Development of the Likelihood Procedures for the Spatial Point Patterns and Their Applications

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For the patterns of certain objects which are distributed in the plane or in the space, there are requirements to analyse them statistically in many fields of applied science such as ecology, geography, medicine, material technology, physics, chemistry and so on. In the statistical analysis and statistical modelling of spatial patterns, there appear new features which are different from those in the other fields of statistics and which are difficult to manage by the prevailing methods. Thus, spatial data need an original methodology for its own. The likelihood-based procedures for spatial data will be of course desirable, but until now they are not sufficiently developed because of the difficulties in constructing and in practically computing likelihood functions in many cases.Several years ago, we have realized likelihood procedure for estimating interaction potentials of spatial point patterns, by overcoming the difficulty through an application of approximation methods. In this research project, we aimed at developing our procedure to be applicable for wider classes of spatial patterns and models. As a result, it became possible, in case the condition of environment is inhomogeneous, to estimate both the interaction potential between objects and the spatial variation of environment. Moreover, when repulsive interactions are expected, we extended a set of individual one-parameter soft-core models to a single two-parameter soft-core model and made possible to use wider class of models. Several other researches concerning spatial point patterns were also done during the period of this project. There are many problems untached including the application to respective fileds of science, and we hope to consider them to be the future themes of our research.