Table of Contents for WMTSA

Wavelet Methods for Time Series Analysis has a total of 594+xxvi pages. The main part of the book consists of eleven chapters and an appendix that gives full solutions to the 114 exercises that are embedded within Chapters 2 to 11. The first chapter provides an introduction to wavelets via the continuous wavelet transform (CWT). The next two chapters go over background material on Fourier theory and orthonormal transforms, after which there are three chapters devoted to developing the discrete wavelet transform (DWT) - and two variations thereof - from the ground level and up. Chapter 7 gives the necessary statistical background for the material covered in Chapter 8, 9 and 10. The final chapter discusses the connections between the CWT and the DWT.

Preface (3 pages)
Conventions and Notation (9 pages)
Chapter 1: Introduction to Wavelets (19 pages) Introduction The Essence of a Wavelet The Essence of Wavelet Analysis Beyond the CWT: the Discrete Wavelet Transform
Chapter 2: Review of Fourier Theory and Filters (21 pages) Introduction Complex Variables and Complex Exponentials Fourier Transform of Infinite Sequences Convolution/Filtering of Infinite Sequences Fourier Transform of Finite Sequences Circular Convolution/Filtering of Finite Sequences Periodized Filters Summary of Fourier Theory Exercises
Chapter 3: Orthonormal Transforms of Time Series (15 pages) Introduction Basic Theory for Orthonormal Transforms The Projection Theorem Complex-Valued Transforms The Orthonormal Discrete Fourier Transform Summary Exercises
Chapter 4: The Discrete Wavelet Transform (103 pages) Introduction Qualitative Description of the DWT The Wavelet Filter The Scaling Filter First Stage of the Pyramid Algorithm Second Stage of the Pyramid Algorithm General Stage of the Pyramid Algorithm The Partial Discrete Wavelet Transform Daubechies Wavelet and Scaling Filters: Form and Phase Coiflet Wavelet and Scaling Filters: Form and Phase Example: Electrocardiogram Data Practical Considerations Summary Exercises
Chapter 5: The Maximal Overlap Discrete Wavelet Transform (47 pages) Introduction Effect of Circular Shifts on the DWT MODWT Wavelet and Scaling Filters Basic Concepts for MODWT Definition of jth Level MODWT Coefficients Pyramid Algorithm for the MODWT MODWT Analysis of `Bump' Time Series Example: Electrocardiogram Data Example: Subtidal Sea Level Fluctuations Example: Nile River Minima Example: Ocean Shear Measurements Practical Considerations Summary Exercises
Chapter 6: The Discrete Wavelet Packet Transform (49 pages) Introduction Basic Concepts Example: DWPT of Solar Physics Data The Best Basis Algorithm Example: Best Basis for Solar Physics Data Time Shifts for Wavelet Packet Filters Maximal Overlap Discrete Wavelet Packet Transform Example: MODWPT of Solar Physics Data Matching Pursuit Example: Subtidal Sea Levels Summary Exercises
Chapter 7: Random Variables and Stochastic Processes (40 pages) Introduction Univariate Random Variables and Probability Density Functions (PDFs) Random Vectors and PDFs A Bayesian Perspective Stationary Stochastic Processes Spectral Density Estimation Definition and Models for Long Memory Processes Nonstationary 1/f-Type Processes Simulation of Stationary Processes Simulation of Stationary Autoregressive Processes Exercises
Chapter 8: The Wavelet Variance (45 pages) Introduction Definition and Rationale for the Wavelet Variance Basic Properties of the Wavelet Variance Estimation of the Wavelet Variance Confidence Intervals for the Wavelet Variance Spectral Estimation via the Wavelet Variance Example: Atomic Clock Deviates Example: Subtidal Sea Level Fluctuations Example: Nile River Minima Example: Ocean Shear Measurements Summary Exercises
Chapter 9: Analysis and Synthesis of Long Memory Processes (53 pages) Introduction Discrete Wavelet Transform of a Long Memory Process Simulation of a Long Memory Process Maximum Likelihood Estimators (MLEs) for Stationary Fractionally Differenced (FD) Processes MLEs for Stationary or Nonstationary FD Processes Least Squares Estimation for FD Processes Testing for Homogeneity of Variance Example: Atomic Clock Deviates Example: Nile River Minima Summary Exercises
Chapter 10: Wavelet-Based Signal Estimation (63 pages) Introduction Signal Representation via Wavelets Signal Estimation via Thresholding Stochastic Signal Estimation via Scaling Stochastic Signal Estimation via Shrinkage IID Gaussian Wavelet Coefficients Uncorrelated Non-Gaussian Wavelet Coefficients Correlated Gaussian Wavelet Coefficients Clustering and Persistence of Wavelet Coefficients Summary Exercises
Chapter 11: Wavelet Analysis of Finite Energy Signals (43 pages) Introduction Translation and Dilation Scaling Functions and Approximation Spaces Approximation of Finite Energy Signals Two-Scale Relationships for Scaling Functions Scaling Functions and Scaling Filters Wavelet Functions and Detail Spaces Wavelet Functions and Wavelet Filters Multiresolution Analysis of Finite Energy Signals Vanishing Moments Spectral Factorization and Filter Coefficients Summary Exercises
Appendix: Answers to Embedded Exercises (51 pages)
References (13 pages)
Author Index (4 pages)
Subject Index (26 pages)

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