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8 - The Wavelet Variance

Published online by Cambridge University Press:  05 December 2013

Donald B. Percival
Affiliation:
University of Washington
Andrew T. Walden
Affiliation:
Imperial College of Science, Technology and Medicine, London
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Summary

Introduction

As we saw in Chapters 4 and 5, one important use for the discrete wavelet transform (DWT) and its variant, the maximal overlap DWT (MODWT), is to decompose the sample variance of a time series on a scale-by-scale basis. In this chapter we explore wavelet-based analysis of variance (ANOVA) in more depth by defining a theoretical quantity known as the wavelet variance (sometimes called the wavelet spectrum). This theoretical variance can be readily estimated based upon the DWT or MODWT and has been successfully used in a number of applications; see, for example, Gamage (1990), Bradshaw and Spies (1992), Flandrin (1992), Gao and Li (1993), Hudgins et al. (1993), Kumar and Foufoula-Georgiou (1993, 1997), Tewfik et al. (1993), Wornell (1993), Scargle (1997), Torrence and Compo (1998) and Carmona et al. (1998). The definition for the wavelet variance and rationales for considering it are given in Section 8.1, after which we discuss a few of its basic properties in Section 8.2. We consider in Section 8.3 how to estimate the wavelet variance given a time series that can be regarded as a realization of a portion of length N of a stochastic process with stationary backward differences. We investigate the large sample statistical properties of wavelet variance estimators and discuss methods for determining an approximate confidence interval for the true wavelet variance based upon the estimated wavelet variance (Section 8.4).

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Publisher: Cambridge University Press
Print publication year: 2000

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  • The Wavelet Variance
  • Donald B. Percival, University of Washington, Andrew T. Walden, Imperial College of Science, Technology and Medicine, London
  • Book: Wavelet Methods for Time Series Analysis
  • Online publication: 05 December 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9780511841040.009
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  • The Wavelet Variance
  • Donald B. Percival, University of Washington, Andrew T. Walden, Imperial College of Science, Technology and Medicine, London
  • Book: Wavelet Methods for Time Series Analysis
  • Online publication: 05 December 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9780511841040.009
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • The Wavelet Variance
  • Donald B. Percival, University of Washington, Andrew T. Walden, Imperial College of Science, Technology and Medicine, London
  • Book: Wavelet Methods for Time Series Analysis
  • Online publication: 05 December 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9780511841040.009
Available formats
×