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

Abstract

Abstract-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.

Random Cascades on Wavelet Trees and Their Use in Analyzing and Modeling Natural Images

“…We limit our treatment to those aspects required for subsequent development; the reader is referred to other literature (e.g., [8,12,18,33] for further details of these models, and their application to a variety of 1-D and 2-D statistical inference problems.…”

“…(2) are called multiscale autoregressive (MAR) processes. It has been shown that the MAR framework can effectively model a wide range of Gaussian stochastic processes, including one-dimensional Markov processes [31,33], 1/f -like processes [8,11,12,32,60], and Markov random fields [32,33]. An additional benefit of the MAR framework is that it leads to extremely efficient algorithms for estimating the process x(s) on the basis of noisy observations of the form y(s) = C(s)x(s) + v(s), where v(s) is a zero-mean, white noise process with covariance R(s).…”

“…Simultaneously, wavelet transforms and other multiresolution representations have profoundly influenced image processing and low-level computer vision (e.g., [34]). Moreover, multiscale theory has proven useful in modeling and synthesizing a variety of stochastic processes (e.g., [12,33,60]). …”

Spatially Homogeneous Dynamic Textures

“…It has been shown that this framework can capture a very rich class of phenomena, ranging from one-dimensional Markov processes to 1/f -like processes [13] and Markov random fields (MRFs) [14].…”

Graphical models for statistical inference and data assimilation

“…Finally, imposing a simple statistical structure also makes it feasible to estimate the parameters of the model [80,70]. First, the statistical structure imposed greatly reduces the number of parameters necessary for the model, and additional prior knowledge can be brought to bear on how the parameters are themselves interrelated.…”