Some Novel Methods in Wavelet Data Analysis: Wavelet Anova, F-test Shrinkage, and ?-Minimax Wavelet Shrinkage

It is well known that given a single time series, or image, it might not be possible to utilize various statistical techniques that for their implementation require more than a single observation at a fixed point in time/space. If the researcher has repeated measurements in form of functions/images, than wavelets can be successfully employed in a functional-type data analysis. In the first part of this paper we focus on wavelet based functional ANOVA procedure in which the noise is separated from the signal for different treatments, and at the same time the treatment responses are additionally split on the mean response and the treatment effect, in the spirit of traditional ANOVA. The key properties utilized are abilities of wavelets to decorrelate and regularize the inputs. Different strategies for (multivariate) shrinkage separation of treatment effects and significance testing in the wavelet domain are discussed. In the second part of this we propose a method for wavelet-filtering of noisy images when prior information about their L 2 -energy is available but the researcher has only a single measurement. Assuming the independence model, according to which the wavelet coefficients are treated individually, we propose a level dependent shrinkage rule that turns out to be the ?-minimax rule for a suitable class, say ?, of realistic priors on the wavelet coefficients. Both methods are illustrated and evaluated on test-functions and images with controlled signal-to-noise ratios