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MATHEMATICAL STATISTICS FOR DATA ANALYSIS IN HIGH DIMENSION, LOW SAMPLE SIZE CONTEXT AND ITS APPLICATIONS

Research Project

Project/Area Number 18300092
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Statistical science
Research InstitutionUniversity of Tsukuba

Principal Investigator

AOSHIMA Makoto  University of Tsukuba, 大学院・数理物質科学研究科, 教授 (90246679)

Co-Investigator(Kenkyū-buntansha) AKAHIRA Masafumi  筑波大学, 副学長 (70017424)
KOIKE Ken-ichi  筑波大学, 大学院・数理物質科学研究科, 准教授 (90260471)
OHYAUCHI Nao  筑波大学, 大学院・数理物質科学研究科, 助教 (40375374)
TASAKI Hiroyuki  筑波大学, 大学院・数理物質科学研究科, 准教授 (30179684)
KAWAMURA Kazuhiro  筑波大学, 大学院・数理物質科学研究科, 准教授 (40204771)
TAKAHASHI Hideto  筑波大学, 大学院・人間総合科学研究科, 准教授 (80261808)
MINAMI Nariyuki  慶應義塾大学, 医学部, 教授 (10183964)
Project Period (FY) 2006 – 2009
Project Status Completed (Fiscal Year 2009)
Budget Amount *help
¥18,120,000 (Direct Cost: ¥14,700,000、Indirect Cost: ¥3,420,000)
Fiscal Year 2009: ¥5,590,000 (Direct Cost: ¥4,300,000、Indirect Cost: ¥1,290,000)
Fiscal Year 2008: ¥5,590,000 (Direct Cost: ¥4,300,000、Indirect Cost: ¥1,290,000)
Fiscal Year 2007: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2006: ¥3,300,000 (Direct Cost: ¥3,300,000)
Keywords多変量解析 / 機械学習 / パターン認識 / モデル選択 / ノイズ / 生体生命情報学 / マイクロアレイ / 高次元データ / 高次元小標本 / 判別分析 / クラスター分析 / モデル選択基準 / ダイバージェンス / 異常値 / 混合正規分布 / 統計科学 / 主成分分析 / 固有値問題 / 次元推定 / 次元縮約 / 次元の呪い / 高次元漸近理論 / マイクロアレイデータ / 多変量データ解析 / 標本数 / 標本数決定 / 領域推定 / 共分散構造 / 漸近理論 / ベイズ法 / ランダム行列 / 逐次区間推定 / 情報不等式
Research Abstract

We developed the high-dimension asymptotic theory for High Dimension, Low Sample Size (HDLSS) datasets under a general setup such as non-Gaussian distributions. We found several geometric structures of HDLSS datasets. We showed that the naive PCA is inconsistent in the HDLSS context. We proposed effective inference methods called (1) the noise-reduction methodology, and (2) the cross-data-matrix methodology. By using those methodologies, we gave consistent estimation for intrinsic dimensionality, eigenvalues, their limiting distributions, PC directions and PC scores in the HDLSS context. We applied those methodologies to the discriminant analysis and the cluster analysis in HDLSS data situations from a microarray study of prostate cancer.

Report

(6 results)
  • 2009 Annual Research Report   Final Research Report ( PDF )
  • 2008 Annual Research Report   Self-evaluation Report ( PDF )
  • 2007 Annual Research Report
  • 2006 Annual Research Report
  • Research Products

    (47 results)

All 2010 2009 2008 2007 2006 Other

All Journal Article (27 results) (of which Peer Reviewed: 18 results) Presentation (16 results) Book (1 results) Remarks (3 results)

  • [Journal Article] Asymptotic second-order consistency for two-stage estimation methodologies and its applications2010

    • Author(s)
      Aoshima, M., Yata, K.
    • Journal Title

      Ann.Inst.Statist.Math. 62(in press)

    • NAID

      120007137458

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Intrinsic dimensionality estimation of high dimension, low sample size data with d-asymptotics2010

