Wavelets in statistics: beyond the standard assumptions
Bernard W. Silverman · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences · 1999
The original application of wavelets in statistics was to the estimation of a curve given observations of the curve plus white noise at 2J regularly spaced points. The rationale for the use of wavelet methods in this context is reviewed briefly. Various extensions of the standard statistical methodology are discussed. These include curve estimation in the presence of correlated and non–stationary noise, the estimation of (0−1) functions, the handling of irregularly spaced data and data with heavy–tailed noise, and deformable templates in image and shape analysis. Important tools are a Bayesian approach, where a suitable prior is placed on the wavelet expansion, encapsulating the notion that most of the wavelet coefficients are zero; the use of the non–decimated, or translation–invariant, wavelet transform; and a fast algorithm for finding all the within–level covariances within the table of wavelet coefficients of a sequence with arbitrary band–limited covariance structure. Practical applications drawn from neurophysiology, meteorology and palaeopathology are presented. Finally, some directions for possible future research are outlined.