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 2.8 Exercises 3. Orthonormal Transforms of Time Series 3.0 Introduction 3.1 Basic Theory for Orthonormal Transforms 3.2 The Projection Theorem 3.3 Complex-Valued Transforms 3.4 The Orthonormal Discrete Fourier Transform Comments and Extensions to Section 3.4 3.5 Summary 3.6 Exercises 4.The Discrete Wavelet Transform 4.0 Introduction 4.1 Qualitative Description of the DWT Key Facts and Definitions in Section 4.1 Comments and Extensions to Section 4.1 4.2 The Wavelet Filter Key Facts and Definitions in Section 4.2 Comments and Extensions to Section 4.2 4.3 The Scaling Filter Key Facts and Definitions in Section 4.3 Comments and Extensions to Section 4.3 4.4 First Stage of the Pyramid Algorithm Key Facts and Definitions in Section 4.4 Comments and Extensions to Section 4.4 4.5 Second Stage of the Pyramid Algorithm Key Facts and Definitions in Section 4.5 Comments and Extensions to Section 4.5 4.6 General Stage of the Pyramid Algorithm Key Facts and Definitions in Section 4.6 Comments and Extensions to Section 4.6 4.7 The Partial Discrete Wavelet Transform 4.8 Daubechies Wavelet and Scaling Filters:Form and Phase Key Facts and Definitions in Section 4.8 Comments and Extensions to Section 4.8 4.9 Coiflet Wavelet and Scaling Filters:Form and Phase 4.10 Example:Electrocardiogram Data Comments and Extensions to Section 4.10 4.11 Practical Considerations Comments and Extensions to Section 4.11 4.12 Summary 4.13 Exercises 5.The Maximal Overlap Discrete Wavelet Transform 5.0 Introduction 5.1 Effect of Circular Shifts on the DWT 5.2 MODWT Wavelet and Scaling Filters 5.3 Basic Concepts for MODWT Key Facts and Definitions in Section 5.3 5.4 Definition of jth Level MODWT Coefficients Key Facts and Definitions in Section 5.4 Comments and Extensions to Section 5.4 5.5 Pyramid Algorithm for the MODWT Key Facts and Definitions in Section 5.5 Comments and Extensions to Section 5.5 5.6 MODWT Analysis of ‘Bump’Time Series 5.7 Example:Electrocardiogram Data 5.8 Example:Subtidal Sea Level Fluctuations 5.9 Example:Nile River Minima 5.10 Example:Ocean Shear Measurements 5.11 Practical Considerations 5.12 Summary 5.13 Exercises 6.The Discrete Wavelet Packet Transform 6.0 Introduction 6.1 Basic Concepts Comments and Extensions to Section 6.1 6.2 Example:DWPT of Solar Physics Data 6.3 The Best Basis Algorithm Comments and Extensions to Section 6.3 6.4 Example:Best Basis for Solar Physics Data 6.5 Time Shifts for Wavelet Packet Filters Comments and Extensions to Section 6.5 6.6 Maximal Overlap Discrete Wavelet Packet Transform 6.7 Example:MODWPT of Solar Physics Data 6.8 Matching Pursuit 6.9 Example:Subtidal Sea Levels Comments and Extensions to Section 6.9 6.10 Summary 6.11 Exercises 7.Random Variables and Stochastic Processes 7.0 Introduction 7.1 Univariate Random Variables and PDFs 7.2 Random Vectors and PDFs 7.3 A Bayesian Perspective 7.4 Stationary Stochastic Processes 7.5 Spectral Density Estimation Comments and Extensions to Section 7.5 7.6 Definition and Models for Long Memory Processes Comments and Extensions to Section 7.6 7.7 Nonstationry 1/f-Type Processes Comments and Extensions to Section 7.7 7.8 Simulation of Stationary Process Comments and Extensions to Section 7.8 7.9 Simulation of Stationary Autoregressive Processes 7.10 Exercises 8.The Wavelet Variance 8.0 Introduction 8.1 Definition and Rationale for the Wavelet Variance Comments and Extensions to Section 8.1 8.2 Basic Properties of the Wavelet Variance Comments and Extensions to Section 8.2 8.3 Estimation of the Wavelet Variance Comments and Extensions to Section 8.3 8.4 Confidence Intervals for the Wavelet Variance Comments and Extensions to Section 8.4 8.5 Spectral Estimation via the Wavelet Variance Comments and Extensions to Section 8.5 8.6 Example:Atomic Clock Deviates 8.7 Example:Subtidal Sea Level Fluctuations 8.8 Example:Nile River Minima 8.9 Example:Ocean Shear Measurements 8.10 Summary 8.11 Exercises 9.Analysis and Synthesis of Long Memory Processes 9.0 Introduction 9.1 Discrete Wavelet Transform of a Long Memory Process Comments and Extensions to Section 9.1 9.2 Simulation of a Long Memory Process Comments and Extensions to Section 9.2 9.3 MLEs for Stationary FD Processes Comments and Extensions to Section 9.3 9.4 MLEs for Stationary or Nonstationary FD Processes Comments and Extensions to Section 9.4 9.5 Least Squares Estimation for FD Processes Comments and Extensions to Section 9.5 9.6 Testing for Homogeneity of Variance Comments and Extensions to Section 9.6 9.7 Example:Atomic Clock Deviates 9.8 Example:Nile River Minima 9.9 Summary 9.10 Exercises 10.Wavelet-Based Signal Estimation 10.0 Introduction 10.1 Signal Representation via Wvelets 10.2 Signal Estimation via Thresholding 10.3 Stochastic Signal Estimation via Scaling 10.4 Stochastic Signal Estimation via Shrinkage Comments and Extensions to Section 10.4 10.5 IID Gaussian Wavelet Coefficients Comments and Extensions to Section 10.5 10.6 Uncorrelated Non-Gaussian Wavelet Coefficients Comments and Extensions to Section 10.6 10.7 Correlated Gaussian Wavelet Coefficients Comments and Extensions to Section 10.7 10.8 Clustering and Persistence of Wavelet Coefficients 10.9 Summary 10.10 Exercises 11.Wavelet Analysis of Finite Energy Signals 11.0 Introduction 11.1 Translation and Dilation 11.2 Scaling Functions and Approximation Spaces Comments and Extensions to Section 11.2 11.3 Approximation of Finite Energy Signals Comments and Extensions to Section 11.3 11.4 Two-Scale Relationships for Scaling Functions 11.5 Scaling Functions and Scaling Filters Comments and Extensions to Section 11.5 11.6 Wavelet Functions and Detail Spaces 11.7 Wavelet Functions and Wavelet Filters 11.8 Multiresolution Analysis of Finite Energy Signals 11.9 Vanishing Moments Comments and Extensions to Section 11.9 11.10 Spectral Factorization and Filter Coefficients Comments and Extensions to Section 11.10 11.11 Summary 11.12 Exercises Appendix.Answers to Embedded Exercises References Author Index Subject Index
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