乱数の理論と擬似乱数検定

• Project/Area Number: 24540153
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Gifu University (2014-2015); The Institute of Statistical Mathematics (2012-2013)
• Principal Investigator: 高橋 勇人 岐阜大学, 教育推進・学生支援機構, 准教授 (50518959)
• Co-Investigator(Kenkyū-buntansha): 釜江 哲朗 大阪市立大学, 理学(系)研究科(研究院), 教授 (80047258) ; 梅野 健 京都大学, 情報学研究科, 教授 (10358872)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Discontinued (Fiscal Year 2015)
• Budget Amount: ¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000); Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 乱数 / 条件付き確率 / マルチンレフ乱数 / 擬似乱数 / 検定 / 乱数の検定 / 乱数検定 / 擬似乱数生成 / 国際情報交換 フランス

Outline of Annual Research Achievements

条件付き確率に関する乱数の定義は確率論及び統計学で最も重要な分野の一つである．高橋は条件付き確率がパラメータに関して相対的に計算可能でない場合の条件付き確率に関する乱数の定義の一般化について研究し，一般化が成立する例を示した．また正規数の部分列が再び正規数となる選出関数を特徴付けたものとしてKamae-Weissの定理があるがそれのアルゴズム乱数に対応する結果を示した．ここでの結果はvan Lambalgen(Ph. D. thesis 1987)で予想されていたものと矛盾するかどうかは不明であるがこの問題に対する自然な結果を初めて与えている．
擬似乱数の生成，検定は応用上重要である．
釜江は記号列のランダムネスを一つの数値としていかに表現するかを研究した．また，記号列の一文字当たりのエントロピーを測定し，様々な性質を導いた．そしてこれが乱数生成や乱数検定に用いられることを示した．
梅野は離散フーリエ変換の代わりに原始根を用いるユニタリー変換に基づく新たな乱数検定法を作り，いくつかの非乱数系列のセットで，離散フーリエ変換検定（DFT検定）では非乱数性が検出されないにも関わらず，この新しい方法では非乱数性の特徴を検出できることが解った．また，乱数検定に用いる乱数生成については，ガウス分布を拡張したq-Gauss分布を持つ乱数生成方法を，Box-Muller法を拡張しエルゴード力学系で生成する一般的な方法を考案した．

