First-order planar autoregressive model

Abstract

This paper establishes the conditions of existence of a stationary solution to the first order autoregressive equation on a plane as well as properties of the stationarity solution. The first-order autoregressive model on a plane is defined by the equation $X_{i,j} = a X_{i-1,j} + b X_{i,j-1} + c X_{i-1,j-1} + \epsilon_{i,j}.$ A stationary solution $X$ to the equation exists if and only if $(1-a-b-c) (1-a+b+c) (1+a-b+c) (1+a+b-c) > 0$. The stationary solution $X$ satisfies the causality condition with respect to the white noise $\epsilon$ if and only if $1-a-b-c>0$, $1-a+b+c>0$, $1+a-b+c>0$ and $1+a+b-c>0$. A sufficient condition for X to be purely nondeterministic is provided. An explicit expression for the autocovariance function of $X$ at some points is provided. With Yule-Walker equations, this allows to compute the autocovariance function everywhere. In addition, all situations are described where different parameters determine the same autocovariance function of $X$.