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

在三向列联表的序贯精确条件检验中，研究了提升问题。所研究的问题是如何从时间t-1构造推理框架(与给定数据具有相同边际的所有列联表的集合)。利用r-邻域定理，证明了利用马尔可夫基，可以由t-1时刻的框架构造出t时刻的框架。另一方面，对于实值三维数据，即3-张量，我们研究了它的秩和最大秩.特别地，我们证明了Atkinson关于无条件复数域的命题，并证明了它在实数域上具有一定的条件。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，并明确了绝对非奇性与行列式多项式的正性之间的联系。我们得到了如何寻找和构造绝对非奇异张量的方法。此外，我们还提出了利用微分几何不变量和在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，在随机矩阵最大特征值分布理论以及基于地理空间数据的森林砍伐建模和图像分类中也取得了一些结果。

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.