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Study of multi-variable hypergeometric differential equations for statistics

Research Project

Project/Area Number 17K05279
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionKobe 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 *help
¥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.

Report

(4 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (17 results)

All 2020 2019 2018 2017 Other

All Int'l Joint Research (1 results) Journal Article (2 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 2 results,  Open Access: 1 results) Presentation (11 results) (of which Int'l Joint Research: 1 results,  Invited: 3 results) Remarks (3 results)

  • [Int'l Joint Research] Johan Radon Institute(オーストリア)

    • Related Report
      2018 Research-status Report
  • [Journal Article] Holonomic Gradient Method for Two Way Contingency Tables2020

    • Author(s)
      Y.Tachibana;Y.Goto;T.Koyama;N.Takayama
    • Journal Title

      Algebraic Statistics

      Volume: -

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Distribution of the ratio of two Wishart matrices and cumulative probability evaluation by the holonomic gradient method2018

    • 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

    • Related Report
      2017 Research-status Report
    • 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
    • Related Report
      2019 Annual Research Report
  • [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
    • Related Report
      2019 Annual Research Report
  • [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
    • Related Report
      2019 Annual Research Report
  • [Presentation] Hypergeometric functions and statistics2019

    • Author(s)
      N.Takayama
    • Organizer
      Differential Systems: from theory to computer mathematics
    • Related Report
      2019 Annual Research Report
    • 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
    • Related Report
      2019 Annual Research Report
  • [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
    • Related Report
      2018 Research-status Report
  • [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
    • Related Report
      2018 Research-status Report
  • [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
    • Related Report
      2018 Research-status Report
  • [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
    • Related Report
      2017 Research-status Report
  • [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
    • Related Report
      2017 Research-status Report
    • 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
    • Related Report
      2017 Research-status Report
    • Invited
  • [Remarks] References of HGM and HGD

    • URL

      http://www.math.kobe-u.ac.jp/OpenXM/Math/hgm/ref-hgm.html

    • Related Report
      2019 Annual Research Report
  • [Remarks] Open Message Exchange for Mathematics

    • URL

      http://www.openxm.org

    • Related Report
      2018 Research-status Report
  • [Remarks] References for HGM

    • URL

      http://www.math.kobe-u.ac.jp/OpenXM/Math/hgm/ref-hgm.html

    • Related Report
      2018 Research-status Report

URL: 

Published: 2017-04-28   Modified: 2021-02-19  

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