2010 Fiscal Year Final Research Report
A study of analysis of high dimensional array data through computational algebraic statistical methods and it's application to statistical image analysis
| Project/Area Number |
20340021
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kyushu University |
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Principal Investigator |
SAKATA Toshio Kyushu University, 芸術工学研究院, 教授 (20117352)
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| Co-Investigator(Kenkyū-buntansha) |
SUMI Toshio 九州大学, 芸術工学研究院, 准教授 (50258513)
MIYAZAKI Mitsuhiro 京都教育大学, 教育学部, 准教授 (90219767)
NISHII Ryuei 九州大学, 数理学府, 教授 (40127684)
KURIKI Satoshi 統計数理研究所, 大学共同利用機関などの部局, 教授 (90195545)
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| Project Period (FY) |
2008 – 2010
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| Keywords | 計算代数統計学 / テンソル型データ解析 / テンソルの最大階数 / 絶対正則テンソル / 絶対正則テンソルの同値性 / チューブ法 / ARXモデル |
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| Research Abstract |
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.
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