Overcomplete systems of wavelet and related local bases for adaptive signal representation and estimation
Rainer von Sachs · 1999
this paper we will discuss the usefulness of overcomplete systems of basis functions, such as wavelets or localized sine and cosine functions, for the adaptive parsimonious representation and estimation of statistical signals which show an inhomogeneous behaviour over time. Typical examples can be found in electrical engineering, sound and speech processing, geophysics, biomedicine, etc. We illustrate this along two methods which aim at estimation of time-dependent autocovariance or spectrum of non-stationary processes: the "Auto-SLEX" method of Ombao et al (1999) and the approach on "locally stationary wavelet processes" by Nason et al (1998). Common to both is on one hand the inherent redundancy of the used overcomplete system which is the key to adaptation. On the other hand, to allow for rigorous modeling and statistical estimation, a control of this redundancy is needed which is the price for the flexibility of overcomplete systems in comparison with orthogonal ones. Two di#erent though related possibilities of a controlled overcompleteness will be investigated here: on one hand collections of ortho-basis from which a certain "best" element (best adapted to the signal) is to be searched for, on the other hand, translation--invariant (and hence overcomplete) representations for the autocovariance function of a possible non--stationary stochastic signal. The idea of using an overcomplete collection (i.e. a "library") of orthogonal basis functions to best represent the content of a given signal goes back to Coifman and Wickerhauser (1991). From a general, not necessarily statistical, point of view one tries to match the predominant signal features by a set of "independent", i.e. orthogonal, components. This so-called search of a "Best Basis (BB)", achieved by a comput...