Research Products

• [Int'l Joint Research] CNRS(France)
• [Journal Article] Spectral measure of the Thue-Morse sequence and the dynamical system and random walk related to it (2016)
  • Author(s): Teturo Kamae and Li Peng
  • Journal Title: Ergodic Theory and Dynamical Systems Volume: 未定 Issue: 4 Pages: 1247-1259
  • DOI: 10.1017/etds.2014.121
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Abelian maximal pattern complexity of words (2015)
  • Author(s): Teturo Kamae (with STEVEN WIDMER and LUCA Q. ZAMBONI)
  • Journal Title: Ergodic Theory and Dynamical Systems Volume: 35 Issue: 1 Pages: 142-151
  • DOI: 10.1017/etds.2013.51
  • Peer Reviewed / Open Access
• [Journal Article] An easy criterion for randomness (2015)
  • Author(s): Teturo Kamae (with Yu-Mei XUE)
  • Journal Title: Sankhya A Volume: 77-1 Issue: 1 Pages: 126-152
  • DOI: 10.1007/s13171-014-0054-3
  • Peer Reviewed / Open Access
• [Journal Article] A characterization of eventually periodicity (2015)
  • Author(s): Teturo Kamae (with Dong Han Kim)
  • Journal Title: Theoretical Computer Science Volume: 581 Pages: 1-8
  • DOI: 10.1016/j.tcs.2015.02.039
  • Peer Reviewed / Open Access
• [Journal Article] Hausdorff dimension of the level sets of self-affine functions (2015)
  • Author(s): Teturo Kamae (wit Li PENG)
  • Journal Title: Journal of Mathematical Analysis and Applications Volume: 423 Issue: 2 Pages: 1400-1409
  • DOI: 10.1016/j.jmaa.2014.10.058
  • Peer Reviewed / Open Access
• [Journal Article] Local time of self-affine sets of Brownian motion type and the jigsaw puzzle problem (2014)
  • Author(s): Teturo Kamae (with Yu-Mei XUE)
  • Journal Title: Journal of Mathematical Analysis and Applications Volume: 419 Issue: 1 Pages: 79-93
  • DOI: 10.1016/j.jmaa.2014.04.018
  • Peer Reviewed / Open Access
• [Journal Article] Infinite Self-Shuffling Words (2014)
  • Author(s): Teturo Kamae (with Emilie Charlier, Svetlana Puzynina and Luca Q. Zamboni)
  • Journal Title: Journal of Combinatorial Theory, Series A Volume: 128 Pages: 1-40
  • DOI: 10.1016/j.jcta.2014.07.008
  • Peer Reviewed / Open Access
• [Journal Article] Algorithmic analogies to Kamae-Weiss Theorem on Normal Numbers (2013)
  • Author(s): Hayato Takahashi
  • Journal Title: Lecture Notes in Artificial Intelligence Volume: 7070 Pages: 411-416
  • DOI: 10.1007/978-3-642-44958-1_32
  • ISBN: 9783642449574, 9783642449581
  • Peer Reviewed
• [Journal Article] Using Deterministic Chaos for superefficient Monte Carlo Simulations (2013)
  • Author(s): Chen-An Yang, Kung Yao, Ken Umeno and Ezio Biglieri
  • Journal Title: IEEE Circuits and Systems Magazine Volume: 13 Issue: 4 Pages: 26-35
  • DOI: 10.1109/mcas.2013.2283966
  • Peer Reviewed
• [Journal Article] Chaotic Method for Generating q-Gaussian Random Variables (2013)
  • Author(s): K. Umeno and A. H. Sato
  • Journal Title: IEEE Transactions on Information Theory Volume: 59 Issue: 5 Pages: 3199-3209
  • DOI: 10.1109/tit.2013.2241174
  • NAID: 110009712440
  • Peer Reviewed
• [Presentation] Algebraicity of formal power series over a finite field (2016)
  • Author(s): Teturo Kamae
  • Organizer: 数論とエルゴード理論
  • Place of Presentation: 金沢大学
  • Year and Date: 2016-02-07
• [Presentation] An example of non-computable conditional probability for generalized Lambalgen's theorem (2016)
  • Author(s): Hayato Takahashi
  • Organizer: 11th Conference on Logic Computability and Randomness
  • Place of Presentation: Hawaii, USA
  • Year and Date: 2016-01-04
  • Int'l Joint Research
• [Presentation] Algebraicity of formal power series over a finite field (2015)
  • Author(s): Teturo Kamae
  • Organizer: Sino-Japanese Workshop on Dynamic Systems and Fractals
  • Place of Presentation: 寧波，中国
  • Year and Date: 2015-12-27
  • Int'l Joint Research
• [Presentation] Self-affine functions of Brownian motion type (2015)
  • Author(s): Teturo Kamae
  • Organizer: エルゴード理論とその周辺
  • Place of Presentation: 慶応大学
  • Year and Date: 2015-11-28
• [Presentation] Randomness and Approximation of measures (2015)
  • Author(s): Hayato Takahashi
  • Organizer: 第３８回情報理論とその応用シンポジウム
  • Place of Presentation: 岡山県倉敷市児島
  • Year and Date: 2015-11-24
• [Presentation] How to measure randomness (2015)
  • Author(s): Teturo Kamae
  • Organizer: Morning Side連続講演
  • Place of Presentation: 北京，中国
  • Year and Date: 2015-07-28
• [Presentation] Entropy estimate (2015)
  • Author(s): Teturo Kamae
  • Organizer: Workshop on Measurable and Topological Dynamics, NIMS
  • Place of Presentation: 大田，韓国
  • Year and Date: 2015-06-30
  • Int'l Joint Research
• [Presentation] Hausdorff dimension of the level sets of self-affine functions (2015)
  • Author(s): Teturo Kamae
  • Organizer: 華中科技大学談話会
  • Place of Presentation: 武漢，中国
  • Year and Date: 2015-05-19
• [Presentation] Criterion of randomness (2015)
  • Author(s): 釜江哲朗
  • Organizer: 数論とエルゴード理論
  • Place of Presentation: 金沢大学
  • Year and Date: 2015-02-07
• [Presentation] Entropy estimation (2014)
  • Author(s): 釜江哲朗
  • Organizer: エルゴード理論とその周辺
  • Place of Presentation: 熊本大学
  • Year and Date: 2014-12-03
• [Presentation] Generalization of van Lambalgen's theorem and blind randomness for conditional probability (2013)
  • Author(s): H. Takahashi
  • Organizer: 8th conference on Logic Computability and Randomness
  • Place of Presentation: Moscow, Russia
• [Presentation] Algorithmic randomness of pair of sequences and blind randomness for conditional probability (2013)
  • Author(s): H. Takahashi
  • Organizer: エルゴード理論，情報理論，計算機科学とその周辺
  • Place of Presentation: 大阪大学理学部
  • Invited
• [Presentation] An easy criterion for randomness (2013)
  • Author(s): 釜江哲朗
  • Organizer: 数論とエルゴード理論
  • Place of Presentation: 金沢大学サテライト・プラザ
• [Presentation] An easy criterion of randomness and a pseudorandom number generator (2013)
  • Author(s): T. Kamae
  • Organizer: エルゴード理論，情報理論，計算機科学とその周辺
  • Place of Presentation: 大阪大学理学部
  • Invited
• [Presentation] 離散フーリエ変換に基づく乱数検定について (2013)
  • Author(s): 尾崎亮輔(京都大学), 梅野健(京都大学)
  • Organizer: 日本応用数理学会
  • Place of Presentation: 東洋大学白山キャンパス6号館2階
• [Presentation] Algorithmic randomness and stochastic selection function (2012)
  • Author(s): H. Takahashi
  • Organizer: 7th Conference on Computability, Complexity and Randomness
  • Place of Presentation: Isaac Newton Institute for Mathematical Sciences