Auto‐Regressive Model for Nonstationary Stochastic Processes

Abstract

An autoregressive model for univariate, one‐dimensional, nonstationary, Gaussian random processes with evolutionary power spectra is introduced. At the same time, an efficient technique for numerically generating sample functions of such nonstationary processes is developed. The technique uses a recursive equation which: (1) Reflects the nature of the nonstationarity of the process whose sample functions are to be generated; and (2) involves a normalized univariate, one‐dimensional white noise sequence. The coefficients of the recursive equation are determined using the autocorrelation function of the process, which in turn is calculated from the evolutionary power spectrum at every time instant. Using the recursive equation with those coefficients, sample functions over a specified domain can be generated with substantial computational ease. Univariate, one‐dimensional, nonstationary processes with three different forms of the evolutionary power spectrum are modeled, and their sample functions are generated with the aid of an 11/750 VAX/VMS computer. The results indicate that the sample functions generated by the method presented reflect the prescribed probabilistic characteristics extremely well. This is seen from the closeness between the analytically prescribed autocorrelation functions and the corresponding sample autocorrelation functions computed from the generated sample functions.