基本信息
- 原书名：Wavelet Methods for Time Series Analysis
- 原出版社： Cambridge University Press
内容简介
目录
Preface
Conventions and Notation
1. Introduction to Wavelets
1.0 Introduction
1.1 The Essence of a Wavelet
Comments and Extensions to Section 1.1
1.2 The Essence of Wavelet Analysis
Comments and Extensions to Section 1.2
1.3 Beyond the CWT: the Discrete Wavelet Transform
Comments and Extensions to Section 1.3
2. Review of Fourier Theory and Filters
2.0 Introduction
2.1 Complex Variables and Complex Exponentials
2.2 Fourier Transform of Infinite Sequences
2.3 Convolution]Filtering of Infinite Sequences
2.4 Fourier Transform of Finite Sequences
2.5 Circular Convolution/Filtering of Finite Sequences
2.6 Periodized Filters
Comments and Extensions to Section 2.6
2.7 Summary of Fourier Theory
Conventions and Notation
1. Introduction to Wavelets
1.0 Introduction
1.1 The Essence of a Wavelet
Comments and Extensions to Section 1.1
1.2 The Essence of Wavelet Analysis
Comments and Extensions to Section 1.2
1.3 Beyond the CWT: the Discrete Wavelet Transform
Comments and Extensions to Section 1.3
2. Review of Fourier Theory and Filters
2.0 Introduction
2.1 Complex Variables and Complex Exponentials
2.2 Fourier Transform of Infinite Sequences
2.3 Convolution]Filtering of Infinite Sequences
2.4 Fourier Transform of Finite Sequences
2.5 Circular Convolution/Filtering of Finite Sequences
2.6 Periodized Filters
Comments and Extensions to Section 2.6
2.7 Summary of Fourier Theory
前言
The last decade has seen an explosion of interest in wavelets, a subject area that has
coalesced from roots in mathematics, physics, electrical engineering and other disciplines. As a result, wavelet methodology has had a significant impact in areas as diverse as differential equations, image processing and statistics. This book is an introduction to wavelets and their application in the analysis of discrete time series typical of those acquired in the physical sciences. While we present a thorough introduction to the basic theory behind the discrete wavelet transform (DWT), our goal is to bridge the gap between theory and practice by
· emphasizing what the DWT actually means in practical terms;
· showing how the DWT can be used to create informative descriptive statistics for time series analysts;
· discussing how stochastic models can be used to assess the statistical properties of quantities computed from the DWT; and
· presenting substantive examples of wavelet analysis of time series representative
of those encountered in the physical sciences.
To date, most books on wavelets describe them in terms of continuous functions and often introduce the reader to a plethora of different types of wavelets. We concentrate on developing wavelet methods in discrete time via standard filtering and matrix transformation ideas. We purposely avoid overloading the reader by focusing almost exclusively on the class of wavelet filters described in Daubechies (1992),which are particularly convenient and useful for statistical applications; however, the understanding gained from a study of the Daubechies class of wavelets will put the reader in a excellent position to work with other classes of interest. For pedagogical purposes, this book in fact starts (Chapter 1) and ends (Chapter 11) with discussions of the continuous case. This organization allows us at the beginning to motivate ideas from a historical perspective and then at the end to link ideas arising in the discrete
analysis to some of the widely known results for continuous time wavelet analysis.
Topics developed early on in the book (Chapters 4 and 5) include the DWT and the 'maximal overlap' discrete wavelet transform (MODWT), which can be regarded as a generalization of the DWT with certain quite appealing properties. As a whole, these two chapters provide a self-contained introduction to the basic properties of wavelets, with an emphasis both on algorithms for computing the DWT and MODWT and also on the use of these transforms to provide informative descriptive statistics for time series. In particular, both transforms lead to both a scale-based decomposition of the sample variance of a time series and also a scale-based additive decomposition known as a multiresolution analysis. A generalization of the DWT and MODWT that are
known in the literature as 'wavelet packet' transforms, and the decomposition of time series via matching pursuit, are among the subjects of Chapter 6. In the second part of the book, we combine these transforms with stochastic models to develop waveletbased statistical inference for time series analysis. Specific topics covered in this part of the book include
· the wavelet variance, which provides a scale-based analysis of variance complementary to traditional frequency-based spectral analysis (Chapter 8);
· the analysis and synthesis of 'long memory processes,' i.e., processes with slowly decaying correlations (Chapter 9); and
· signal estimation via 'thresholding' and 'denoising' (Chapter 10).