    • Author(s)
      Yata, K., Aoshima, M.
    • Journal Title

      Commun.Statist.-Theory and Meth. 39(in press)

    • NAID

      120007137805

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Sequential estimation procedures for end points of support in a non-regular distribution2010

    • Author(s)
      Koike, K.
    • Journal Title

      Commun.Statist.-Theory and Meth. 39(in press)

    • NAID

      120007137763

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Asymptotic second-order consistency for two-stage estimation methodologies and its applications2010

    • Author(s)
      Makoto Aoshima
    • Journal Title

      Ann.Inst.Statist.Math. 62(印刷中)

    • NAID

      120007137458

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Intrinsic dimensionality estimation of high dimension, low sample size data with d-asymptotics2010

    • Author(s)
      矢田和善
    • Journal Title

      Commun.Statist.-Theory Meth. 39(印刷中)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The first and second order large-deviation efficiency for an exponential family and certain curved exponential models2010

    • Author(s)
      Masafumi Akahira
    • Journal Title

      Commun.Statist.-Theory Meth. 39(印刷中)

    • NAID

      120007136924

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Sequential estimation procedures for end points of support in a non-regular distribution2010

    • Author(s)
      小池健一
    • Journal Title

      Commun.Statist.-Theory Meth. 39(印刷中)

    • NAID

      120007137763

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] PCA consistency for non-Gaussian data in high dimension, low sample size context2009

    • Author(s)
      Yata, K., Aoshima, M.
    • Journal Title

      Commun.Statist.-Theory and Meth. 38

      Pages: 2634-2652

    • NAID

      120007131251

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Double shrink methodologies to determine the sample size via covariance structures2009

    • Author(s)
      矢田和善
    • Journal Title

      J.Statist.Plan.Infer. 139

      Pages: 81-99

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] PCA consistency for non-Gaussian data in high dimension, low sample size context2009

    • Author(s)
      矢田和善
    • Journal Title

      Commun.Statist.-Theory Meth. 38

      Pages: 2634-2652

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] 高次元小標本における固有値の推定とその応用2009

    • Author(s)
      矢田和善
    • Journal Title

      数理解析研究所講究録 1621

      Pages: 112-129

    • Related Report
      2009 Annual Research Report 2008 Annual Research Report
  • [Journal Article] 非正則分布における分布の台の端点の逐次推測について2009

    • Author(s)
      小池健一
    • Journal Title

      数理解析研究所講究録 1621

      Pages: 104-111

    • Related Report
      2009 Annual Research Report
  • [Journal Article] Double shrink methodologies to determine the sample size via covariance structures2009

    • Author(s)
      Kazuyoshi Yata
    • Journal Title

      J. Statist. Plan. Infer. 139

      Pages: 81-99

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Asymptotic second-order consistency for two-stage estimatioh methodologies and its applications2009

    • Author(s)
      Makoto Aoshima
    • Journal Title

      Ann. Inst. Statist. Math. (印刷中)

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Sequential estimation procedures for end points of support in a non-regular distribution2009

    • Author(s)
      ken-ichi Koike
    • Journal Title

      Commun. Statist.- Theory Meth. (印刷中)

    • NAID

      120007137763

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] 非正則推定における情報量の概念とその役割2008

    • Author(s)
      赤平昌文
    • Journal Title

      日本統計学会誌 37

      Pages: 329-342

    • NAID

      110006650705

    • Related Report
      2009 Final Research Report 2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Bayes estimation under the LINEX loss2008

    • Author(s)
      大谷内奈穂
    • Journal Title

      数理解析研究所講究録 1603

      Pages: 25-37

    • Related Report
      2008 Annual Research Report
  • [Journal Article] Double shrink methodologies to determine the sample size via covariance structures2008

    • Author(s)
      Yata, K.
    • Journal Title

      Journal of Statistical Planning and Inference (印刷中)

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A Bayesian view of the Hammersley-Chapman-Robbins-type inequality2007

    • Author(s)
      Akahira, M., Ohyauchi, N.
    • Journal Title

      Statistics 41

      Pages: 137-144

    • NAID

      120007136926

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] A Bayesian view of the Hammersley-Chapman-Robbins-type inequality2007

