2007 Fiscal Year Final Research Report Summary
Time series analysis of complex phenomena and a study of %paragon property from the viewpoint of stochastic processes
| Project/Area Number |
17340024
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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 | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Ehime University (2007)
The University of Tokyo (2005-2006) |
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Principal Investigator |
MATSUURA Masaya Ehime University, Graduate School of Science and Engineering, Associate professor (70334258)
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| Co-Investigator(Kenkyū-buntansha) |
YUMOTO Kiyofumi Kyushu University, Space Environment Research Center, Professor (20125686)
TAKEO Minoru The University of Tbkyo, Earthquake Reseatrch Institute, Professor (00197279)
KATO Amami Kinki University, School of Medicine, Professor (00233776)
HORITA Takehiko Osaka Prefecture University, Department of Mathematirel Sciences, Associate Professor (90222281)
OKABE Yasunori Meiji University, School of Science and Technology, Specially Appointed Professor (30028211)
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| Project Period (FY) |
2005 – 2007
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| Keywords | KM2O-Langevin equation / Nonlinear Information analysis / Test(ABN-S) / Test (ABN-EP) / Separation property / Deep low frequency earthquake / Nikkei 225 / Electromaenetic wave |
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| Research Abstract |
The aim of this research project is to develop a method by which we can extract nonlinear structures of time series data and derive mathematical models of time evolutions. Our basic idea is "from data to mathematical laws and models", which means that it is necessary to examine whether the preconditions of the mathematical theorems are satisfied before applying them to data analysis. This can be realized by the theory of the KM20-Langevin equations. So far, we have proposed several types of tests: Test (S)- stationarity test, Test (ABN)-abnormality test, Test (D)-determinacy test. We have applied these methods to seismic time series of deep low frequency earthquakes and discovered the so called "separation property", which can be seen as one of the characteristic properties of deep low frequency earthquakes. Therefore it is quite important to characterize "separation property" from a mathematical viewpoint.
In this research project, we have obtained the following results.
1. We have formulated "separation property" as a mathematical concept and proved that if the finite dimensional distributions of a stochastic process are symmetric, the process satisfies separation property. Moreover, we have derived a kind of expression theorem of discrete time stochastic processes.
2. In connection with "separation properties", it is extremely important to detect abnormalities of time series. However, our abnormality test Test (ABN)is not sufficient enough for this purpose. Therefore, we have newly proposed a method for detecting abnormality, called Test (RSK), by utilizing nonlinear prediction errors. We also verified the effectiveness of the method.
3. We have also analyzed time series of stock prices, which do not satisfy separation property, and found that polynomial transformations of degree two play an important role in describing the dynamics of these time series. Mathematical interpretation of this fact is a future task.
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