This book is written 'from the ground level and up.' We have attempted to. make the book as self-contained as possible (to this end, Chapters 2, 3 and 7 contain reviews of, respectively, relevant Fourier and filtering theory; key ideas in the orthonormal transforms of time series; and important concepts involving random variables and stochastic processes). The text should thus be suitable for advanced undergraduates,but is primarily intended for graduate students and researchers in statistics, electrical engineering, physics, geophysics, astronomy, oceanography and other physical sciences. Readers with a strong mathematical background can skip Chapters 2 and 3 after a quick perusal. Those with prior knowledge of the DWT can make use of the Key Facts
and Definitions toward the end of various sections in Chapters 4 and 5 to assess how much of these sections they need to study. Drafts of this book have been used as a textbook for a graduate course taught at the University of Washington for the past five years, but we have also designed it to be a self-study work-book by including a large number of exercises embedded within the body of the chapters (particularly Chapters 2 to 5), with solutions provided in the Appendix. Working the embedded exercises will provide readers with a means of progressively understanding the material. For use as a course textbook, we have also provided additional exercises at the end of each chapter (instructors wishing to obtain a solution guide for the exercises should follow
the guidance given on the Web site detailed below).
The wavelet analyses of time series that are described in Chapters 4 and 5 can readily be carried out once the basic algorithms for computing the DWT and MODWT (and their inverses) are implemented. While these can be immediately and readily coded up using the pseudo-code in the Comments and Extensions to Sections 4.6 and 5.5, links to existing software in S-Plus and Lisp can be found by consulting the Web site for this book, which currently is at
http://www, staff, washington, edu/dbp/wmtsa. html
(alternatively the reader can go to the site for Cambridge University Press - currently at http://wax/, cup. org - and search for the page describing this book, which should have a link to the Web site). The reader should also consult this Web site to obtain a current errata sheet, updates to references at the end of the book that are yet to appear, and references to additional material. Additionally readers can use the Web site to download the coefficients for Various scaling filters (as discussed in Sections 4.8 and 4.9), the values for all the time series used as examples in this book, and certain computed values that can be used to check computer code. To facilitate preparation of overheads for courses and seminars, the Web site also allows access to pdf files with all the figures and tables in the book (please note that these figures and tables are the copyright of Cambridge University Press and must not be further distributed or used
coalesced from roots in mathematics, physics, electrical engineering and other disciplines. As a result, wavelet methodology has had a significant impact in areas as diverse as differential equations, image processing and statistics. This book is an introduction to wavelets and their application in the analysis of discrete time series typical of those acquired in the physical sciences. While we present a thorough introduction to the basic theory behind the discrete wavelet transform (DWT), our goal is to bridge the gap between theory and practice by
· emphasizing what the DWT actually means in practical terms;
· showing how the DWT can be used to create informative descriptive statistics for time series analysts;
· discussing how stochastic models can be used to assess the statistical properties of quantities computed from the DWT; and
· presenting substantive examples of wavelet analysis of time series representative
of those encountered in the physical sciences.
