First-order planar autoregressive model
Sergiy Shklyar
TL;DR
This paper characterizes when a first-order planar autoregressive model $X_{i,j} = a X_{i-1,j} + b X_{i,j-1} + c X_{i-1,j-1} + \varepsilon_{i,j}$ admits a stationary solution, identifying the necessary-and-sufficient condition $D>0$ with $D = (1-a-b-c)(1-a+b+c)(1+a-b+c)(1+a+b-c)$. It then derives the explicit spectral density $f_X(\nu_1,\nu_2) = \sigma_\varepsilon^2 / g(\nu_1,\nu_2)$ and the autocovariance function via a Fourier integral, together with Yule–Walker equations that govern cross-lag covariances. The paper establishes a causal representation $X_{i,j} = \sum_{k,l\ge 0} \psi_{k,l} \varepsilon_{i-k,j-l}$ under four positivity inequalities, and provides conditions for stability and uniqueness of the solution. It also analyzes symmetry and nonidentifiability, showing equivalence classes of parameters yielding the same autocovariance and, in special cases, multiple parameterizations with identical distributions. Additionally, a sufficient condition for pure nondeterminism is given, along with a counterexample to a prior claim, and the appendices provide constructive proofs for spectral-density-based field existence, summation, and integration formulas.
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 $X_{i,j} = a X_{i-1,j} + b X_{i,j-1} + c X_{i-1,j-1} + ε_{i,j}.$ 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 $ε$ 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 $X$.