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

在生态学、地理学、医学、材料技术、物理、化学等应用科学的许多领域中，对于某些物体在平面或空间上的分布规律，都需要对其进行统计分析。在空间格局的统计分析和统计建模中，出现了不同于其他统计领域的新特征，这些特征是现有方法难以管理的。因此，空间数据本身需要一种独创的方法。空间数据的基于似然的程序当然是可取的，但到目前为止，由于在许多情况下构造和实际计算似然函数的困难，它们还没有得到充分发展。几年前，我们通过应用近似方法克服了估计空间点模式相互作用势的困难，实现了估计空间点模式相互作用势的似然程序。在这个研究项目中，我们的目标是发展我们的程序，使其适用于更广泛的空间模式和模型。这样，在环境条件不均匀的情况下，既可以估计物体之间的相互作用势，又可以估计环境的空间变化。此外，当预期排斥相互作用时，我们将一组单独的单参数软核模型扩展到单个双参数软核模型，并使使用更广泛的模型成为可能。在此项目期间，还进行了其他一些关于空间点模式的研究。包括在各自科学领域的应用，还有很多问题没有解决，我们希望将它们作为我们未来研究的主题。

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.