To date, most books on wavelets describe them in terms of continuous functions and often introduce the reader to a plethora of different types of wavelets. We concentrate on developing wavelet methods in discrete time via standard filtering and matrix transformation ideas. We purposely avoid overloading the reader by focusing almost exclusively on the class of wavelet filters described in Daubechies (1992),which are particularly convenient and useful for statistical applications; however, the understanding gained from a study of the Daubechies class of wavelets will put the reader in a excellent position to work with other classes of interest. For pedagogical purposes, this book in fact starts (Chapter 1) and ends (Chapter 11) with discussions of the continuous case. This organization allows us at the beginning to motivate ideas from a historical perspective and then at the end to link ideas arising in the discrete
analysis to some of the widely known results for continuous time wavelet analysis.
Topics developed early on in the book (Chapters 4 and 5) include the DWT and the 'maximal overlap' discrete wavelet transform (MODWT), which can be regarded as a generalization of the DWT with certain quite appealing properties. As a whole, these two chapters provide a self-contained introduction to the basic properties of wavelets, with an emphasis both on algorithms for computing the DWT and MODWT and also on the use of these transforms to provide informative descriptive statistics for time series. In particular, both transforms lead to both a scale-based decomposition of the sample variance of a time series and also a scale-based additive decomposition known as a multiresolution analysis. A generalization of the DWT and MODWT that are
known in the literature as 'wavelet packet' transforms, and the decomposition of time series via matching pursuit, are among the subjects of Chapter 6. In the second part of the book, we combine these transforms with stochastic models to develop waveletbased statistical inference for time series analysis. Specific topics covered in this part of the book include
· the wavelet variance, which provides a scale-based analysis of variance complementary to traditional frequency-based spectral analysis (Chapter 8);
· the analysis and synthesis of 'long memory processes,' i.e., processes with slowly decaying correlations (Chapter 9); and
· signal estimation via 'thresholding' and 'denoising' (Chapter 10).
This book is written 'from the ground level and up.' We have attempted to. make the book as self-contained as possible (to this end, Chapters 2, 3 and 7 contain reviews of, respectively, relevant Fourier and filtering theory; key ideas in the orthonormal transforms of time series; and important concepts involving random variables and stochastic processes). The text should thus be suitable for advanced undergraduates,but is primarily intended for graduate students and researchers in statistics, electrical engineering, physics, geophysics, astronomy, oceanography and other physical sciences. Readers with a strong mathematical background can skip Chapters 2 and 3 after a quick perusal. Those with prior knowledge of the DWT can make use of the Key Facts
and Definitions toward the end of various sections in Chapters 4 and 5 to assess how much of these sections they need to study. Drafts of this book have been used as a textbook for a graduate course taught at the University of Washington for the past five years, but we have also designed it to be a self-study work-book by including a large number of exercises embedded within the body of the chapters (particularly Chapters 2 to 5), with solutions provided in the Appendix. Working the embedded exercises will provide readers with a means of progressively understanding the material. For use as a course textbook, we have also provided additional exercises at the end of each chapter (instructors wishing to obtain a solution guide for the exercises should follow
the guidance given on the Web site detailed below).
The wavelet analyses of time series that are described in Chapters 4 and 5 can readily be carried out once the basic algorithms for computing the DWT and MODWT (and their inverses) are implemented. While these can be immediately and readily coded up using the pseudo-code in the Comments and Extensions to Sections 4.6 and 5.5, links to existing software in S-Plus and Lisp can be found by consulting the Web site for this book, which currently is at
http://www, staff, washington, edu/dbp/wmtsa. html
(alternatively the reader can go to the site for Cambridge University Press - currently at http://wax/, cup. org - and search for the page describing this book, which should have a link to the Web site). The reader should also consult this Web site to obtain a current errata sheet, updates to references at the end of the book that are yet to appear, and references to additional material. Additionally readers can use the Web site to download the coefficients for Various scaling filters (as discussed in Sections 4.8 and 4.9), the values for all the time series used as examples in this book, and certain computed values that can be used to check computer code. To facilitate preparation of overheads for courses and seminars, the Web site also allows access to pdf files with all the figures and tables in the book (please note that these figures and tables are the copyright of Cambridge University Press and must not be further distributed or used
