The modeling and estimation of statistically self-similar processes in a multiresolution framework

Statistically self-similar (SSS) processes can be used to describe a variet y of physical phenomena, yet modeling these phenomena has proved challengin g. Most of the proposed models for SSS and approximately SSS processes have power spectra that behave as 1/f(gamma), such as fractional Brownian motio n (fBm), fractionally differenced noise, and wavelet-based syntheses. The m ost flexible framework is perhaps that based on wavelets, which provides a powerful tool for the synthesis and estimation of 1/f processes, but assume s a particular distribution of the measurements. An alternative framework i s the class of multiresolution processes proposed by Chou ef RI. [1994], wh ich has already been shown to be useful for the identification of the param eters of fBm, These multiresolution processes are defined by an autoregress ion in scale that makes them naturally suited to the representation of SSS (and approximately SSS) phenomena, both stationary and nonstationary, Also, this multiresolution framework is accompanied by an efficient estimator, l ikelihood calculator, and conditional simulator that make no assumptions ab out the distribution of the measurements. In this paper, we show how to use the multiscale framework to represent SSS (or approximately SSS) processes such as fBm and fractionally differenced Gaussian noise. The multiscale mo dels are realized by using canonical correlations (CC) and by exploiting th e selfsimilarity and possible stationarity or stationary increments of the desired process, A number of examples are provided to demonstrate the utili ty of the multiscale framework in simulating and estimating SSS processes.