The Stationary Wavelet Transform and some Statistical Applications

Abstract

Wavelets are of wide potential use in statistical contexts. The basicsof the discrete wavelet transform are reviewed using a filter notationthat is useful subsequently in the paper. A `stationary wavelettransform', where the coefficient sequences are not decimated ateach stage, is described. Two different approaches to the constructionof an inverse of the stationary wavelet transform are set out. Theapplication of the stationary wavelet transform as an exploratorystatistical method is discussed, together with its potential usein nonparametric regression. A method of local spectral densityestimation is developed. This involves extensions to the waveletcontext of standard time series ideas such as the periodogram andspectrum. The technique is illustrated by its application to datasets from astronomy and veterinary anatomy.