical statistical computing topics in optimization, numerical in-

tegration, density estimation and smoothing, as well as the Monte Carlo and

MCMC methods of computational statistics. We chose the title “Statistical

Computing with R” for this book, which is both computational statistics and

statistical computing, and perhaps emphasizes Monte Carlo and resampling

methods more than the title would suggest.

R is a statistical computing environment based on the S language. The soft-

ware is free under the terms of the Free Software Foundation’s GNU General

Public License. It is available for a wide variety of platforms including among

others Linux, Windows, and MacOS. See http://www.r-project.org/ for a

description. All examples in the text are implemented in R.

This book is designed for graduate students or advanced undergraduates

with preparation in calculus, linear algebra, probability and mathematical

statistics. The text will be suitable for an introductory course in computa-

tional statistics, and may also be used for independent study. In addition,

because of the computational nature of the material, this book serves as an

excellent tutorial on the R language, providing examples that illustrate pro-

gramming concepts in the context of practical computational problems. The

text does not assume previous expertise in any particular programming lan-

guage.

The presentation will focus on implementation rather than theory, but the

connection to the mathematical ideas and theoretical foundations will be made

clear. The ﬁrst chapter provides an overview of computational statistics and

a brief introduction to the R statistical computing environment. The second

chapter is a summary and review of some basic concepts in probability and

classical statistical inference. Each of the remaining chapters covers a topic

in computational statistics.

xvi

The selection of topics includes the traditional core material of computa-

tional statistics: simulating random variables from probability distributions,

Monte Carlo integration and variance reduction methods, Monte Carlo and

MCMC methods, bootstrap and jackknife, density estimation, and visualiza-

tion of multivariate data. Although R includes random generators for the

commonly used probability distributions, there is instructive value in study-

ing the algorithms for generating them. Research problems often involve

distributions that are non-standard, generalized, or not implemented. Meth-

ods for generating mixtures and multivariate data are also covered. The text

concludes with a chapter on numerical methods in R.

A large number of examples and exercises are included. All examples are

fully implemented in the R statistical computing environment, and the R

code for examples in the book can be downloaded from the author’s web

site at personal.bgsu.edu/

mrizzo. In an eﬀort to keep the material self-

contained, most examples and exercises use datasets available in the R distri-

bution (base plus recommended packages), or simulated data. Some functions

and datasets in contributed packages available on CRAN are used, which can

be installed by functions provided in R.

Books in print have a long lifetime, while software is constantly evolving.

By the time this book is in a reader’s hands, one or more newer versions of R

will have been released. Every eﬀort has been made to check the code samples

under the current version of R; comments, suggestions, and corrections are

always welcome.

Acknowledgements

This book was inspired at least in part by the excellent statistical computing

package R, and the author would like to acknowledge the team of developers

for continuing to support and improve this software.

I would like to thank several reviewers who made invaluable suggestions and

comments, especially Jim Albert, Hua Fang, Herb McGrath, Xiaoping Shen,

and G´abor Sz´ekely. I would also like to acknowledge the contribution of my

students who used a preliminary draft of the text at Ohio University and

provided much helpful feedback, with special thanks to Roxana Hritcu, Nihar

Shah, and Jinfei Zhang. Editor Bob Stern, Project Editor Marsha Hecht, and

Project Coordinator Amber Donley of Taylor & Francis / CRC Press have

been very helpful throughout the entire project. Finally, I would like to thank

my family for their constant support and encouragement.

Maria L. Rizzo

Department of Mathematics and Statistics

Bowling Green State University

Chapter 1

Introduction

1.1 Computational Statistics and Statistical Computing

Computational statistics and statistical computing are two areas within

statistics that may be broadly described as computational, graphical, and

numerical approaches to solving statistical problems. Statistical computing

traditionally has more emphasis on numerical methods and algorithms, such

as optimization and random number generation, while computational statis-

tics may encompass such topics as exploratory data analysis, Monte Carlo

methods, and data partitioning, etc. However, most researchers who apply

computationally intensive methods in statistics use both computational statis-

tics and statistical computing methods; there is much overlap and the terms

are used diﬀerently in diﬀerent contexts and disciplines. Gentle [113] and

Givens and Hoeting [121] use “computational statistics” to encompass all the

relevant topics that should be covered in a modern introductory text, so that

“statistical computing” is somewhat absorbed under this more broad deﬁni-

tion of computational statistics. On the other hand, journals and professional

organizations seem to use both terms to cover similar areas. Some examples

are the International Association for Statistical Computing (IASC), part of

the International Statistical Insititute, and the Statistical Computing section

of the American Statistical Association.

This book encompasses parts of both of these subjects, because a ﬁrst

course in computational methods for statistics necessarily includes both. Some

examples of topics covered are described below.

Monte Carlo methods refer to a diverse collection of methods in statistical

inference and numerical analysis where simulation is used. Many statistical

problems can be approached through some form of Monte Carlo integration.

