Wavelet Methods for Time Series Analysis

Transcription

1 Wavelet Methods for Time Series Analysis Donald B. Percival UNIVERSITY OF WASHINGTON, SEATTLE Andrew T. Walden IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY AND MEDICINE, LONDON CAMBRIDGE UNIVERSITY PRESS

2 Contents Preface Conventions and Notation xiii xvii 1. Introduction to Wavelets Introduction The Essence of a Wavelet 2 Comments and Extensions to Section The Essence of Wavelet Analysis 5 Comments and Extensions to Section Beyond the CWT: the Discrete Wavelet Transform 12 Comments and Extensions to Section Review of Fourier Theory and Filters 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 32 Comments and Extensions to Section Summary of Fourier Theory Exercises 39 vn

3 viii Contents 3. Orthonormal Transforms of Time Series Introduction Basic Theory for Orthonormal Transforms The Projection Theorem Complex-Valued Transforms The Orthonormal Discrete Fourier Transform 46 Comments and Extensions to Section Summary Exercises The Discrete Wavelet Transform Introduction Qualitative Description of the DWT 57 Key Facts and Definitions in Section Comments and Extensions to Section The Wavelet Filter 68 Key Facts and Definitions in Section Comments and Extensions to Section The Scaling Filter 75 Key Facts and Definitions in Section Comments and Extensions to Section First Stage of the Pyramid Algorithm 80 Key Facts and Definitions in Section Comments and Extensions to Section Second Stage of the Pyramid Algorithm 88 Key Facts and Definitions in Section General Stage of the Pyramid Algorithm 93 Key Facts and Definitions in Section Comments and Extensions to Section The Partial Discrete Wavelet Transform : Daubechies Wavelet and Scaling Filters: Form and Phase 105 Key Facts and Definitions in Section Comments and Extensions to Section Coiflet Wavelet and Scaling Filters: Form and Phase Example: Electrocardiogram Data 125 Comments and Extensions to Section Practical Considerations 135 Comments and Extensions to Section Summary Exercises 156

4 Contents ix 5. The Maximal Overlap Discrete Wavelet Transform Introduction Effect of Circular Shifts on the DWT MODWT Wavelet and Scaling Filters Basic Concepts for MODWT 164 Key Facts and Definitions in Section Definition of jth Level MODWT Coefficients 169 Key Facts and Definitions in Section Comments and Extensions to Section Pyramid Algorithm for the MODWT 174 Key Facts and Definitions in Section Comments and Extensions to Section 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 The Discrete Wavelet Packet Transform Introduction Basic Concepts 207 Comments and Extensions to Section Example: DWPT of Solar Physics Data The Best Basis Algorithm 221 Comments and Extensions to Section Example: Best Basis for Solar Physics Data Time Shifts for Wavelet Packet Filters 229 Comments and Extensions to Section Maximal Overlap Discrete Wavelet Packet Transform Example: MODWPT of Solar Physics Data Matching Pursuit Example: Subtidal Sea Levels 243 Comments and Extensions to Section Summary Exercises Random Variables and Stochastic Processes Introduction Univariate Random Variables and PDFs Random Vectors and PDFs A Bayesian Perspective Stationary Stochastic Processes Spectral Density Estimation 269

5 x Contents Comments and Extensions to Section Definition and Models for Long Memory Processes 279 Comments and Extensions to Section Nonstationary 1//-Type Processes 287 Comments and Extensions to Section Simulation of Stationary Processes 290 Comments and Extensions to Section Simulation of Stationary Autoregressive Processes Exercises The Wavelet Variance Introduction Definition and Rationale for the Wavelet Variance 295 Comments and Extensions to Section Basic Properties of the Wavelet Variance 304 Comments and Extensions to Section Estimation of the Wavelet Variance 306 Comments and Extensions to Section Confidence Intervals for the Wavelet Variance 311 Comments and Extensions to Section Spectral Estimation via the Wavelet Variance 315 Comments and Extensions to Section Example: Atomic Clock Deviates Example: Subtidal Sea Level Fluctuations Example: Nile River Minima Example: Ocean Shear Measurements Summary Exercises Analysis and Synthesis of Long Memory Processes Introduction Discrete Wavelet Transform of a Long Memory Process 341 Comments and Extensions to Section Simulation of a Long Memory Process 355 Comments and Extensions to Section MLEs for Stationary FD Processes 361 Comments and Extensions to Section MLEs for Stationary or Nonstationary FD Processes 368 Comments and Extensions to Section Least Squares Estimation for FD Processes 374 Comments and Extensions to Section Testing for Homogeneity of Variance 379 Comments and Extensions to Section Example: Atomic Clock Deviates Example: Nile River Minima Summary 388

6 Contents xi 9.10 Exercises Wavelet-Based Signal Estimation Introduction Signal Representation via Wavelets Signal Estimation via Thresholding Stochastic Signal Estimation via Scaling Stochastic Signal Estimation via Shrinkage 408 Comments and Extensions to Section IID Gaussian Wavelet Coefficients 417 Comments and Extensions to Section Uncorrelated Non-Gaussian Wavelet Coefficients 432 Comments and Extensions to Section Correlated Gaussian Wavelet Coefficients 440 Comments and Extensions to Section Clustering and Persistence of Wavelet Coefficients Summary Exercises Wavelet Analysis of Finite Energy Signals Introduction Translation and Dilation Scaling Functions and Approximation Spaces 459 Comments and Extensions to Section a 11.3 Approximation of Finite Energy Signals 462 Comments and Extensions to Section Two-Scale Relationships for Scaling Functions Scaling Functions and Scaling Filters 469 Comments and Extensions to Section Wavelet Functions and Detail Spaces Wavelet Functions and Wavelet Filters Multiresolution Analysis of Finite Energy Signals Vanishing Moments 483 Comments and Extensions to Section Spectral Factorization and Filter Coefficients 487 Comments and Extensions to Section Summary Exercises 500 Appendix. Answers to Embedded Exercises 501 References 552 Author Index 565 Subject Index 569