MATLAB在统计计算中的应用——Computational Statistics Handbook解析

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"Computational Statistics Handbook with MATLAB 是一本详细介绍如何使用MATLAB进行统计计算的教材，适合于想要结合实践学习统计学和MATLAB编程的读者。书中涵盖了广泛的统计计算实例，帮助读者理解并应用概率和统计理论。" 本书由Wendy L. Martinez和Angel R. Martinez合著，出版于2002年，由CHAPMAN&HALL/CRC出版社发行。书中内容来源于权威且备受推崇的来源，并已获得适当引用和授权。尽管作者和出版社已尽力提供可靠的数据和信息，但他们不承担因使用书中材料而导致的任何后果。重要的是，本书禁止未经许可的任何形式复制或传播，包括电子和机械方式，如影印、缩微胶片、录音或任何信息存储和检索系统。如果需要进行此类复制，必须事先得到CRC Press LLC的书面许可。对于用于一般分布、推广、创建新作品或转售的复制行为，需要向CRC Press LLC申请特定的许可。书中深入探讨了统计计算的关键概念，包括但不限于概率理论、假设检验、回归分析、时间序列分析、贝叶斯方法等。MATLAB作为一种强大的计算工具，被用来演示和实现这些统计方法，使得学习过程更为直观和实用。读者可以通过这些实例学习到如何在实际问题中应用统计模型，并提升MATLAB编程技能。此外，书中可能还包含解决问题的技巧、数据处理方法以及MATLAB函数的使用，旨在帮助读者不仅理解统计学的理论，还能掌握其在MATLAB环境中的实际操作。无论是初学者还是有经验的统计工作者，都可以从这本书中受益，加深对统计计算的理解，并提升在MATLAB平台上的工作效率。《Computational Statistics Handbook with MATLAB》是学习统计计算与MATLAB结合的宝贵资源，通过实例驱动的学习方法，读者可以逐步提升统计分析和编程能力。

2 Computational Statistics Handbook with M

ATLAB

gain some useful information from them. In contrast, the traditional

approach has been to first design the study based on research questions and

then collect the required data.

Because the storage and collection is so cheap, the data sets that analysts

must deal with today tend to be very large and high-dimensional. It is in sit-

uations like these where many of the classical methods in statistics are inad-

equate. As examples of computational statistics methods, Wegman [1988]

includes parallel coordinates for high dimensional data representation, non-

parametric functional inference, and data set mapping where the analysis

techniques are considered fixed.

Efron and Tibshirani [1991] refer to what we call computational statistics as

computer-intensive statistical methods. They give the following as examples for

these types of techniques: bootstrap methods, nonparametric regression,

generalized additive models and classification and regression trees. They

note that these methods differ from the classical methods in statistics because

they substitute computer algorithms for the more traditional mathematical

method of obtaining an answer. An important aspect of computational statis-

tics is that the methods free the analyst from choosing methods mainly

because of their mathematical tractability.

Volume 9 of the Handbook of Statistics: Computational Statistics [Rao, 1993]

covers topics that illustrate the “... trend in modern statistics of basic method-

ology supported by the state-of-the-art computational and graphical facili-

ties...” It includes chapters on computing, density estimation, Gibbs

sampling, the bootstrap, the jackknife, nonparametric function estimation,

statistical visualization, and others.

We mention the topics that can be considered part of computational statis-

tics to help the reader understand the difference between these and the more

traditional methods of statistics. Table 1.1 [Wegman, 1988] gives an excellent

comparison of the two areas.

1.2 An Overview of the Book

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The focus of this book is on methods of computational statistics and how to

implement them. We leave out much of the theory, so the reader can concen-

trate on how the techniques may be applied. In many texts and journal arti-

cles, the theory obscures implementation issues, contributing to a loss of

interest on the part of those needing to apply the theory. The reader should

not misunderstand, though; the methods presented in this book are built on

solid mathematical foundations. Therefore, at the end of each chapter, we

4 Computational Statistics Handbook with M

ATLAB

will be used in later chapters of the book. Chapter 3 covers some of the basic

ideas of statistics and sampling distributions. Since many of the methods in

computational statistics are concerned with estimating distributions via sim-

ulation, this chapter is fundamental to the rest of the book. For the same rea-

son, we present some techniques for generating random variables in

Chapter 4.

Some of the methods in computational statistics enable the researcher to

explore the data before other analyses are performed. These techniques are

especially important with high dimensional data sets or when the questions

to be answered using the data are not well focused. In Chapter 5, we present

some graphical exploratory data analysis techniques that could fall into the

category of traditional statistics (e.g., box plots, scatterplots). We include

them in this text so statisticians can see how to implement them in MATLAB

and to educate scientists and engineers as to their usage in exploratory data

analysis. Other graphical methods in this chapter do fall into the category of

computational statistics. Among these are isosurfaces, parallel coordinates,

the grand tour and projection pursuit.

In Chapters 6 and 7, we present methods that come under the general head-

ing of resampling. We first cover some of the general concepts in hypothesis

testing and confidence intervals to help the reader better understand what

follows. We then provide procedures for hypothesis testing using simulation,

including a discussion on evaluating the performance of hypothesis tests.

