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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley Series in Computational Statistics
- Sprache: Englisch
A comprehensive introduction to sampling-based methods in statistical computing
The use of computers in mathematics and statistics has opened up a wide range of techniques for studying otherwise intractable problems. Sampling-based simulation techniques are now an invaluable tool for exploring statistical models. This book gives a comprehensive introduction to the exciting area of sampling-based methods.
An Introduction to Statistical Computing introduces the classical topics of random number generation and Monte Carlo methods. It also includes some advanced methods such as the reversible jump Markov chain Monte Carlo algorithm and modern methods such as approximate Bayesian computation and multilevel Monte Carlo techniques
An Introduction to Statistical Computing:
- Fully covers the traditional topics of statistical computing.
- Discusses both practical aspects and the theoretical background.
- Includes a chapter about continuous-time models.
- Illustrates all methods using examples and exercises.
- Provides answers to the exercises (using the statistical computing environment R); the corresponding source code is available online.
- Includes an introduction to programming in R.
This book is mostly self-contained; the only prerequisites are basic knowledge of probability up to the law of large numbers. Careful presentation and examples make this book accessible to a wide range of students and suitable for self-study or as the basis of a taught course.
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Seitenzahl: 492
Veröffentlichungsjahr: 2013
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Contents
Cover
Series Page
Title Page
Copyright Page
List of Algorithms
Preface
Nomenclature
1: Random number generation
1.1 Pseudo random number generators
1.2 Discrete distributions
1.3 The inverse transform method
1.4 Rejection sampling
1.5 Transformation of random variables
1.6 Special-purpose methods
1.7 Summary and further reading
Exercises
2: Simulating statistical models
2.1 Multivariate normal distributions
2.2 Hierarchical models
2.3 Markov chains
2.4 Poisson processes
2.5 Summary and further reading
Exercises
3: Monte Carlo methods
3.1 Studying models via simulation
3.2 Monte Carlo estimates
3.3 Variance reduction methods
3.4 Applications to statistical inference
3.5 Summary and further reading
Exercises
4: Markov Chain Monte Carlo methods
4.1 The Metropolis–Hastings method
4.2 Convergence of Markov Chain Monte Carlo methods
4.3 Applications to Bayesian inference
4.4 The Gibbs sampler
4.5 Reversible Jump Markov Chain Monte Carlo
4.6 Summary and further reading
Exercises
5: Beyond Monte Carlo
5.1 Approximate Bayesian Computation
5.2 Resampling methods
5.3 Summary and further reading
Exercises
6: Continuous-time models
6.1 Time discretisation
6.2 Brownian motion
6.3 Geometric Brownian motion
6.4 Stochastic differential equations
6.5 Monte Carlo estimates
6.6 Application to option pricing
6.7 Summary and further reading
Exercises
Appendix A: Probability reminders
A.1 Events and probability
A.2 Conditional probability
A.3 Expectation
A.4 Limit theorems
A.5 Further reading
Appendix B: Programming in R
B.1 General advice
B.2 R as a Calculator
B.3 Programming principles
B.4 Random number generation
B.5 Summary and further reading
Exercises
Appendix C: Answers to the exercises
C.1 Answers for Chapter 1
C.2 Answers for Chapter 2
C.3 Answers for Chapter 3
C.4 Answers for Chapter 4
C.5 Answers for Chapter 5
C.6 Answers for Chapter 6
C.7 Answers for Appendix B
References
Index
WILEY SERIES IN COMPUTATIONAL STATISTICS
Consulting Editors:
Paolo GiudiciUniversity of Pavia, Italy
Geof H. GivensColorado State University, USA
Bani K. MallickTexas A & M University, USA
Wiley Series in Computational Statistics is comprised of practical guides and cutting edge research books on new developments in computational statistics. It features quality authors with a strong applications focus. The texts in the series provide detailed coverage of statistical concepts, methods and case studies in areas at the interface of statistics, computing, and numerics.
With sound motivation and a wealth of practical examples, the books show in concrete terms how to select and to use appropriate ranges of statistical computing techniques in particular fields of study. Readers are assumed to have a basic understanding of introductory terminology.
The series concentrates on applications of computational methods in statistics to fields of bioinformatics, genomics, epidemiology, business, engineering, finance and applied statistics.
