Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure
- 96 Downloads
A homogeneous spatial point pattern is regarded as one of thermal equilibrium configurations whose points interact on each other through a certain pairwise potential. Parameterizing the potential function, the likelihood is then defined by the Gibbs canonical ensemble. A Monte Carlo simulation method is reviewed to obtain equilibrium point patterns which correspond to a given potential function. An approximate log likelihood function for gas-like patterns is derived in order to compute the maximum likelihood estimates efficiently. Some parametric potential functions are suggested, and the Akaike Information Criterion is used for model selection. The feasibility of our procedure is demonstrated by some computer experiments. Using the procedure, some real data are investigated.
KeywordsPartition Function Potential Function Akaike Information Criterion Monte Carlo Simulation Method Maximum Likelihood Procedure
Unable to display preview. Download preview PDF.
- Akaike, H. (1977). On entropy maximization principle,Applications of Statistics (Ed. P. R. Krishnaiah), North-Holland, Amsterdam, 27–41.Google Scholar
- Baudin, M. (1980). Likelihood and nearest neighbor distance properties of multidimensional Poisson cluster processes, submitted toJ. Appl. Prob. Google Scholar
- Fisher, L. (1972). A survey of the mathematical theory of multidimensional point processes,Stochastic Point Processes: statistical analysis, theory and applications (ed. P. A. W. Lewis), Wiley, New York, 468–513.Google Scholar
- Hasegawa, M. and Tanemura, M. (1978). Mathematical models on spatial patterns of territories,Proceedings of the international symposium on mathematical topics in biology, Kyoto, Japan, Sept. 11–12, 1978, 39–48.Google Scholar
- Howell, T. R., Araya, B. and Millie, W. R. (1974). Breeding biology of the Gray Gull,Larus modestus, Univ. Calif. Publ. Zool,104, 1–57.Google Scholar
- Matérn, B. (1960). Spatial variation,Meddelanden fran Statens Skogsforskningsinstitut,49, No. 5, 1–144.Google Scholar
- Numata, M. (1961). Forest vegetation in the vicinity of Choshi-coastal flora and vegetation at Choshi, Chiba Prefecture, IV (in Japanese),Bull. Choshi Marine Laboratory, No. 3, Chiba University, 28–48.Google Scholar
- Numata, M. (1964). Forest vegetation, particularly pine stands in the vicinity of Choshi-flora and vegetation at Choshi, Chiba Prefecture, VI (in Japanese),Bull. Choshi Marine Laboratory, No. 6, Chiba University, 27–37.Google Scholar
- Ripley, B. D. (1977). Modelling spatial patterns (with discussion),J. R. Statist. Soc., B,39, 172–212.Google Scholar
- Wood, W. W. (1968). Monte Carlo studies of simple liquid models,Physics of Simple Liquids (eds. H. N. V. Temperley, J. S. Rowlinson and G. S. Rushbrooke), Chap. 5, North-Holland, Amsterdam, 115–230.Google Scholar