Wavelet Methods for Time Series Analysis,9780521685085

Wavelet Methods for Time Series Analysis

by
ISBN13: 9780521685085
ISBN10: 0521685087
Format: Paperback
Pub. Date: 2006-02-27
Publisher(s): Cambridge University Press
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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

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