Persival D.B., Walden A.T. Wavelet Method for Time Series Analysis

Cambridge University Press, 2000. — 611 p. — ISBN 0521640687, 9780521640688.The analysis of time series data is essential to many areas of science, engineering, finance and economics. 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, R and other languages. Students and researchers wishing to use wavelet methods to analyze time series will find this book essential.Conventions and Notation
Introduction to WaveletsThe Essence of a Wavelet
The Essence of Wavelet Analysis
Beyond the CWT: the Discrete Wavelet Transform
Review of Fourier Theory and FiltersComplex 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
Orthonormal Transforms of Time SeriesBasic Theory for Orthonormal Transforms
The Projection Theorem
Complex-Valued Transforms
The Orthonormal Discrete Fourier TransformExercises
The Discrete Wavelet TransformQualitative 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 ConsiderationsExercises
The Maximal Overlap Discrete Wavelet TransformEffect 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 ConsiderationsExercises
The Discrete Wavelet Packet TransformBasic 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 LevelsExercises
Random Variables and Stochastic ProcessesUnivariate 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
The Wavelet VarianceDefinition 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 MeasurementsExercises
Analysis and Synthesis of Long Memory ProcessesDiscrete 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 MinimaExercises
Wavelet-Based Signal EstimationSignal 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 CoefficientsExercises
Wavelet Analysis of Finite Energy SignalsTranslation 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 CoefficientsExercisesAppendix: Answers to Embedded ExercisesAuthor Index
Subject Index