This is followed by the bootstrap method, where the data set is used as an

estimate of the population and subsequent sampling is done from the sam-

ple. We show how to get bootstrap estimates of standard error, bias and con-

fidence intervals. Chapter 7 continues with two closely related methods

called jackknife and cross-validation.

One of the important applications of computational statistics is the estima-

tion of probability density functions. Chapter 8 covers this topic, with an

emphasis on the nonparametric approach. We show how to obtain estimates

using probability density histograms, frequency polygons, averaged shifted

histograms, kernel density estimates, finite mixtures and adaptive mixtures.

Chapter 9 uses some of the concepts from probability density estimation

and cross-validation. In this chapter, we present some techniques for statisti-

cal pattern recognition. As before, we start with an introduction of the classi-

cal methods and then illustrate some of the techniques that can be considered

part of computational statistics, such as classification trees and clustering.

In Chapter 10 we describe some of the algorithms for nonparametric

regression and smoothing. One nonparametric technique is a tree-based

method called regression trees. Another uses the kernel densities of

Chapter 8. Finally, we discuss smoothing using loess and its variants.

An approach for simulating a distribution that has become widely used

over the last several years is called Markov chain Monte Carlo. Chapter 11

covers this important topic and shows how it can be used to simulate a pos-

terior distribution. Once we have the posterior distribution, we can use it to

estimate statistics of interest (means, variances, etc.).

Chapter 1: Introduction 5

We conclude the book with a chapter on spatial statistics as a way of show-

ing how some of the methods can be employed in the analysis of spatial data.

We provide some background on the different types of spatial data analysis,

but we concentrate on spatial point patterns only. We apply kernel density

estimation, exploratory data analysis, and simulation-based hypothesis test-

ing to the investigation of spatial point processes.

We also include several appendices to aid the reader. Appendix A contains

a brief introduction to MATLAB, which should help readers understand the

code in the examples and exercises. Appendix B is an index to notation, with

definitions and references to where it is used in the text. Appendices C and D

include some further information about projection pursuit and MATLAB

source code that is too lengthy for the body of the text. In Appendices E and

F, we provide a list of the functions that are contained in the MATLAB Statis-

tics Toolbox and the Computational Statistics Toolbox, respectively. Finally,

in Appendix G, we include a brief description of the data sets that are men-

tioned in the book.

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The explanation of the algorithms in computational statistics (and the under-

standing of them!) depends a lot on notation. In most instances, we follow the

notation that is used in the literature for the corresponding method. Rather

than try to have unique symbols throughout the book, we think it is more

important to be faithful to the convention to facilitate understanding of the

theory and to make it easier for readers to make the connection between the

theory and the text. Because of this, the same symbols might be used in sev-

eral places.

In general, we try to stay with the convention that random variables are

capital letters, whereas small letters refer to realizations of random variables.

For example, X is a random variable, and x is an observed value of that ran-

dom variable. When we use the term log, we are referring to the natural log-

arithm.

A symbol that is in bold refers to an array. Arrays can be row vectors, col-

umn vectors or matrices. Typically, a matrix is represented by a bold capital

letter such as

, while a vector is denoted by a bold lowercase letter such as

. When we are using explicit matrix notation, then we specify the dimen-

sions of the arrays. Otherwise, we do not hold to the convention that a vector

always has to be in a column format. For example, we might represent a vec-

tor of observed random variables as or a vector of parameters as

,,()

µσ,()

6 Computational Statistics Handbook with M

ATLAB

1.3 M

ATLAB

Code

Along with the algorithmic explanation of the procedures, we include

MATLAB commands to show how they are implemented. Any MATLAB

commands, functions or data sets are in courier bold font. For example, plot

denotes the MATLAB plotting function. The commands that are in the exam-

ples can be typed in at the command line to execute the examples. However,

we note that due to typesetting considerations, we often have to continue a

MATLAB command using the continuation punctuation (...). However,

users do not have to include that with their implementations of the algo-

rithms. See Appendix A for more information on how this punctuation is

used in MATLAB.

Since this is a book about computational statistics, we assume the reader

has the MATLAB Statistics Toolbox. In Appendix E, we include a list of func-

tions that are in the toolbox and try to note in the text what functions are part

of the main MATLAB software package and what functions are available

only in the Statistics Toolbox.

The choice of MATLAB for implementation of the methods is due to the fol-

lowing reasons:

• The commands, functions and arguments in MATLAB are not cryp-

tic. It is important to have a programming language that is easy to

understand and intuitive, since we include the programs to help

teach the concepts.

• It is used extensively by scientists and engineers.

• Student versions are available.

• It is easy to write programs in MATLAB.

• The source code or M-files can be viewed, so users can learn about

the algorithms and their implementation.

• User-written MATLAB programs are freely available.

• The graphics capabilities are excellent.

It is important to note that the MATLAB code given in the body of the book

is for learning purposes. In many cases, it is not the most efficient way to pro-

gram the algorithm. One of the purposes of including the MATLAB code is

to help the reader understand the algorithms, especially how to implement

them. So, we try to have the code match the procedures and to stay away

from cryptic programming constructs. For example, we use for loops at

times (when unnecessary!) to match the procedure. We make no claims that

our code is the best way or the only way to program the algorithms.

In some cases, the MATLAB code is contained in an appendix, rather than

in the corresponding chapter. These are applications where the MATLAB

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