Evolutionary Spectra and Non-Stationary Processes

Abstract

We develop an approach to the spectral analysis of non-stationary processes which is based on the concept of "evolutionary spectra"; that is, spectral functions which are time dependent, and have a physical interpretation as local energy distributions over frequency. It is shown that the notion of evolutionary spectra generalizes the usual definition of spectra for stationary processes, and that, under certain conditions, the evolutionary spectrum at each instant of time may be estimated from a single realization of a process. By such means it is possible to study processes with continuously changing "spectral patterns".1. INTRODUCTION IN the classical approach to statistical spectral analysis it is always assumed that the process under study, Xt, is stationary, at least up to the second order. That is, we assume that E(Xt) = 1-', a constant (independent of t) which we may take to be zero, and that, for each sand t, the covariance Rs,t = E{(Xs-I-'HXt-I-')*} (1.1) (* denoting the complex conjugate) is a function ofls-tl only. In this case it is well known that Rs,t has a spectral representation of the formwhere F(w) is some function having the properties of a distribution function, and the range of integration is (-00,00) for a continuous parameter process, and (-7T,7T) in the discrete case.Corresponding to (1.2), {XI} has a spectral representation of the formwhere Z(w) is an orthogonal process with E{ldZ(w)12} = dF(w). When {Xt} represents some physical process, the spectral density function few) = F'(w) (when it exists) describes the distribution (over the frequency range) of the energy (per unit time) dissipated by the process, and given a sample record of {XI}, there are several methods of estimatingf(w) (see, e.g., Grenander and Rosenblatt, 1957, Ch. 4).In practice, however, it often happens that the assumption of stationarity is a very doubtful one. For example, records of atmospheric turbulence exhibit marked changes over periods of time, and in such cases classical spectral analysis based on a stationary model can hardly be carried through with conviction. The question arises, therefore, 1965]

Oil price and stock market co-movement: What can we learn from time-scale approaches?

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Asymptotic Properties of Quasi-Maximum Likelihood Estimators for ARMA Models with Time-Dependent Coefficients

“…Subba Rao (1970) is interested in AR models using estimators based on the evolutionary spectral analysis of Priestley (1965). Mélard (1977) fits (marginally) heteroscedastic ARMA models called ARMAG models.…”

Simulation of non-stationary ground motion processes (II)

“…Evolutionary power spectrum (Priestley, 1965(Priestley, , 1967, as a direct extension of power spectrum of stationary stochastic processes, has clear and definite physical meaning, and has been widely applied especially to ground motion records analysis. However, because the complex modulating function A(t, co) cannot be solely determined (Hammond, 1968) in evolutionary power spectrum estimation, uniformly modulated processes are usually assumed for non-stationary ground motion processes, i.e., complex modulating function is only a real function A(t) of time (Liu, 1970;Bendat and Piersol, 2000).…”

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