In parametric bootstrap, samples are generated from a given probability dis-

tribution to compute probabilities, gain information about sampling distrib-

utions of statistics such as bias and standard error, assess the performance of

procedures in statistical inference, and to compare the performance of compet-

ing methods for the same problem. Resampling methods such as the ordinary

bootstrap and jackknife are nonparametric methods that can be applied when

the distribution of the random variable or a method to simulate it directly is

unavailable. The need for Monte Carlo analysis also arises because in many

2 Statistical Computing with R

problems, an asymptotic approximation is unsatisfactory or intractable. The

convergence to the limit distribution may be too slow, or we require results

for ﬁnite samples; or the asymptotic distribution has unknown parameters.

MonteCarlomethodsarecoveredinChapters5,6,7,8,and9. Theﬁrsttool

needed in a simulation is a method for generating psuedo random samples;

these methods are covered in Chapter 3.

Markov Chain Monte Carlo (MCMC)methodsarebasedonanalgorithm

to sample from a speciﬁed target probability distribution that is the stationary

distribution of a Markov chain. These methods are widely applied for prob-

lems arising in Bayesian analysis, and in such diverse ﬁelds as computational

physics and computational ﬁnance. Markov Chain Monte Carlo methods are

covered in Chapter 9.

Several special topics also deserve an introduction in a survey of compu-

tationally intensive methods. Density estimation (Chapter 10) provides a

nonparametric estimate of a density, which has many applications in addition

to estimation ranging from exploratory data analysis to cluster analysis. Com-

putational methods are essential for the visualization of multivariate data and

reduction of dimensionality. The increasing interest in massive and streaming

data sets, and high dimensional data arising in applications of biology and en-

gineering, for example, demand improved and new computational approaches

for multivariate analysis and visualization. Chapter 4 is an introduction to

methods for visualization of multivariate data. A review of selected topics in

numerical methods for optimization and numerical integration is presented in

Chapter 11.

Many references can be recommended for further reading on these topics.

Gentle [113] and the volume edited by Gentle, et al. [114] have thorough cov-

erage of topics in computational statistics. Givens and Hoeting [121] is a

recent graduate text on computational statistics and statistical computing.

Martinez and Martinez [192] is an accessible introduction to computational

statistics, with numerous examples in Matlab. Texts on statistical computing

include the classics by Kennedy and Gentle [161] and Thisted [269], and a

more recent survey of methods in statistical computing is covered in Kundu

and Basu [165]. For statistical applications of numerical analysis see Lange

[168] or Monahan [202]. Books that primarily cover Monte Carlo methods

or resampling methods include Davison and Hinkley [63], Efron and Tibshi-

rani [84], Hjorth [143], Liu [179], and Robert and Casella [228]. On density

estimation see Scott [244] and Silverman [252].

Introduction 3

1.2 The R Environment

The R environment is a suite of software and programming language based

on S, for data analysis and visualization. “What is R” is one of the frequently

asked questions included in the online documentation for R. Here is an excerpt

from the R FAQ [217]:

R is a system for statistical computation and graphics. It consists

of a language plus a run-time environment with graphics, a de-

bugger, access to certain system functions, and the ability to run

programs stored in script ﬁles.

The home page of the R project is http://www.r-project.org/,andthe

current R distribution and documentation are available on the Comprehensive

R Archive Network (CRAN). The CRAN master site is at TU Wien, Austria,

http://cran.R-project.org/. The R distribution includes the base and

recommended packages with documentation. A help system and several ref-

erence manuals are installed with the program.

R is based on the S langauge. Some details about diﬀerences between R

and S are given in the R FAQ [147]. Venables and Ripley [278] is a good

resource for applied statistics with S, Splus, and R. Other references on the S

language include [24, 41, 42, 277].

An excellent starting point is the manual Introduction to R [279]. Some

introductory books using R include Dalgaard [62] and Verzani [280]. On

programming methods see Chambers [41], and Venables and Ripley [277, 278].

Other texts that feature Splus, S, and/or R may also be helpful (see e.g.

Crawley [57] or Everitt and Hothorn [88]). Albert [5] is an introductory text

on Bayesian computation. On statistical models see Faraway [90, 91], Fox,

[97], Harrell [131], and Pinhiero and Bates [211]. Many more references can

be found through links on the R project home page.

Programming is discussed as needed in the chapters that follow. In this

text, new functions or programming methods are explained in remarks called

“R notes” as they arise. Readers are always encouraged to consult the R

help system and manuals [147, 279, 217]. For platform speciﬁc details about

installation and interacting with the graphical user interface the best resource

is the R manual [218] and current information at www.r-project.org.

In the remainder of this chapter, we cover some basic information aimed

to help a new user get started with R. Topics include basic syntax, using the

online help, datasets, ﬁles, scripts, and packages. There is a brief overview of

basic graphics functions. Also see Appendix B on working with data frames.

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