# Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure

## Summary

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.

## Keywords

Partition Function Potential Function Akaike Information Criterion Monte Carlo Simulation Method Maximum Likelihood Procedure## Preview

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## References

- [1]Akaike, H. (1977). On entropy maximization principle,
*Applications of Statistics*(Ed. P. R. Krishnaiah), North-Holland, Amsterdam, 27–41.Google Scholar - [2]Bartlett, M. S. (1964). The spectral analysis of two-dimensional point processes,
*Biometrika*,**44**, 299–311.MathSciNetCrossRefGoogle Scholar - [3]Baudin, M. (1980). Likelihood and nearest neighbor distance properties of multidimensional Poisson cluster processes, submitted to
*J. Appl. Prob.*Google Scholar - [4]Besag, J. and Diggle, P. J. (1977). Simple Monte Carlo tests for spatial pattern,
*Appl. Statist.*,**26**, 327–333.CrossRefGoogle Scholar - [5]Diggle, P. J. (1979). On parameter estimation and goodness-of-fit testing for spatial patterns,
*Biometrics*,**35**, 87–101.CrossRefGoogle Scholar - [6]Feynman, R. P. (1972).
*Statistical Mechanics: A Set of Lectures*, Benjamin, Reading.zbMATHGoogle Scholar - [7]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 - [8]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 - [9]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 - [10]Matérn, B. (1960). Spatial variation,
*Meddelanden fran Statens Skogsforskningsinstitut*,**49**, No. 5, 1–144.Google Scholar - [11]Mayer, J. E. and Mayer, M. G. (1940).
*Statistical Mechanics*, John Wiley, New York.zbMATHGoogle Scholar - [12]Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equation of state calculations by fast computing machines.
*J. Chem. Phys.*,**21**, 1087–1092.CrossRefGoogle Scholar - [13]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 - [14]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 - [15]Ogata, Y. (1978). The asymptotic behaviour of maximum likelihood estimators for stationary point processes,
*Ann. Inst. Statist. Math.*,**30**, A, 243–261.MathSciNetCrossRefGoogle Scholar - [16]Ogata, Y. (1980). Maximum likelihood estimates of incorrect Markov models for time series and the derivation of AIC,
*J. Appl. Prob.*,**17**, 59–72.MathSciNetCrossRefGoogle Scholar - [17]Ogata, Y. (1981). On Lewis' simulation method for point processes,
*IEEE Trans. Inform. Theory*, IT-27,**1**, 23–31.CrossRefGoogle Scholar - [18]Ripley, B. D. (1977). Modelling spatial patterns (with discussion),
*J. R. Statist. Soc.*, B,**39**, 172–212.Google Scholar - [19]Sakamoto, Y. and Akaike, H. (1978). Analysis of cross classified data by AIC,
*Ann. Inst. Statist. Math.*,**30**, B, 185–197.MathSciNetCrossRefGoogle Scholar - [20]Tanemura, M. and Hasegawa, M. (1980). Geometrical models of territory. I. Models for synchronous and asynchronous settlement of territories,
*J. Theor. Biol.*,**82**, 477–496.MathSciNetCrossRefGoogle Scholar - [21]Vere-Jones, D. (1970). Stochastic models for earthquake occurrences (with discussion),
*J. R. Statist. Soc.*, B,**32**, 1–62.zbMATHGoogle Scholar - [22]Vere-Jones, D. (1978). Space time correlations for microearthquakes—a pilot study,
*Suppl. Adv. Appl. Prob.*,**10**, 73–87.CrossRefGoogle Scholar - [23]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