Statistical Inference on Spatial Random Network

在“空间统计”领域，聚类点模式（即具有非均匀性的点模式）的参数化方法研究不足。为了打破这一现状，我们开始研究假定由随机网络结构产生的空间点模式作为非齐次点模式的一类。本文的目的是构建与上述假设相对应的空间统计模型，建立一种参数估计方法，并验证模型和方法的有效性。我们的最终目标是将我们的模型和方法应用于观察到的实际数据。作为一种可能的参数化方法，我们提出了以下基于贝叶斯过程的方法。让我们假设观测到的数据由点x的结构表示，并且数据发生在三维空间中的多晶网络的边界上。我们的假设肯定是正确的……设y为网络的未观测生成点。此外，我们假设，在网络的边缘，未观测点z与每个观测点x有一对一的对应关系，使得x=z+ε。这里我们假设ε服从正态分布。最后，我们假设通过y生成的网络由Voronoi镶嵌网络组成。在未观测变量的合适先验分布条件下，采用马尔可夫链蒙特卡罗（MCMC）方法获得后验分布。对计算机模拟得到的二维人工数据，我们的方法是有效的。在这个阶段，我们的结果在1999年在加拿大举行的国际会议上发表，等等。同时，我们准备出版FORTRAN程序，该程序是为二维动态构建Voronoi镶嵌而设计的。有些结果已经发表，有些结果正在准备发表。少

In the field of "Spatial Statistics", the research of the parametric methods for clustered point patterns (namely, the point patterns with inhomogeneity) had been insufficient. In order to break the above status, we started to study the spatial point patterns which are assumed to be generated from the random network structures as one of the classes of inhomogeneous point patterns. Our purposes were to construct the models of spatial statistics corresponding to the above assumption, to develop a method of parametric estimating and to confirm the validity of both the model and the method. Our final object was to apply our model and method to the observed real data.As a possible parametric method, we proposed the following approach based on Bayesian procedure. Let us assume that the observed data is represendted by the configuration of points x and that the data was ocurred from the boundaries of, say, polycrystalline network in three-dimensional space. There is surely the case our assump … More tions are realistic. Let y be the unobserved generating points of the netowrk. Furthermore, we assume, on the edge of the network, the unobserved point z which has one to one correspondence to each observation x such that x=z+ε. Here we assume that ε obeys the normal distribution. Finally we suppose that the network which is generated through y is composed of Voronoi tessellation network. Under the suitable prior distributions of unobserved variables, we used Markov Chain Monte Carlo (MCMC) method for obtaining the posterior distributions.We have confirmed that our procedure is valid for the two-dimensional artificial data which was produced by computer simulation. At this stage, our results are presented at the International Conference which was held in Canada in 1999 and so on. At the same time, we prepared the publication of the FORTRAN programs wich were devised for the dynamic construction of Voronoi tessellation in two dimensions.Certain results are already published and some results are under the preparation for publication. Less