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A study of analysis of high dimensional array data through computational algebraic statistical methods and it's application to statistical image analysis

计算代数统计方法分析高维阵列数据及其在统计图像分析中的应用研究

基本信息

批准号：
20340021
负责人：
SAKATA Toshio
金额：
$ 5.91万
依托单位：
Kyushu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2008
资助国家：
日本
起止时间：
2008 至 2010
项目状态：
已结题

项目摘要

In the sequential exact conditional test for three-way contingency tables lifting problem is studied. The problem studied is how to construct the inferential frame (the set of all contingency tables with the same marginals as a given datum) at the time t from that of the time t-1. We made clear by r-neighborhood theorem that the frame at the time t is constructible from the frame at the time t-1 by using Markov basis. On the other hand for the real valued three dimensional datum, that is, 3-tensor, we studied the rank and the maximal rank. Especially, we proved Atkinson's claim for the complex number fields with no condition and proved it over the real number field with some condition. We called tensors, which does not satisfy the condition, as absolutely nonsingular tensors. For studying absolutely nonsingular tensors we devised the determinant polynomial of tensors and made clear the link between the absolutely non-singularity and the positivity of the determinant polynomial. We obtained methods how to find and how to construct absolutely nonsingular tensor. Also, we proposed methods to detect non equivalence between them by using differential geometric invariants and the integrations over the orthogonal or the unitary group. In addition, some results were obtained in the distribution theory of the largest eigenvalue of a random matrix and in the deforestation modeling and the image classification, based on geo-spatial data.

在三向列联表序贯精确条件检验中，研究了提升问题。研究的问题是如何从时间t-1的推理框架构造时间t的推理框架（所有列联表的集合，其边缘与给定的数据相同）。我们通过r-邻域定理证明了时间t的标架可以由时间t-1的标架利用马尔可夫基构造。另一方面，对于真实的三维数据，即三维张量，我们研究了它的秩和最大秩。特别地，我们证明了Atkinson在复数域上的无条件要求和在真实的数域上的有条件要求。我们称不满足条件的张量为绝对非奇异张量。为了研究绝对非奇异张量，我们设计了张量的行列式多项式，并阐明了绝对非奇异性与行列式多项式的正性之间的联系。给出了绝对非奇异张量的构造方法。同时，我们还提出了利用微分几何不变量和正交群或酉群上的积分来判定它们之间不等价的方法。此外，在随机矩阵的最大特征值的分布理论和在森林砍伐建模和图像分类，基于地理空间数据，取得了一些成果。