ARMA Model for Two‐Dimensional Processes

Abstract

An autoregressive moving‐average model (ARMA) for univariate two‐dimensional homogeneous Gaussian processes is introduced. At the same time, an efficient technique for numerically generating sample functions of such two‐dimensional random processes is developed. The technique uses a recursive equation whose coefficient matrices are determined in accordance with the prescribed autocorrelation function. Using the recursive equation with these coefficient matrices, we can generate with substantial computational ease, sample functions over a large two‐dimensional domain; in principle, we can generate sample functions over an infinite domain. In the present study, sample functions of two‐dimensional, homogeneous, Gaussian random processes with four different analytical forms of the autocorrelation function are generated with the aid of a digital computer. The results indicate that the sample functions 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.