Studies of Statistical Distributions of Voronoi Cells for Higher Dimensional Poisson Point Processes

当空间中给定了多个粒子的位形时，阐明粒子位形的统计性质是空间统计学的重要课题。研究给定粒子构型的Voronoi胞体的统计分布是这一领域的有效方法之一。众所周知，给定配置的自然零假设是Poisson点过程。相应地，在我们的方法中，澄清Poisson点过程的Voronoi胞元(即Poisson Voronoi胞元)的统计性质是必不可少的。在本研究项目中，我们旨在通过计算机模拟尽可能准确地获得Poisson Voronoi胞元的统计性质，即使在更高维空间也是如此。在项目期间，我们成功地获得了五维Poisson Voronoi盒，其中d=2的样本量为1000万，d=3、4、5时的样本量为500万。然后我们制作了泊松Voronoi细胞的体积、表面积和其他特征的直方图，并对它们进行了三参数广义Gamma分布的拟合，得到了这些特征的统计分布的实验表达式。我们还检查了一些理论矩的值与我们的实验结果，发现很好的重合。对于d=4和5的Poisson Voronoi胞格，我们发现了一些新的事实。我们的部分结果以论文的形式发表，并在国际和国内会议上发表。

When a configuration of many particles is given in the space, it is the important subject of "Spatial Statistics" to clarify the statistical properties of the particle configuration. The approach in which the statistical distributions of Voronoi cells for a given particle configuration are studied is one of the effective methods in this area. It is known that a natural null hypothesis for a given configuration is a Poisson point processes. Correspondingly, it is indispensable to clarify the statistical properties of Voronoi cells for Poisson point processes (i.e., Poisson Voronoi cells) in our approach. In this research project, we aimed at obtaining the statistical properties of Poisson Voronoi cells accurately as far as possible through computer simulations even in higher dimensional space. During the period of the project, we succeeded to obtain Poisson Voronoi cells up to five-dimensions where the sample size is ten million for d = 2, and five million for d = 3,4,5,respectively. We then made histograms of volume, surface area and other characteristics of Poisson Voronoi cells, fitted a 3-parameter generalized Gamma distribution to them, and obtained experimental expressions for the statistical distributions of these characteristics. We also checked some theoretical moment values against those of our experimental results and found good coincidences. We found some new facts which arise for Poisson Voronoi cells for d = 4 and 5. Parts of our results were published as papers, and presented at international and national meetings.