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(λt)]=t^2Hλ^HR(λ),
t>0 for some -x<H<∞. 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