Wavelet Methods for Time Series Analysis

by Donald B. Percival , Andrew T. Walden

• ISBN13: 9780521685085
• ISBN10: 0521685087
• Format: Paperback
• Copyright: 2006-02-27
• Publisher: Cambridge University Press

Summary

This introduction to wavelet analysis 'from the ground level and up', and to wavelet-based statistical analysis of time series focuses on practical discrete time techniques, with detailed descriptions of the theory and algorithms needed to understand and implement the discrete wavelet transforms. Numerous examples illustrate the techniques on actual time series. The many embedded exercises - with complete solutions provided in the Appendix - allow readers to use the book for self-guided study. Additional exercises can be used in a classroom setting. A Web site offers access to the time series and wavelets used in the book, as well as information on accessing software in S-Plus and other languages. Students and researchers wishing to use wavelet methods to analyze time series will find this book essential.

Table of Contents

Preface	xiii
Conventions and Notation	xvii
Introduction to Wavelets	1	(19)
Introduction	1	(1)
The Essence of a Wavelet	2	(3)
Comments and Extensions to Section 1.1	4	(1)
The Essence of Wavelet Analysis	5	(7)
Comments and Extensions to Section 1.2	12	(1)
Beyond the CWT: the Discrete Wavelet Transform	12	(8)
Comments and Extensions to Section 1.3	19	(1)
Review of Fourier Theory and Filters	20	(21)
Introduction	20	(1)
Complex Variables and Complex Exponentials	20	(1)
Fourier Transform of Infinite Sequences	21	(3)
Convolution/Filtering of Infinite Sequences	24	(4)
Fourier Transform of Finite Sequences	28	(1)
Circular Convolution/Filtering of Finite Sequences	29	(3)
Periodized Filters	32	(3)
Comments and Extensions to Section 2.6	35	(1)
Summary of Fourier Theory	35	(4)
Exercises	39	(2)
Orthonormal Transforms of Time Series	41	(15)
Introduction	41	(1)
Basic Theory for Orthonormal Transforms	41	(3)
The Projection Theorem	44	(1)
Complex-Valued Transforms	45	(1)
The Orthonormal Discrete Fourier Transform	46	(7)
Comments and Extensions to Section 3.4	52	(1)
Summary	53	(1)
Exercises	54	(2)
The Discrete Wavelet Transform	56	(103)
Introduction	56	(1)
Qualitative Description of the DWT	57	(11)
Key Facts and Definitions in Section 4.1	67	(1)
Comments and Extensions to Section 4.1	68	(1)
The Wavelet Filter	68	(7)
Key Facts and Definitions in Section 4.2	74	(1)
Comments and Extensions to Section 4.2	75	(1)
The Scaling Filter	75	(5)
Key Facts and Definitions in Section 4.3	78	(1)
Comments and Extensions to Section 4.3	79	(1)
First Stage of the Pyramid Algorithm	80	(8)
Key Facts and Definitions in Section 4.4	86	(1)
Comments and Extensions to Section 4.4	87	(1)
Second Stage of the Pyramid Algorithm	88	(5)
Key Facts and Definitions in Section 4.5	93	(1)
General Stage of the Pyramid Algorithm	93	(11)
Key Facts and Definitions in Section 4.6	99	(1)
Comments and Extensions to Section 4.6	100	(4)
The Partial Discrete Wavelet Transform	104	(1)
Daubechies Wavelet and Scaling Filters: Form and Phase	105	(18)
Key Facts and Definitions in Section 4.8	116	(1)
Comments and Extensions to Section 4.8	117	(6)
Coiflet Wavelet and Scaling Filters: Form and Phase	123	(2)
Example: Electrocardiogram Data	125	(10)
Comments and Extensions to Section 4.10	134	(1)
Practical Considerations	135	(15)
Comments and Extensions to Section 4.11	145	(5)
Summary	150	(6)
Exercises	156	(3)
The Maximal Overlap Discrete Wavelet Transform	159	(47)
Introduction	159	(1)
Effect of Circular Shifts on the DWT	160	(3)
MODWT Wavelet and Scaling Filters	163	(1)
Basic Concepts for MODWT	164	(5)
Key Facts and Definitions in Section 5.3	168	(1)
Definition of jth Level MODWT Coefficients	169	(5)
Key Facts and Definitions in Section 5.4	173	(1)
Comments and Extensions to Section 5.4	174	(1)
Pyramid Algorithm for the MODWT	174	(5)
Key Facts and Definitions in Section 5.5	177	(1)
Comments and Extensions to Section 5.5	177	(2)
MODWT Analysis of `Bump' Time Series	179	(3)
Example: Electrocardiogram Data	182	(3)
Example: Subtidal Sea Level Fluctuations	185	(5)
Example: Nile River Minima	190	(3)
Example: Ocean Shear Measurements	193	(2)
Practical Considerations	195	(5)
Summary	200	(4)
Exercises	204	(2)
The Discrete Wavelet Packet Transform	206	(49)
Introduction	206	(1)
Basic Concepts	207	(11)
