Integrative Study of Voronoi-Delaunay Spatial Tessellation and Spatial Statistics

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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