    • Author(s)
      Akahira, M.
    • Journal Title

      Statistics 41

      Pages: 137-144

    • NAID

      120007136926

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Sequential point estimation of the location parameter in the location-scale family of non-regular distributions2007

    • Author(s)
      Koike, K.
    • Journal Title

      Sequential Analysis 26

      Pages: 383-393

    • NAID

      120007130860

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The construction of combined Bayesian-frequentist confidence intervals for a positive parameter2007

    • Author(s)
      Akahira, M.
    • Journal Title

      Statistica (印刷中)

    • NAID

      120007137095

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Sequential interval estimation of a location parameter with the fixed width in the non-regular case2007

    • Author(s)
      Koike, K.
    • Journal Title

      Sequential Analysis 26・1

      Pages: 63-70

    • NAID

      120007130864

    • Related Report
      2006 Annual Research Report
  • [Journal Article] A Bayesian view of the Hammersley-Chapman-Robbins type inequality2007

    • Author(s)
      Akahira, M.
    • Journal Title

      Statistics (印刷中)

    • NAID

      120007136926

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Theory of point processes and some basic notions in energy level statistics2007

    • Author(s)
      Minami, N.
    • Journal Title

      CRM Proceedings and Lecture Notes (印刷中)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Second-order efficiency for two-stage estimation of a linear function of normal mean vectors when covariance matrices have some structures2006

    • Author(s)
      Aoshima, M.
    • Journal Title

      Sequential Analysis 25・3

      Pages: 327-345

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Geometry of reflective submanifolds in Reimannian symmetric spaces2006

    • Author(s)
      Tasaki, H.
    • Journal Title

      Journal of the Mathematical Society of Japan 58・1

      Pages: 275-297

    • Related Report
      2006 Annual Research Report
  • [Presentation] Effective clustering in non-regular cases2010

    • Author(s)
      小林裕子
    • Organizer
      京都大学数理解析研究所研究会
    • Place of Presentation
      京都大学
    • Year and Date
      2010-03-09
    • Related Report
      2009 Annual Research Report
  • [Presentation] 高次元小標本における判別分析とクラスター分析2009

    • Author(s)
      矢田和善
    • Organizer
      日本学術振興会科学研究費による研究集会
    • Place of Presentation
      筑波大学
    • Year and Date
      2009-12-15
    • Related Report
      2009 Annual Research Report
  • [Presentation] Effective PCA for high-dimension, low-sample-size data with singular value decomposition of cross data matrix2009

    • Author(s)
      矢田和善
    • Organizer
      統計関連学会連合大会
    • Place of Presentation
      同志社大学
    • Year and Date
      2009-09-07
    • Related Report
      2009 Annual Research Report
  • [Presentation] Eigenvalue estimation in HDLSS context and its applications2008

    • Author(s)
      Aoshima, M.
    • Organizer
      日本学術振興会日露共同研究プロジェクト研究集会
    • Place of Presentation
      広島大学
    • Year and Date
      2008-11-04
    • Related Report
      2009 Final Research Report
  • [Presentation] Eigenvalue estimation in HDLSS context and its applications2008

    • Author(s)
      Makoto Aoshima
    • Organizer
      日本学術振興会日露共同研究プロジェクト
    • Place of Presentation
      広島大学
    • Year and Date
      2008-11-04
    • Related Report
      2008 Annual Research Report
  • [Presentation] 高次元小標本におけるベイズ的アプローチに基づく固有空間の推定2008

    • Author(s)
      矢田和善
    • Organizer
      京都大学数理解析研究所研究会
    • Place of Presentation
      京都大学
    • Year and Date
      2008-10-23
    • Related Report
      2008 Annual Research Report
  • [Presentation] 高次元小標本における固有値の推定とその応用2008