Comments and Extensions to Section 6.1	217	(1)
Example: DWPT of Solar Physics Data	218	(3)
The Best Basis Algorithm	221	(5)
Comments and Extensions to Section 6.3	226	(1)
Example: Best Basis for Solar Physics Data	226	(3)
Time Shifts for Wavelet Packet Filters	229	(2)
Comments and Extensions to Section 6.5	231	(1)
Maximal Overlap Discrete Wavelet Packet Transform	231	(3)
Example: MODWPT of Solar Physics Data	234	(5)
Matching Pursuit	239	(4)
Example: Subtidal Sea Levels	243	(4)
Comments and Extensions to Section 6.9	247	(1)
Summary	247	(6)
Exercises	253	(2)
Random Variables and Stochastic Processes	255	(40)
Introduction	255	(1)
Univariate Random Variables and PDFs	256	(2)
Random Vectors and PDFs	258	(6)
A Bayesian Perspective	264	(2)
Stationary Stochastic Processes	266	(3)
Spectral Density Estimation	269	(10)
Comments and Extensions to Section 7.5	278	(1)
Definition and Models for Long Memory Processes	279	(8)
Comments and Extensions to Section 7.6	285	(2)
Nonstationary 1/f-Type Processes	287	(3)
Comments and Extensions to Section 7.7	289	(1)
Simulation of Stationary Processes	290	(2)
Comments and Extensions to Section 7.8	292	(1)
Simulation of Stationary Autoregressive Processes	292	(1)
Exercises	293	(2)
The Wavelet Variance	295	(45)
Introduction	295	(1)
Definition and Rationale for the Wavelet Variance	295	(9)
Comments and Extensions to Section 8.1	301	(3)
Basic Properties of the Wavelet Variance	304	(2)
Comments and Extensions to Section 8.2	306	(1)
Estimation of the Wavelet Variance	306	(5)
Comments and Extensions to Section 8.3	308	(3)
Confidence Intervals for the Wavelet Variance	311	(4)
Comments and Extensions to Section 8.4	315	(1)
Spectral Estimation via the Wavelet Variance	315	(2)
Comments and Extensions to Section 8.5	317	(1)
Example: Atomic Clock Deviates	317	(7)
Example: Subtidal Sea Level Fluctuations	324	(2)
Example: Nile River Minima	326	(1)
Example: Ocean Shear Measurements	327	(8)
Summary	335	(2)
Exercises	337	(3)
Analysis and Synthesis of Long Memory Processes	340	(53)
Introduction	340	(1)
Discrete Wavelet Transform of a Long Memory Process	341	(14)
Comments and Extensions to Section 9.1	350	(5)
Simulation of a Long Memory Process	355	(6)
Comments and Extensions to Section 9.2	361	(1)
MLEs for Stationary FD Processes	361	(7)
Comments and Extensions to Section 9.3	366	(2)
MLEs for Stationary or Nonstationary FD Processes	368	(6)
Comments and Extensions to Section 9.4	373	(1)
Least Squares Estimation for FD Processes	374	(5)
Comments and Extensions to Section 9.5	378	(1)
Testing for Homogeneity of Variance	379	(4)
Comments and Extensions to Section 9.6	382	(1)
Example: Atomic Clock Deviates	383	(3)
Example: Nile River Minima	386	(2)
Summary	388	(3)
Exercises	391	(2)
Wavelet-Based Signal Estimation	393	(64)
Introduction	393	(1)
Signal Representation via Wavelets	394	(4)
Signal Estimation via Thresholding	398	(9)
Stochastic Signal Estimation via Scaling	407	(1)
Stochastic Signal Estimation via Shrinkage	408	(9)
Comments and Extensions to Section 10.4	415	(2)
IID Gaussian Wavelet Coefficients	417	(15)
Comments and Extensions to Section 10.5	429	(3)
Uncorrelated Non-Gaussian Wavelet Coefficients	432	(8)
Comments and Extensions to Section 10.6	439	(1)
Correlated Gaussian Wavelet Coefficients	440	(10)
Comments and Extensions to Section 10.7	449	(1)
Clustering and Persistence of Wavelet Coefficients	450	(2)
Summary	452	(3)
Exercises	455	(2)
Wavelet Analysis of Finite Energy Signals	457	(44)
Introduction	457	(1)
Translation and Dilation	457	(2)
Scaling Functions and Approximation Spaces	459	(3)
Comments and Extensions to Section 11.2	462	(1)
Approximation of Finite Energy Signals	462	(2)
Comments and Extensions to Section 11.3	464	(1)
Two-Scale Relationships for Scaling Functions	464	(5)
Scaling Functions and Scaling Filters	469	(3)
Comments and Extensions to Section 11.5	472	(1)
Wavelet Functions and Detail Spaces	472	(4)
Wavelet Functions and Wavelet Filters	476	(2)
Multiresolution Analysis of Finite Energy Signals	478	(5)
Vanishing Moments	483	(4)
Comments and Extensions to Section 11.9	486	(1)
Spectral Factorization and Filter Coefficients	487	(7)
Comments and Extensions to Section 11.10	494	(1)
Summary	494	(6)
Exercises	500	(1)
Appendix. Answers to Embedded Exercises	501	(51)
References	552	(13)
Author Index	565	(4)
Subject Index	569