# zbMATH — the first resource for mathematics

Wavelet methods for time series analysis. (English) Zbl 0963.62079
Cambridge Series in Statistical and Probabilistic Mathematics. 4. Cambridge: Cambridge University Press. xxv, 594 p. (2000).
This book is an introduction to wavelets and their applications in time series analysis. The authors motivate the use of wavelets by the continuous wavelet transform (CWT). However, most of the text is devoted to the discrete wavelet transform (DWT) since this is a tool for discrete time series analysis. A review of Fourier transform and filters is presented. The basic theory for orthonormal transforms is explained, which includes the orthonormal discrete Fourier transform. Then the authors describe DWT and the wavelet and scaling filters. An example of the DWT analysis concerns some electrocardiogram measurements. A chapter is devoted to the maximal overlap DWT. Then the discrete wavelet packet transform is described as any one of a collection of orthonormal transforms. This transform depends on a wavelet filter and its associated scaling filter.
A method for defining an optimal transform for a given time series is considered. Basic concepts of random variables and stochastic processes are introduced, from the definition of a random variable to some special time series models. The authors define the wavelet variance (sometimes called the wavelet spectrum). Large sample statistical properties of wavelet variance estimators are discussed. It is described how estimates of the wavelet variance can be turned into estimates of the spectral density. Particular attention is paid to the analysis and synthesis of long memory processes using the DWT. In this case the DWT creates a new set of random variables, namely, the wavelet coefficients, that are approximately uncorrelated so that their statistical analysis is easier. Another important application of the DWT is signal estimation (or denoising). The wavelet-based techniques are adaptive to a wide variety of signals. Wavelet analysis of finite energy signals concerns the multiresolution view of the CWT and the relation between the CWT and the DWT.
Theoretical results are illustrated on real data. The book contains many embedded exercises with full solutions provided in the Appendix. Additional exercises can be found at the end of each chapter and a solution guide is available on the Web site introduced in the book. The Web site gives also access to the time series and wavelets used in the book. The text is suitable for students and researchers in physical sciences, particularly in electrical engineering, physics, geophysics, astronomy, and hydrology.
Reviewer: J.Anděl (Praha)

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics 62P35 Applications of statistics to physics 94A12 Signal theory (characterization, reconstruction, filtering, etc.) 62M15 Inference from stochastic processes and spectral analysis
##### Software:
wavemulcor; wmtsa