The Modeling and Estimation of Statistically Self-Similar Processes in a Multiresolution Framework

Statistically self-similar (SSS) processes can be used to describe a variety of physical phenomena, yet modeling these phenomena has proved challenging. Most of the proposed models for SSS and approximately SSS processes have power spectra that behave as 1=f , such as fractional Brownian motion (fBm), fractionally differenced noise, and wavelet-based syntheses. The most flexible framework is perhaps that based on wavelets, which provides a powerful tool for the synthesis and estimation of 1=f processes, but assumes a particular distribution of the measurements. An alternative framework is the class of multiresolution processes proposed by Chou et al. [1994], which has already been shown to be useful for the identification of the parameters of fBm. These multiresolution processes are defined by an autoregression 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, likelihood calculator, and conditional simulator that make no assumptions about 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 models are realized by using canonical correlations (CC) and by exploiting the selfsimilarity and possible stationarity or stationary increments of the desired process. A number of examples are provided to demonstrate the utility of the multiscale framework in simulating and estimating SSS processes.