Study of multi-variable hypergeometric differential equations for statistics

• Project/Area Number: 17K05279
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kobe University
• Principal Investigator: Takayama Nobuki 神戸大学, 理学研究科, 教授 (30188099)
• Project Period (FY): 2017-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2017: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: 多変数超幾何関数 / ホロノミック勾配法 / 二元分割表 / Wishart行列 / A超幾何関数 / 分割表 / ねじれコホモロジー / オイラー標数法 / ランダム行列 / ２元分割表 / A-超幾何関数 / 数値計算 / non-central Wishart / 解析学

Outline of Final Research Achievements

We gave a new algorithm to translate an A-hypergeometric system to a Pfaffian equation. A Pfaffian equation for the matrix 2F1 is derived. We show that the expectation of a random manifold defined by Wishart matrix and its maximal eigenvalue is expressed by an integral. We studied holonomic systems for the integral under several conditions on the Wishart matrix and performed a numerical analysis of them. An error free method to solve systems of difference equations is given. It utilizes the Chinese remainder theorem. It is demonstrated that the method is useful to evaluate the normalizing constant and its derivatives for two way contingency tables.

Academic Significance and Societal Importance of the Research Achievements

Holonomic 系の Pfaffian 方程式を導出するという古典的な問題に関して, 新しい計算アルゴリズムを与えるとともに, 行列超幾何関数や対称性が高いEuler標数の期待値関数については理論的な結果を得た. これらは統計分布の正規化定数の数値評価問題に適用可能である.

Research Products

• [Int'l Joint Research] Johan Radon Institute(オーストリア)
• [Journal Article] Holonomic Gradient Method for Two Way Contingency Tables (2020)
  • Author(s): Y.Tachibana;Y.Goto;T.Koyama;N.Takayama
  • Journal Title: Algebraic Statistics Volume: -
  • Peer Reviewed
• [Journal Article] Distribution of the ratio of two Wishart matrices and cumulative probability evaluation by the holonomic gradient method (2018)
  • Author(s): Hashiguchi Hiroki、Takayama Nobuki、Takemura Akimichi
  • Journal Title: Journal of Multivariate Analysis Volume: 165 Pages: 270-278
  • DOI: 10.1016/j.jmva.2018.01.002
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Presentation] GKZ 超幾何系(A-超幾何系)の Pfaffian system の計算法 (2020)
  • Author(s): 高山信毅
  • Organizer: 超幾何方程式研究会2020
• [Presentation] Jupyter への asir カーネルの実装 (2019)
  • Author(s): 高山信毅
  • Organizer: 日本数式処理学会第28回大会
• [Presentation] HGM の不安定性をどう回避するか? (2019)
  • Author(s): 高山信毅
  • Organizer: 確率・統計・行列ワークショップ
• [Presentation] Hypergeometric functions and statistics (2019)
  • Author(s): N.Takayama
  • Organizer: Differential Systems: from theory to computer mathematics
  • Int'l Joint Research / Invited
• [Presentation] Python を用いた asir 実行環境の実装 (2019)
  • Author(s): 池田;高山
  • Organizer: Computer Algebra --- Theory and its Applications
• [Presentation] Twisted cohomology 群の交点数を求めるアルゴリズム (2019)
  • Author(s): 高山信毅
  • Organizer: 日本数学会2019年度年会
• [Presentation] 計算代数の direct sampler への応用 (2018)
  • Author(s): 高山信毅
  • Organizer: Computer algebra --- theory and its applications
• [Presentation] 超幾何方程式 E(k,n)の contiguity relation を用いた乱数生成 (2018)
  • Author(s): 高山信毅
  • Organizer: 第１２回玉原特殊多様体研究集会
• [Presentation] 分割表の条件付き最尤推定のための Risa/Asir パッケージ (2018)
  • Author(s): 高山信毅
  • Organizer: Risa/Asir Conference, 2018
• [Presentation] Random matrix と多変数特殊関数の数値計算 (2017)
  • Author(s): 高山信毅
  • Organizer: 複素微分方程式の楽しみ
  • Invited
• [Presentation] Non-central complex Wishart 行列の最大固有値の CDF の数値計算 (2017)
  • Author(s): 高山信毅
  • Organizer: 確率, 統計, 行列ワークショップ 松本 2017
  • Invited
• [Remarks] References of HGM and HGD
  • URL: http://www.math.kobe-u.ac.jp/OpenXM/Math/hgm/ref-hgm.html
• [Remarks] Open Message Exchange for Mathematics
  • URL: http://www.openxm.org
• [Remarks] References for HGM
  • URL: http://www.math.kobe-u.ac.jp/OpenXM/Math/hgm/ref-hgm.html