    • Author(s)
      矢田和善
    • Organizer
      日本学術振興会科学研究費による研究集会
    • Place of Presentation
      大阪大学
    • Year and Date
      2008-09-30
    • Related Report
      2008 Annual Research Report
  • [Presentation] Intrinsic dimensionality estimation of high dimension low sample size data with d-asymptotics2008

    • Author(s)
      矢田和善
    • Organizer
      統計関連学会連合大会
    • Place of Presentation
      慶應義塾大学
    • Year and Date
      2008-09-09
    • Related Report
      2008 Annual Research Report
  • [Presentation] Intrinsic dimensionality estimation of high dimension, low sample size data with geometric representation2008

    • Author(s)
      Aoshima, M.
    • Organizer
      International IISA Conference, University of Connecticut
    • Place of Presentation
      Connecticut, U.S.A.
    • Year and Date
      2008-05-24
    • Related Report
      2009 Final Research Report
  • [Presentation] Intrinsic dimensionality estimation of high dimension,low sample size data with geometric representation2008

    • Author(s)
      Makoto Aoshima
    • Organizer
      International IISA Conference
    • Place of Presentation
      Univ. of Connecticut U.S.A.
    • Year and Date
      2008-05-24
    • Related Report
      2008 Annual Research Report
  • [Presentation] Information inequality bounds in non-regular estimation2007

    • Author(s)
      大谷内奈穂
    • Organizer
      日本数学会秋季総合分科会, 統計数学分科会
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-23
    • Related Report
      2009 Final Research Report
  • [Presentation] Sequential estimation of a location parameter for the location-scale family of distributions in non-regular case2007

    • Author(s)
      Koike, K.
    • Organizer
      The 56th Session of the International Statistical Institute, Lisboa Congress Centre
    • Place of Presentation
      Lisboa, Portugal
    • Year and Date
      2007-08-27
    • Related Report
      2009 Final Research Report
  • [Presentation] Asymptotic second-order consistency for two-stage estimation methodologies and its applications2007

    • Author(s)
      Aoshima, M.
    • Organizer
      The 56th Session of the International Statistical Institute, Lisboa Congress Centre
    • Place of Presentation
      Lisboa, Portugal
    • Year and Date
      2007-08-25
    • Related Report
      2009 Final Research Report
  • [Presentation] Asymptotic second-order consistency for two-stage estimation methodologies and its applications2007

    • Author(s)
      Aoshima, M.
    • Organizer
      The 56th Session of the International Statistical Institute
    • Place of Presentation
      Lisboa Congress Centre Lisboa, Portugal
    • Year and Date
      2007-08-25
    • Related Report
      2007 Annual Research Report
  • [Presentation] Asymptotic second-order consistency for two-stage methodologies via covariance structures2007

    • Author(s)
      Aoshima, M.
    • Organizer
      First International Workshop in Sequential Methodologies
    • Place of Presentation
      Auburn, U.S.A.
    • Year and Date
      2007-07-25
    • Related Report
      2009 Final Research Report
  • [Presentation] The second order large- deviation efficiency for an exponential family of distributions2006

    • Author(s)
      Akahira, M.
    • Organizer
      日本学術振興会日露共同プロジェクト研究集会
    • Place of Presentation
      広島大学
    • Year and Date
      2006-08-07
    • Related Report
      2009 Final Research Report
  • [Book] 知識ベース : 電子情報通信基礎2010

    • Author(s)
      青嶋誠, 他
    • Publisher
      電子情報通信学会(印刷中)
    • Related Report
      2009 Final Research Report
  • [Remarks] ホームページ等 : つくばリポジトリ

    • URL

      https://www.tulips.tsukuba.ac.jp/portal/tulips-r.php

    • Related Report
      2009 Final Research Report
  • [Remarks] つくばリポジトリ

    • URL

      https://www.tulips.tsukuba.ac.jp/portal/tulips-r.php

    • Related Report
      2009 Annual Research Report
  • [Remarks] つくばリポジトリ:

    • URL

      https://www.tulips.tsukuba.ac.jp/portal/tulips-r.php

    • Related Report
      2008 Annual Research Report

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Published: 2006-04-01   Modified: 2016-04-21  

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