Time series analysis for abnormality test and modeling of corrplex system and a study for the derived model from a view point of the theory of stochastic processes

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We have obtained a formula for calculating a nonlinear filter between two discrete time stochastic processes, by using the theory of KM_2O-Langevin equations. Then we have applied it to a concrete non-linear system consisting of signal process and observation process, and given a solution for the non-linear filtering problem by obtaing an algorithm for calculating the non-linear fiter, which had been unsolved since Kalman-Bucy's work.By definning the abnormality of time series as the degree of breakdown of stationarity of time series, we have proposed Test(ABN) for catching certain signs of the abnormality of time series, by using Test(S) for testing the stationarity of time series and the generating system of polynomial type for the nonlinear information spaces associated with the discrete time stochastic processes.By applying Test(ABN) and Test(D) for testing the determinacy of time series to the time series of earthquakes, aurora and brain waves, we have discovered a new qualitative property, to be called separation property, on the region possessing stationarity after the arrival of S-wave for the deep low frequency earthquakes, the occurrence of aurora, and on the total region possessing stationarity of ECoG. This separation property do not appear for the usual earthquakes and EEG.By treating the continuous time stationary process X whose gloval time evolution is governed by [α, β, γ]-Langevin equation, we have derived both the continuous time KM_2O-Langevin equation describing the local time evolution the stochastic process X and the system of equations characterizing the coefficients appearing in the above equation.