First-order planar autoregressive model
Pub. online: 6 August 2024
Type: Research Article
Open Access
Open Access
Received
5 February 2024
5 February 2024
Revised
29 May 2024
29 May 2024
Accepted
20 July 2024
20 July 2024
Published
6 August 2024
6 August 2024
Abstract
This paper establishes the conditions for the existence of a stationary solution to the first-order autoregressive equation on a plane as well as properties of the stationary solution. The first-order autoregressive model on a plane is defined by the equation
A stationary solution X to the equation exists if and only if $(1-a-b-c)(1-a+b+c)(1+a-b+c)(1+a+b-c)\gt 0$. The stationary solution X satisfies the causality condition with respect to the white noise ϵ if and only if $1-a-b-c\gt 0$, $1-a+b+c\gt 0$, $1+a-b+c\gt 0$ and $1+a+b-c\gt 0$. A sufficient condition for X to be purely nondeterministic is provided.
An explicit expression for the autocovariance function of X on the axes is provided. With Yule–Walker equations, this facilitates the computation of the autocovariance function everywhere, at all integer points of the plane. In addition, all situations are described where different parameters determine the same autocovariance function of X.
References
Adu, N., Richardson, G.: Unit roots test: Spatial model with long memory errors. Statist. Probab. Lett. 140, 126–131 (2018). MR3812259. https://doi.org/10.1016/j.spl.2018.05.003
Baran, S.: On the variances of a spatial unit root model. Lith. Math. J. 51(2), 122–140 (2011). MR2805732. https://doi.org/10.1007/s10986-011-9113-9
Baran, S., Pap, G.: Parameter estimation in a spatial unilateral unit root autoregressive model. J. Multivariate Anal. 107, 282–305 (2012). MR2890448. https://doi.org/10.1016/j.jmva.2012.01.022
Baran, S., Pap, G., Van Zuijlen, M.C.A.: Asymptotic inference for an unstable spatial AR model. Statistics 38(6), 465–482 (2004). MR2109629. https://doi.org/10.1080/02331880412331319297
Basu, S., Reinsel, G.C.: Properties of the spatial unilateral first-order ARMA model. Adv. in Appl. Probab. 25(3), 631–648 (1993). MR1234300. https://doi.org/10.2307/1427527
Champagnat, F., Idier, J.: On the correlation structure of unilateral AR processes on the plane. Adv. in Appl. Probab. 32(2), 408–425 (2000). MR1778572. https://doi.org/10.1239/aap/1013540171
Cressie, N.A.C.: Statistics for Spatial Data. Wiley, New York (1993). MR1239641. https://doi.org/10.1002/9781119115151
Jain, A.K.: Advances in mathematical models for image processing. Proc. IEEE 69(5), 502–528 (1981). https://doi.org/10.1109/PROC.1981.12021
Kallianpur, G.: Some remarks on the purely nondeterministic property of second order random fields. In: Arató, M., Vermes, D., Balakrishnan, A.V. (eds.) Stochastic Differential Systems, pp. 98–109. Springer, Berlin (1981). https://doi.org/10.1007/BFb0006413 MR0653652
Martin, R.J.: A subclass of lattice processes applied to a problem in planar sampling. Biometrika 66(2), 209–217 (1979). MR0548186. https://doi.org/10.1093/biomet/66.2.209
Martin, R.J.: The use of time-series models and methods in the analysis of agricultural field trials. Comm. Statist. Theory Methods 19(1), 55–81 (1990). MR1060398. https://doi.org/10.1080/03610929008830187
Martin, R.J.: Some results on unilateral ARMA lattice processes. J. Statist. Plann. Inference 50(3), 395–411 (1996). MR1394140. https://doi.org/10.1016/0378-3758(95)00066-6
Paulauskas, V.: On unit root for spatial autoregressive models. J. Multivariate Anal. 98(1), 209–226 (2007). MR2292924. https://doi.org/10.1016/j.jmva.2006.08.001
Pickard, D.K.: Unilateral Markov fields. Adv. in Appl. Probab. 12(3), 655–671 (1980). MR0578842. https://doi.org/10.2307/1426425
Shumway, R.H., Stoffer, D.S.: Time Series Analysis and Its Applications, 4th edn. Springer Texts in Statistics. Springer, Cham (2017). MR3642322. https://doi.org/10.1007/978-3-319-52452-8
Tjøstheim, D.: Statistical spatial series modelling. Adv. in Appl. Probab. 10(1), 130–154 (1978). MR0471224. https://doi.org/10.2307/1426722
Tjøstheim, D.: Statistical spatial series modelling II: Some further results on unilateral lattice processes. Adv. in Appl. Probab. 15(3), 562–584 (1983). MR0706617. https://doi.org/10.2307/1426619
Tjøstheim, D.: Correction: Statistical spatial series modelling. Adv. in Appl. Probab. 16(1), 220 (1984). MR0732140. https://doi.org/10.2307/1427234
Tory, E.M., Pickard, D.K.: Unilateral Gaussian fields. Adv. in Appl. Probab. 24(1), 95–112 (1992). MR1146521. https://doi.org/10.2307/1427731
Unwin, D.J., Hepple, L.W.: The statistical analysis of spatial series. J. R. Stat. Soc. D 23(3/4), 211–227 (1974). https://doi.org/10.2307/2987581
Whittle, P.: On stationary processes in the plane. Biometrika 41(3-4), 434–449 (1954). MR0067450. https://doi.org/10.1093/biomet/41.3-4.434
