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

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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.

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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.

    Article  MathSciNet  Google Scholar 

  3. Baudin, M. (1980). Likelihood and nearest neighbor distance properties of multidimensional Poisson cluster processes, submitted toJ. Appl. Prob.

  4. Besag, J. and Diggle, P. J. (1977). Simple Monte Carlo tests for spatial pattern,Appl. Statist.,26, 327–333.

    Article  Google Scholar 

  5. Diggle, P. J. (1979). On parameter estimation and goodness-of-fit testing for spatial patterns,Biometrics,35, 87–101.

    Article  Google Scholar 

  6. Feynman, R. P. (1972).Statistical Mechanics: A Set of Lectures, Benjamin, Reading.

    MATH  Google 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.

  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.

    MATH  Google 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.

    Article  Google 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.

    Article  MathSciNet  Google 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.

    Article  MathSciNet  Google Scholar 

  17. Ogata, Y. (1981). On Lewis' simulation method for point processes,IEEE Trans. Inform. Theory, IT-27,1, 23–31.

    Article  Google 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.

    Article  MathSciNet  Google 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.

    Article  MathSciNet  Google Scholar 

  21. Vere-Jones, D. (1970). Stochastic models for earthquake occurrences (with discussion),J. R. Statist. Soc., B,32, 1–62.

    MATH  Google Scholar 

  22. Vere-Jones, D. (1978). Space time correlations for microearthquakes—a pilot study,Suppl. Adv. Appl. Prob.,10, 73–87.

    Article  Google 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 

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Authors
  1. Yosihiko Ogata
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  2. Masaharu Tanemura
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The Institute of Statistical Mathematics

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Ogata, Y., Tanemura, M. Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure. Ann Inst Stat Math 33, 315–338 (1981). https://doi.org/10.1007/BF02480944

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  • DOI: https://doi.org/10.1007/BF02480944

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