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.

在顺序的确切条件测试中，研究了三个偶性表提升问题。研究的问题是如何在时间t的时间t构建推论帧（所有与给定基准相同边缘表的集合的集合）。我们通过r-nighborhood定理清楚地表明，在T-1时T-1使用Markov基础，t框架在T框架上是可构造的。另一方面，对于实际有价值的三维基准，即3汤匙，我们研究了等级和最大等级。特别是，我们证明了阿特金森对没有条件的复杂数字字段的主张，并在具有某种条件的实际数字字段上证明了这一点。我们称不满足条件的张量为绝对非量的张量。为了研究绝对的非发音张量，我们设计了张量的决定性多项式，并明确了绝对非象征性与决定元多项式的阳性之间的联系。我们获得了如何找到以及如何构造绝对非张量的方法。此外，我们提出了通过使用差异几何不变剂以及正交或单一组的差异几何不变的方法来检测它们之间无等的方法。此外，基于地理空间数据，在随机矩阵的最大特征值以及森林砍伐建模和图像分类的分布理论中获得了一些结果。