Time series analysis of complex phenomena and a study of %paragon property from the viewpoint of stochastic processes

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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.