Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T18:01:42.607Z Has data issue: false hasContentIssue false
  • English
  • Français

Properties of the spatial unilateral first-order ARMA model

Part of: Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Sabyasachi Basu  and
Gregory C. Reinsel
Sabyasachi Basu*
Affiliation:
Southern Methodist University
Gregory C. Reinsel*
Affiliation:
University of Wisconsin-Madison
*
∗ Postal address: Department of Statistical Science, Southern Methodist University, Dallas, TX 75275, USA.
∗∗ Postal address: Department of Statistics, University of Wisconsin, Madison, WI 53706, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

For two-dimensional spatial data, a spatial unilateral autoregressive moving average (ARMA) model of first order is defined and its properties studied. The spatial correlation properties for these models are explicitly obtained, as well as simple conditions for stationarity and conditional expectation (interpolation) properties of the model. The multiplicative or linear-by-linear first-order spatial models are seen to be a special case which have proved to be of practical use in modeling of two-dimensional spatial lattice data, and hence the more general models should prove to be useful in applications. These unilateral models possess a convenient computational form for the exact likelihood function, which gives proper treatment to the border cell values in the lattice that have a substantial effect in estimation of parameters. Some simulation results to examine properties of the maximum likelihood estimator and a numerical example to illustrate the methods are briefly presented.

Keywords

SPATIAL UNILATERAL MODELSPATIAL CORRELATIONSEXACT LIKELIHOODCONDITIONAL EXPECTATION

MSC classification

Primary: 62M30: Spatial processes
Secondary: 62M05: Markov processes 60G10: Stationary processes
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 3 , September 1993 , pp. 631 - 648
Copyright
Copyright © Applied Probability Trust 1993 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bartlett, M. S. (1971) Physical nearest neighbour models and non-linear time series. J. Appl. Prob. 8, 222–232.CrossRefGoogle Scholar
Bartlett, M. S. (1975) The Statistical Analysis of Spatial Pattern. Chapman and Hall, London.Google Scholar
Basu, S. and Reinsel, G. C. (1992a) Regression models with spatially correlated errors. Technical Report, Department of Statistics, University of Wisconsin-Madison.Google Scholar
Basu, S. and Reinsel, G. C. (1992b) A note on properties of spatial Yule-Walker estimators. J. Statist. Comput. Simul. 41, 243–255.CrossRefGoogle Scholar
Besag, J. E. (1972) On the correlation structure of some two dimensional stationary processes. Biometrika 59, 43–48.CrossRefGoogle Scholar
Besag, J. (1974) Spatial interaction and the statistical analysis of lattice systems. J. R. Statist. Soc. B 36, 192–236.Google Scholar
Box, G. E. P. and Jenkins, G. M. (1976) Time Series Analysis: Forecasting and Control, revised edn. Holden-Day, Oakland.Google Scholar
Bronars, S. G. and Jansen, D. W. (1987) The geographic distribution of unemployment in the U.S.: a spatial-time series analysis. J. Econometrics 36, 251–279.CrossRefGoogle Scholar
Chellappa, R. (1985) Two-dimensional discrete Gaussian Markov random field models for image processing. In Progress in Pattern Recognition 2, ed. Kanal, L. N. and Rosenfeld, A., pp. 79–112, North-Holland, Amsterdam.Google Scholar
Cliff, A. D. and Ord, J. K. (1981) Spatial Processes: Models and Applications. Pion, London.Google Scholar
Dahlhaus, R. and Künsch, H. (1987) Edge effects and efficient parameter estimation for stationary random fields. Biometrika 74, 877–882.CrossRefGoogle Scholar
Geman, S. and Geman, D. (1984) Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741.CrossRefGoogle ScholarPubMed
Guyon, X. (1982) Parameter estimation for a stationary process on a d-dimensional lattice. Biometrika 69, 95–105.CrossRefGoogle Scholar
Haining, R. P. (1978a) Estimating spatial-interaction models. Env. Planning A 10, 305–320.CrossRefGoogle Scholar
Haining, R. P. (1978b) The moving average model for spatial interaction. Trans. Inst. Br. Geog. 3, 202–225.CrossRefGoogle Scholar
Kempton, R. A. and Howes, C. W. (1981). The use of neighbouring plot values in the analysis of variety trials. Appl. Statist. 30, 59–70.CrossRefGoogle Scholar
Martin, R. J. (1979) A subclass of lattice processes applied to a problem in planar sampling. Biometrika 66, 209–217.CrossRefGoogle Scholar
Martin, R. J. (1990) The use of time-series models and methods in the analysis of agricultural field trials. Commun. Statist.-Theory Meth. 19, 55–81.CrossRefGoogle Scholar
Mickens, R. E. (1987) Difference Equations. Van Nostrand Reinhold, New York.Google Scholar
Moore, M. (1988) Spatial linear processes. Commun. Statist.-Stoch. Models 4, 45–75.Google Scholar
Rao, C. R. (1973) Linear Statistical Inference and Its Applications, 2nd edn. Wiley, New York.CrossRefGoogle Scholar
Ripley, B. D. (1981) Spatial Statistics. Wiley, New York.CrossRefGoogle Scholar
Tjøstheim, D. (1978) Statistical spatial series modelling. Adv. Appl. Prob. 10, 130–154.CrossRefGoogle Scholar
Tjøstheim, D. (1981) Autoregressive modeling and spectral analysis of array data in the plane. IEEE Trans. Geosci. Rem. Sensing 19, 15–24.CrossRefGoogle Scholar
Tjøstheim, D. (1983) Statistical spatial series modelling II: Some further results on unilateral processes. Adv. Appl. Prob. 15, 562–584.CrossRefGoogle Scholar
Whittle, P. (1954) On stationary processes in the plane. Biometrika 41, 434–449.CrossRefGoogle Scholar