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

高维泊松点过程的 Voronoi 单元统计分布研究

基本信息

批准号：
13680379
负责人：
TANEMURA Masaharu
金额：
$ 2.37万
依托单位：
The Institute of Statistical Mathematics
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13680379/
关键词：
Poisson point processes Voronoi cell Computer simulation Generalized Gamma distribution Maximum likelihood estimation Higher dimensional space Spatial statistics Stochastic geometry f-vector

项目摘要

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.

当空间中给定多个粒子的位形时，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胞统计分布的方法是该领域的有效方法之一。已知对于给定的配置，自然零假设是泊松点过程。相应地，阐明Poisson点过程的Voronoi胞的统计性质是必不可少的（即，Poisson Voronoi细胞）。在本研究项目中，我们的目标是通过计算机模拟，即使在更高的维空间，尽可能准确地获得泊松Voronoi细胞的统计特性。在项目期间，我们成功地获得了5维Poisson Voronoi单元，其中d = 2的样本量为1000万，d = 3，4，5的样本量分别为500万。然后，我们的体积，表面积和其他特性的Poisson Voronoi细胞的直方图，拟合一个3参数的广义Gamma分布，并获得这些特性的统计分布的实验表达式。我们还将一些理论力矩值与实验结果进行了比较，发现了很好的一致性。我们发现了一些新的事实，出现泊松Voronoi细胞d = 4和5。我们的部分研究结果已作为论文发表，并在国际和国家会议上发表。