Integrative Study of Voronoi-Delaunay Spatial Tessellation and Spatial Statistics

当多个粒子分布在空间中的有限区域时，空间统计学的重要作用是提取给定空间数据的统计特性信息，并提供对与观测数据相对应的科学领域有用的结果。另一方面，有一种分析空间数据的方法，称为Voronoi-Delaunay分析，它产生给定形状的粒子的空间信息，作为通过对观测区域进行细分而获得的Voronoi单元或Delaunay单元形状的统计分布。但是，到目前为止，Voronoi-Delaunay分析作为空间统计的辅助工具是有限的。然而，在本研究项目中，我们的目标是将Voronoi-Delaunay分析作为提取空间数据详细信息的工具，并将分析结果作为手段…，将Voronoi-Delaunay分析积极地整合到空间统计中在Poisson Voronoi细胞的统计分布研究中，我们得到了Poisson Voronoi细胞在4维和5维空间中几个特征的准确统计分布，并将其作为多元空间统计的原型。然后，我们对大规模时空数据进行了Voronoi-Delaunay镶嵌计算。这与日本地震的余震记录有关，数据由每个事件的纬度、经度和时间(3维)组成。数据量约为56万，为了对如此大的数据量进行三维Voronoi-Delaunay剖分的计算，我们使用ISM的超级计算机系统Altix进行主计算，并使用本项目引入的Unix工作站来执行和检查计算机程序。为了对这样的大数据生成Delaunay单元表，我们开发了一个高效的程序。此外，我们还编制了全球地震余震的Voronoi-Delaunay分析。对于这一计算，我们需要一个球面上的Voronoi-Delaunay剖分的计算机程序。我们已经设计了这样的程序。最后，我们对小鼠皮肤组织中朗格汉斯细胞的空间点图案和野外环境中松树的空间点图案进行了Voronoi-Delaunay细分。较少

When a number of particles are distributed in a finite region in the space, it is the important role of spatial statistics to extract the information of statistical properties of the given spatial data and to provide an appropriate result which is useful to the scientific field corresponding to the observed data. On the other hand, there is a method of analyzing spatial data, called Voronoi-Delaunay analysis, which produces the spatial information of a given configuration of particles as a statistical distribution of the shape of Voronoi cells or Delaunay cells which are obtained by the tessellation of observed area. But, hitherto, Voronoi-Delaunay analysis was limited as a supplementary tool of the spatial statistics. In this research project, however, we aimed to positively integrate the Voronoi-Delaunay analysis to the spatial statistics by utilizing the former as a tool for extracting detailed information about the spatial data and by utilizing the results of the analysis as means … More of constructing various statistical models.In the study of statistical distribution of Poisson Voronoi cells, we obtained accurate statistical distributions of several characteristics of Poisson Voronoi cells in 4-D and 5-D spaces, and presented them as prototypes of multivariate spatial statistics. Then, we performed computations of Voronoi-Delaunay tessellation of a large scale spatial-temporal data. This is concerned with records of aftershocks of earthquakes in Japan, and the data is composed of latitude, longitude and time (3-dimensional) for each event. Data size is about 560, 000, and in order to perform the computation of 3-D Voronoi-Delaunay tessellation for such a big size of data, we used the Super-Computer system Altix of ISM for main computation and the UNIX workstation which was introduced by this project for implementing and checking computer programs.In order to make a list of Delaunay cells for such a big data, we have developed an efficient program. Further, we have prepared the Voronoi-Delaunay analysis of aftershocks of earthquakes for whole of the earth. For this computation, we need a computer program of Voronoi-Delaunay tessellation on the sphere. We have already devised such a program.Finally, we have performed Voronoi-Delaunay tessellation for spatial point patterns of Langerhans cells in a skin tissue of mouse and for spatial point patterns of pine trees in a wild situation. Less