Wavelets in Statistics: Some Recent Developments

Abstract

Wavelet methods in statistics are in their infancy as far as the range of problems to which they have been applied is concerned, some elementary aspects of wavelets are reviewed, concentrating on the discrete wavelet transform, because of its relevance to practical and computational statisticians. Several recent areas of research are discussed, concentrating on extensions of the standard paradigm. A Bayesian approach is natural, because of the notion that the wavelet expansion is sparse. Wavelets are easily applied to regression where the errors are correlated. The combination of these ideas is demonstrated on data generated in the study of ion channel gating, with excellent results. The arrangement and distribution of the data can be quite general, because of a fast algorithm that finds details of the variance structure of the discrete wavelet transform. Furthermore, this algorithm naturally leads to simple methods of dealing with outliers and with heteroscedastic data. Finally, an application to deformable templates demonstrates the wide potential of wavelet methods; this arises from a palaeopathological data set in an arthritis study.