Abstract
The likelihood method is developed for the analysis of socalled regular point patterns. Approximating the normalizing factor of Gibbs canonical distribution, we simultaneously estimate two parameters, one for the scale and the other which measures the softness (or hardness), of repulsive interactions between points. The approximations are useful up to a considerably high density. Some real data are analyzed to illustrate the utility of the parameters for characterizing the regular point pattern.
Similar content being viewed by others
References
Akaike, H. (1980). Likelihood and Bayes procedure,Bayesian Statistics, (eds. J. M., Bernardo et al.), 141–166, University Press, Valencia, Spain.
Besag, J., Milne, R. and Zachary, S. (1982). Point process limits of lattice processes,J. Appl. Probab.,19, 210–216.
Dacey, M. F. (1972). Regularity in spatial distributions: a stochastic model of the imperfect central place plane,Statistical Ecology, (eds. G. Patil, E. C. Pielou and W. E. Waters), Vol. 1, 287–309.
Gates, D. J. and Westcott, M. (1986). Clustering estimates for spatial point distributions with unstable potentials,Ann. Inst. Statist. Math.,38, 123–135.
Good, I. J. (1965).The Estimation of Probabilities, M.I.T. Press, Cambridge, Massachusetts.
Good, I. J. and Gaskins, R. A. (1971). Nonparametric roughness penalties for probability densities,Biometrika,58, 255–277.
Hoover, W. G., Gray, S. G. and Johnson, K. W. (1971). Thermodynamic properties of the fluid and solid phases for inverse power potentials,J. Chem. Phys.,55, 1128–1136.
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.
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.
Numata, M. (1964). Forest vegetation, particularly pine stems in the vicinity of Choshi — flora and vegetation at Choshi, Chiba Prefecture, IV,Bull. Choshi Marine Laboratory, No. 6, 27–37, Chiba University (in Japanese).
Ogata, Y. and Katsura, K. (1988). Likelihood analysis of spatial inhomogeneity for marked point patterns,Ann. Inst. Statist. Math.,40, 29–39.
Ogata, Y. and Tanemura, M. (1981a). Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure,Ann. Inst. Statist. Math.,33, 315–338.
Ogata, Y. and Tanemura, M. (1981b). A simple simulation method for quasi-equilibrium point patterns, Research Memorandum No. 210, The Institute of Statistical Mathematics, Tokyo, Japan.
Ogata, Y. and Tanemura, M. (1981c). Approximation of likelihood function in estimating the interaction potentials from spatial point patterns, Research Memorandum No. 216, The Institute of Statistical Mathematics, Tokyo, Japan.
Ogata, Y. and Tanemura, M. (1982). Likelihood analysis of spatial point patterns, Research Memorandum No. 241, The Institute of Statistical Mathematics, Tokyo, Japan.
Ogata, Y. and Tanemura, M. (1984). Likelihood analysis of spatial point patterns,J. Roy. Statist. Soc. Ser. B,46, 496–518.
Ogata, Y. and Tanemura, M. (1985). Estimation of interaction potentials of marked spatial point patterns through the maximum likelihood method,Biometrics,41, 421–433.
Ogata, Y. and Tanemura, M. (1986). Likelihood estimation of interaction potentials and external fields of inhomogeneous spatial point patterns,Pacific Statistical Congress, (eds. I.S., Francis, B. F. J., Manly and F. C., Lam), 150–154, North-Holland, Amsterdam.
Penttinen, A. (1984).Modelling Interactions in Spatial Point Patterns: Parameter Estimation by the Maximum Likelihood Method, Jyväskylä Studies in Computer Science, Economics and Statistics, 7, University of Jyväskylä, Jyväskylä, Finland.
Ree, F. H. and Hoover, W. G. (1967). Seventh virial coefficients for hard spheres and hard disks,J. Chem. Phys.,46, 4181–4197.
Ripley, B. D. (1979). Simulating spatial patterns: dependent samples from a multivariate density,Appl. Statist.,28, 109–112.
Shapiro, M. B., Schein, S. J. and Monasterio, F. M. (1985). Regularity and structure of the spatial pattern of blue cones of macaque retina,J. Amer. Statist. Assoc.,80, 803–814.
Swol, F.van, Woodcock, L. V. and Cape, J. N. (1980). Melting in two dimensions: determination of phase transition boundaries,J. Chem. Phys.,73, 913–922.
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), 115–230, North-Holland, Amsterdam.
Author information
Authors and Affiliations
About this article
Cite this article
Ogata, Y., Tanemura, M. Likelihood estimation of soft-core interaction potentials for Gibbsian point patterns. Ann Inst Stat Math 41, 583–600 (1989). https://doi.org/10.1007/BF00050670
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00050670