The last decade has seen an explosion of interest in wavelets, a subject area that has coalesced from roots in mathematics, physics, electrical engineering and other disciplines. As a result, wavelet methodology has had a significant impact in areas as diverse as differential equations, image processing and statistics. This book is an introduction to wavelets and their application in the analysis of discrete time series typical of those acquired in the physical sciences. While we present a thorough introduction to the basic theory behind the discrete wavelet transform (DWT), our goal is to bridge the gap between theory and practice by

- emphasizing what the DWT actually means in practical terms;
- showing how the DWT can be used to create informative descriptive statistics for time series analysts;
- discussing how stochastic models can be used to assess the statistical properties of quantities computed from the DWT; and
- presenting substantive examples of wavelet analysis of time series representative of those encountered in the physical sciences.

To date, most books on wavelets describe them in terms of continuous functions and often introduce the reader to a plethora of different types of wavelets. We concentrate on developing wavelet methods in discrete time via standard filtering and matrix transformation ideas. We purposely avoid overloading the reader by focusing almost exclusively on the class of wavelet filters described in Daubechies (1992), which are particularly convenient and useful for statistical applications; however, the understanding gained from a study of the Daubechies class of wavelets will put the reader in a excellent position to work with other classes of interest. For pedagogical purposes, this book in fact starts (Chapter 1) and ends (Chapter 11) with discussions of the continuous case. This organization allows us at the beginning to motivate ideas from a historical perspective and then at the end to link ideas arising in the discrete analysis to some of the widely known results for continuous time wavelet analysis.

Topics developed early on in the book (Chapters 4 and 5) include the DWT and the `maximal overlap' discrete wavelet transform (MODWT), which can be regarded as a generalization of the DWT with certain quite appealing properties. As a whole, these two chapters provide a self-contained introduction to the basic properties of wavelets, with an emphasis both on algorithms for computing the DWT and MODWT and also on the use of these transforms to provide informative descriptive statistics for time series. In particular, both transforms lead to both a scale-based decomposition of the sample variance of a time series and also a scale-based additive decomposition known as a multiresolution analysis. A generalization of the DWT and MODWT that are known in the literature as `wavelet packet' transforms, and the decomposition of time series via matching pursuit, are among the subjects of Chapter 6. In the second part of the book, we combine these transforms with stochastic models to develop wavelet-based statistical inference for time series analysis. Specific topics covered in this part of the book include

- the wavelet variance, which provides a scale-based analysis of variance complementary to traditional frequency-based spectral analysis (Chapter 8);
- the analysis and synthesis of `long memory processes,' i.e., processes with slowly decaying correlations (Chapter 9); and
- signal estimation via `thresholding' and `denoising' (Chapter 10).

This book is written `from the ground level and up.' We have attempted to make the book as self-contained as possible (to this end, Chapters 2, 3 and 7 contain reviews of, respectively, relevant Fourier and filtering theory; key ideas in the orthonormal transforms of time series; and important concepts involving random variables and stochastic processes). The text should thus be suitable for advanced undergraduates, but is primarily intended for graduate students and researchers in statistics, electrical engineering, physics, geophysics, astronomy, oceanography and other physical sciences. Readers with a strong mathematical background can skip Chapters 2 and 3 after a quick perusal. Those with prior knowledge of the DWT can make use of the Key Facts and Definitions toward the end of various sections in Chapters 4 and 5 to assess how much of these sections they need to study. This book -- or drafts thereof -- have been used as a textbook for a graduate course taught at the University of Washington for the past ten years, but we have also designed it to be a self-study work-book by including a large number of exercises embedded within the body of the chapters (particularly Chapters 2 to 5), with solutions provided in the Appendix. Working the embedded exercises will provide readers with a means of progressively understanding the material. For use as a course textbook, we have also provided additional exercises at the end of each chapter (instructors wishing to obtain a solution guide for the exercises should follow the guidance given on the Web site detailed below).

The wavelet analyses of time series that are described in Chapters 4 and 5 can readily be carried out once the basic algorithms for computing the DWT and MODWT (and their inverses) are implemented. While these can be immediately and readily coded up using the pseudo-code in the Comments and Extensions to Sections 4.6 and 5.5, there is some existing software in S-Plus, R, MATLAB and Lisp. Additionally errata sheets, the coefficients for various scaling filters (as discussed in Sections 4.8 and 4.8), the values for all the time series used as examples in this book and certain computed values that can be used to check computer code are also available. To facilitate preparation of overheads for courses and seminars, there are also pdf files with all the figures and tables in the book (please note that these figures and tables are the copyright of Cambridge University Press and must not be further distributed or used without written permission).

The book was written using Donald Knuth's superb typesetting system TeX as implemented by Blue Sky Research in their product TeXtures for Apple Macintosh computers. The figures in this book were created using either the plotting system GPL written by W. Hess (whom we thank for many years of support) or S-Plus, the commercial version of the S language developed by J. Chambers and co-workers and marketed by MathSoft, Inc. The computations necessary for the various examples and figures were carried out using either S-Plus or PiTSSa (a Lisp-based object-oriented program for interactive time series and signal analysis that was developed in part by one of us (Percival)).

We thank R. Spindel and the late J. Harlett of the Applied Physics Laboratory, University of Washington, for providing discretionary funding that led to the start of this book. We thank the National Science Foundation, the National Institutes of Health, the Environmental Protection Agency (through the National Research Center for Statistics and the Environment at the University of Washington), the Office of Naval Research and the Air Force Office of Scientific Research for ongoing support during the writing of this book. Our stay at the Isaac Newton Institute for Mathematical Sciences (Cambridge University) during the program on Nonlinear and Nonstationary Signal Processing in 1998 contributed greatly to the completion of this book; we thank the Engineering and Physical Science Research Council (EPSRC) for the support of one of us (Percival) through a Senior Visiting Fellowship while at Cambridge.

We are indebted to those who have commented on drafts of the manuscript or supplied data to us, namely, G. Bardy, J. Bassingthwaighte, A. Bruce, M. Clyde, W. Constantine, A. Contreras Cristan, P. Craigmile, H.-Y. Gao, A. Gibbs, C. Greenhall, M. Gregg, M. Griffin, P. Guttorp, T. Horbury, M. Jensen, W. King, R. D. Martin, E. McCoy, F. McGraw, H. Mofjeld, F. Noraz, G. Raymond, P. Reinhall, S. Sardy, E. Tsakiroglou and B. Whitcher. We are also very grateful to the many graduate students who have given us valuable critiques of the manuscript and exercises and found numerous errors. We would like to thank E. Aldrich, C. Cornish, N. Derby, A. Jach, I. Kang, M. Keim, I. MacLeod, M. Meyer, K. Tanaka and Z. Xuelin for pointing out errors that have been corrected in reprintings of the book. For any remaining errors - which in a work of this size are inevitable - we apologize, and we would be pleased to hear from any reader who finds a mistake so that we can list them on the Web site and correct any future printings (our `paper' and electronic mailing addresses are listed below). Finally we acknowledge two sources of great support for this project, David Tranah, our editor at Cambridge University Press, and our respective families.

Don Percival Andrew Walden Applied Physics Laboratory Department of Mathematics Box 355640 Imperial College of Science, University of Washington Technology and Medicine Seattle, WA 98195-5640 London SW7 2BZ, UK dbp@apl.washington.edu a.walden@ic.ac.uk