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

研究了三向列联表的序贯精确条件检验中的提升问题。所研究的问题是如何从时间t-1构造时间t的推理框架（所有与给定基准具有相同边缘的列联表的集合）。我们通过r邻域定理明确了时刻t的坐标系可以由时刻t-1的坐标系通过马尔可夫基构造出来。另一方面，对于实值三维基准，即3张量，我们研究了秩和最大秩。特别地，我们在没有条件的复数域上证明了Atkinson的论断，并在有条件的实数域上证明了它。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，明确了绝对非奇异与行列式多项式正性之间的联系。给出了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正交群或酉群上的积分来检测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、基于地理空间数据的森林砍伐建模和图像分类等方面也取得了一些成果。

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.