A class of second-order stationary self-similar processes for 1/f phenomena

Abstract

We propose a class of statistically self-similar processes and outline an alternative mathematical framework for the modeling and analysis of 1/f phenomena. The foundation of the proposed class is based on the extensions of the basic concepts of classical time series analysis, in particular, on the notion of stationarity. We consider a class of stochastic processes whose second-order structure is invariant with respect to time scales, i.e., E[X(t)X(/spl lambda/t)]=t/sup 2H//spl lambda//sup H/R(/spl lambda/), t>0 for some -x<H</spl infin/. For H=0, we refer to these processes as wide sense scale stationary. We show that any self-similar process can be generated from scale stationary processes. We establish a relationship between linear scale-invariant system theory and the proposed class that leads to a concrete analysis framework. We introduce new concepts, such as periodicity, autocorrelation, and spectral density functions, by which practical signal processing schemes can be developed. We give several examples of scale stationary processes including Gaussian, non-Gaussian, covariance, and generative models, as well as fractional Brownian motion as a special case. In particular, we introduce a class of finite parameter self-similar models that are similar in spirit to the ordinary ARMA models by which an arbitrary self-similar process can be approximated. Results from our study suggest that the proposed self-similar processes and the mathematical formulation provide an intuitive, general, and mathematically simple approach to 1/f signal processing.