Titles in the Series
Biegler, Biros, Ghattas, Heinkenschloss, Keyes, Mallick, Marzouk, Tenorio, Waanders, Willcox – Large-Scale Inverse Problems and Quantification of Uncertainty
Billard and Diday – Symbolic Data Analysis: Conceptual Statistics and Data Mining
Bolstad – Understanding Computational Bayesian Statistics
Borgelt, Steinbrecher and Kruse – Graphical Models, 2e
Dunne – A Statistical Approach to Neutral Networks for Pattern Recognition
Liang, Liu and Carroll – Advanced Markov Chain Monte Carlo Methods
Ntzoufras – Bayesian Modeling Using WinBUGS
Tufféry – Data Mining and Statistics for Decision Making
This edition first published 2014 © 2014 John Wiley & Sons, Ltd
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Library of Congress Cataloging-in-Publication Data
Voss, Jochen. An introduction to statistical computing : a simulation-based approach / Jochen Voss. − First edition. pages cm. – (Wiley series in computational statistics) Includes bibliographical references and index. ISBN 978-1-118-35772-9 (hardback) 1. Mathematical statistics-Data processing. I. Title. QA276.4.V66 2013 519.501′13-dc23
2013019321
A catalogue record for this book is available from the British Library.
ISBN: 978-1-118-35772-9
List of algorithms
Random number generation
Simulating statistical models
Monte Carlo methods
Markov Chain Monte Carlo methods
Beyond Monte Carlo
Continuous-time models
Preface
This is a book about exploring random systems using computer simulation and thus, this book combines two different topic areas which have always fascinated me: the mathematical theory of probability and the art of programming computers. The method of using computer simulations to study a system is very different from the more traditional, purely mathematical approach. On the one hand, computer experiments normally can only provide approximate answers to quantitative questions, but on the other hand, results can be obtained for a much wider class of systems, including large and complex systems where a purely theoretical approach becomes difficult.
In this text we will focus on three different types of questions. The first, easiest question is about the normal behaviour of the system: what is a typical state of the system? Such questions can be easily answered using computer experiments: simulating a few random samples of the system gives examples of typical behaviour. The second kind of question is about variability: how large are the random fluctuations? This type of question can be answered statistically by analysing large samples, generated using repeated computer simulations. A final, more complicated class of questions is about exceptional behaviour: how small is the probability of the system behaving in a specified untypical way? Often, advanced methods are required to answer this third type of question. The purpose of this book is to explain how such questions can be answered. My hope is that, after reading this book, the reader will not only be able to confidently use methods from statistical computing for answering such questions, but also to adjust existing methods to the requirements of a given problem and, for use in more complex situations, to develop new specialised variants of the existing methods.
This text originated as a set of handwritten notes which I used for teaching the ‘Statistical Computing’ module at the University of Leeds, but now is greatly extended by the addition of many examples and more advanced topics. The material we managed to cover in the ‘Statistical Computing’ course during one semester is less than half of what is now the contents of the book! This book is aimed at postgraduate students and their lecturers; it can be used both for self-study and as the basis of taught courses. With the inclusion of many examples and exercises, the text should also be accessible to interested undergraduate students and to mathematically inclined researchers from areas outside mathematics.
Only very few prerequisites are required for this book. On the mathematical side, the text assumes that the reader is familiar with basic probability, up to and including the law of large numbers; Appendix A summarises the required results. As a consequence of the decision to require so little mathematical background, some of the finer mathematical subtleties are not discussed in this book. Results are presented in a way which makes them easily accessible to readers with limited mathematical background, but the statements are given in a form which allows the mathematically more knowledgeable reader to easily add the required detail on his/her own. (For example, I often use phrases such as ‘every set where full mathematical rigour would require us to write ‘every measurable set .) On the computational side, basic programming skills are required to make use of the numerical methods introduced in this book. While the text is written independent of any specific programming language, the reader will need to choose a language when implementing methods from this book on a computer. Possible choices of programming language include Python, Matlab and C/C++. For my own implementations, provided as part of the solutions to the exercises in Appendix C, I used the R programming language; a short introduction to programming with R is provided in Appendix B.
Writing this book has been a big adventure for me. When I started this project, more than a year ago, my aim was to cover enough material so that I could discuss the topics of multilevel Monte Carlo and reversible jump Markov Chain Monte Carlo methods. I estimated that 350 pages would be enough to cover this material but it quickly transpired that I had been much too optimistic: my estimates for the final page count kept rising and even after several rounds of throwing out side-topics and generally tightening the text, the book is still stretching this limit! Nevertheless, the text now covers most of the originally planned topics, including multilevel Monte Carlo methods near the very end of the book. Due to my travel during the last year, parts of this book have been written on a laptop in exciting places. For example, the initial draft of section 1.5 was written on a coach travelling through the beautiful island of Kyushu, halfway around the world from where I live! All in all, I greatly enjoyed writing this book and I hope that the result is useful to the reader.
This book contains an accompanying website. Please visit www.wiley.com/go/statistical_computing
Jochen VossLeeds, March 2013
