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Statistical spatial series modelling

Published online by Cambridge University Press:  01 July 2016

Dag Tjøstheim
Dag Tjøstheim*
Affiliation:
Royal Norwegian Council for Scientific and Industrial Research (NORSAR)
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Abstract

A random spatial series is a collection of random variables F(xi, · · ·, xn) depending on several spatial coordinates (x1, · · ·, xn). An attempt is made to construct a statistical second-order theory of such series when (x1, · · ·, xn) varies over a regular cartesian lattice. Using the properties of the linear (Hilbert) space associated with the series, the concepts of innovation and purely non-deterministic (p.n.d.) series are introduced. For a p.n.d. series F(x1, · · ·, xn) a unilateral representation is obtained in terms of a white innovations series Z(y1, · · ·, yn) where The representatation is specialized to the homogeneous case and we discuss spectral conditions for p.n.d. The familiar time-series condition ∫ log f(λ) dλ > –∞ on the spectral density f is necessary but not sufficient. A sufficient condition is stated. Motivated by the p.n.d. unilateral representation results we define unilateral arma (autoregressive-moving average) spatial series models. Stability and invertibility conditions are formulated in terms of the location of zero sets of polynomials relative to the unit polydisc in Cn, and a rigorous shift operator formalism is established. For autoregressive spatial series a Yule–Walker-type matrix equation is formulated and it is shown how this can be used to obtain estimates of the autoregressive parameters. It is demonstrated that under mild conditions the estimates are consistent and asymptotically normal.

Keywords

SPATIAL SERIESREGULAR CARTESIAN LATTICEPURELY NON-DETERMINISTICSHIFT OPERATOR FORMALISMUNILATERAL ARMA MODELSYULE–WALKER ESTIMATESASYMPTOTIC PROPERTIES

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 1 , March 1978 , pp. 130 - 154
Copyright
Copyright © Applied Probability Trust 1978 

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Footnotes

This research was supported by the Advanced Research Projects Agency of the U.S. Department of Defense and was monitored by AFTAC, Patrick AFB FL 32925, under Contract No. F08606-77-C-0001.

References

[1] Agterberg, F. P. (1967) Computer techniques in geology. Earth Science Reviews 3, 47–77.Google Scholar
[2] Anderson, T. W. (1971) The Statistical Analysis of Time Series. Wiley, New York.Google Scholar
[3] Barry, R. G. and Perry, A. (1973) Synoptic Climatology. Methuen, London.Google Scholar
[4] Bartlett, M. S. (1975) The Statistical Analysis of Spatial Pattern. Chapman and Hall, London.Google Scholar
[5] Berry, B. L. J. and Marble, D. F., (eds.) (1968) Spatial Analysis: A Reader in Statistical Geography. Prentice-Hall, Englewood Cliffs, N.J.Google Scholar
[6] Besag, J. (1974) Spatial interaction and the statistical analysis of lattice systems. J.R. Statist. Soc. B 36, 192–236.Google Scholar
[7] Cliff, A. D. and Ord, J. K. (1973) Spatial Autocorrelation. Pion, London.Google Scholar
[8] Cramér, H. (1961) On some classes of nonstationary stochastic processes. Proc. 4th Berkeley Symp. Math. Statist. Prob. 2, 57–77.Google Scholar
[9] Davies, J. C. (1973) Statistics and Data Analysis in Geology. Wiley, New York.Google Scholar
[10] Delfiner, P. and Delhomme, J. P. (1975) Optimum interpolation by Kriging. In Nato Advanced Study Institute on Display and Analysis of Spatial Data, Wiley, New York, 96–115.Google Scholar
[11] Fisher, W. D. (1971) Econometric estimation with spatial dependence. Regional and Urban Economics 1, 40–64.Google Scholar
[12] Hoffman, K. (1962) Banach Spaces of Analytic Functions. Prentice-Hall, Englewood Cliffs, N.J.Google Scholar
[13] Kershaw, K. A. (1964) Quantitative and Dynamic Ecology. Arnold, London.Google Scholar
[14] Kung, S. Y., Levy, B., Morf, M. and Kailath, T. (1977) New results in 2D systems theory, II: 2D state-space models–realization and the notions of controllability, observability and minimality. Proc. IEEE 65, 945–960.Google Scholar
[15] Levy, B., Kung, S. Y., and Morf, M. (1976) New results in 2D systems theory—realization and the notions of controllability, observability and minimality. Information Systems Laboratory, Department of Electrical Engineering, Stanford University.Google Scholar
[16] Riesz, F. and Sz.-Nagy, B. (1955) Functional Analysis, 2nd edn. Ungar, New York.Google Scholar
[17] Rozanov, Yu. A. (1967) Stationary Random Processes. Holden-Day, San Francisco.Google Scholar
[18] Rudin, W. (1969) Function Theory in Polydiscs. Benjamin, New York.Google Scholar
[19] Tjøstheim, D. (1976) Spectral representations and density operators for infinite-dimensional homogeneous random fields. Z. Wahrscheinlichkeitsth. 35, 323–336.Google Scholar
[20] Tjøstheim, D. (1977) Autoregressive modelling and spectral estimation for spatial data. Some simulation experiments. Stanford Exploration Project SEP-11, Department of Geophysics, Stanford University.Google Scholar
[21] Unwin, D. J. and Hepple, L. W. (1974) The statistical analysis of spatial series. The Statistician 23, 211–227.Google Scholar
[22] Whittle, P. (1974) On stationary processes in the plane. Biometrika 41, 434–449.Google Scholar
[23] Yaglom, A. M. (1957) Some classes of random fields in n-dimensional space, related to stationary processes. Theory Prob. Appl. 2, 273–320.Google Scholar