| Project/Area Number |
11554004
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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 | 展開研究 |
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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 |
KOHDA Tohru Graduate School of Information Science and Electrical Engineering, Kyushu University, professor, 大学院・システム情報科学研究院, 教授 (20038102)
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| Co-Investigator(Kenkyū-buntansha) |
SAKURAI Koichi Graduate School of Information Science and Electrical Engineering, Kyushu University, associate professor, 大学院・システム情報科学研究院, 助教授 (60264066)
SUGITA Hiroshi Graduate School of Mathematics, Kyushu University, associate professor, 大学院・数理学研究科, 助教授 (50192125)
OOHAMA Yasutada Graduate School of Information Science and Electrical Engineering, Kyushu University, associate professor, 大学院・システム情報科学研究院, 助教授 (20243892)
JITSUMATSU Yutaka Graduate School of Information Science and Electrical Engineering, Kyushu University, research associate, 大学院・システム情報科学研究院, 助手 (60336063)
藤崎 礼志 九州大学, 大学院・システム情報科学研究院, 助手 (80304757)
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| Project Period (FY) |
1999 – 2001
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| Project Status |
Completed (Fiscal Year 2001)
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| Budget Amount *help |
¥6,300,000 (Direct Cost: ¥6,300,000)
Fiscal Year 2001: ¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2000: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1999: ¥2,000,000 (Direct Cost: ¥2,000,000)
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| Keywords | Markov Chain / transition matrix / characteristic equation / correlated property / sequence of symbols / histogram / central limit theorem / Wiener-Hopf equation / マルコフ推移行列 / 遷移確率の推定 / スペクトルの推定 |
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| Research Abstract |
Markov information sources play an important role in modeling subjects to be studied as a stochastic process, e.g., block cipher, speech recognition, recognition of human genes in DNA as well as the well-known Shannon's model of a communication system. The main purpose of this research project is to study how to identify a Markov information source with transition matrix $P$ by only observing a sequence of symbols generated by the source.
Consider an N-state simple Markov source with transition matrix P which takes symbols in S (1, 2, …, N). It is natural to estimate directly all the elements p_<ij>, (i,j=1, …, N) in P by using N^2 histograms of possible strings of length 2. This method, however, requires too many histograms if the number of states N becomes large. Since statistics of sequences generated by the Markov sources are primarily governed by eigenvalues of P, one of simple ways to identify the source is to estimate eigenvalues of P.
In this research we give a simple method to determine sequences of symbols to be observed which give histograms whose number is in the order of N. Furthermore, we derive a nonsymmetric Toeplitz system (or referred to as the Wiener-Hopf equation), in the linear equation form, whose coefficient matrix and constants are functions of means and variances of the histograms. In addition, the solution of this linear equation is shown to determine a characteristic polynomial of P. Numerical simulations show that the identification of 2-state Markov chains are successful ; but the one of 3-state are not. Thus we get a conclusion that further investigation is needed and problems remain on algorithm based on the minimum number of histograms.
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