| Project/Area Number |
15500190
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Statistical science
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| Research Institution | Keio University |
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Principal Investigator |
SHIMIZU Kunio Keio University, Faculty of Science and Technology, Professor, 理工学部, 教授 (60110946)
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| Co-Investigator(Kenkyū-buntansha) |
KATO Takesi Keio University, Faculty of Science and Technology, Assistant Professor, 理工学部, 講師 (40267399)
JIMBO Masakazu Nagoya University, Graduate School of Information Science, Professor, 大学院・情報科学研究科, 教授 (50103049)
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| Project Period (FY) |
2003 – 2004
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| Project Status |
Completed (Fiscal Year 2004)
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| Budget Amount *help |
¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 2004: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2003: ¥1,000,000 (Direct Cost: ¥1,000,000)
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| Keywords | strictly stationary ellipsoidal process / scale mixtures of normals / level crossings / binary time series / trinary time series / number of extremes / number of inflection points / length of excursions |
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| Research Abstract |
In the paper by Tanaka and Shimizu (2001), the expectation formula of level crossings for discrete time series is studied in the case where the underlying process is a strictly stationary ellipsoidal process with zero-mean, unit-variance and autocorrelation function ρ(h), which is actually governed by a scale mixture of normal distributions. The present study assumes the same process and aims to get expectation and variance formulae of the numbers of one-step and two-step level-crossings for an observed discrete sample Y_1,...,Y_N. Transformation into trinary series 1, 0, -1 is used to get the results. Here the number of one-step level-crossings is defined by the number t (2≦t≦N) satisfying that [Y_t≧u and v≦Y_<t-1><u], or [v≦Y_t<u and Y_<t-1>≧u], or [v≦Y_t<u and Y_<t-1><v], or finally [Y_t<v and v≦Y_<t-1><u] for u, v(u>v). Similarly the number of two-step level-crossings is defined by the number t (2≦t≦N) satisfying that [Y_t≧u and Y_<t-1><u] or [ Y_t<v and Y_<t-1>≧u]. The expectation formulae are obtained in the paper by Shimizu and Tanaka (2003) and the paper also provides applications to the expectation formula for the number of two-step level-crossings after taking the first- and second-difference time series in a stationary Gaussian zero-mean, unit-variance and autocorrelation function ρ(h). The numbers generalize the number of extremes (local minima and maxima) and that of inflection points. Asymptotic behavior of the length of excursions is studied in Tanaka and Shimizu (2004). A paper by Shikama and Shimizu which studies variance formulae is submitted to a journal for possible publication and is now under review.
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