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

本文利用KM_2 O-Langevin方程的理论，得到了两个离散时间随机过程之间的非线性滤波器的计算公式。然后将其应用于一个具体的由信号处理和观测过程组成的非线性系统中，给出了自Kalman-Bucy工作以来一直没有解决的非线性滤波问题的解决方法，得到了一个计算非线性滤波器的算法，将时间序列的异常定义为时间序列平稳性的破坏程度，我们已经提出了用于捕获时间序列异常的某些迹象的测试（ABN），利用Test（S）检验时间序列的平稳性和与离散时间随机过程相关的非线性信息空间的多项式型生成系统，利用Test（ABN）和Test（D）摘要为了检验时间序列对地震、极光和脑电波时间序列的确定性，我们发现了深低频地震、极光发生的S波到达后具有平稳性的区域与ECoG具有平稳性的总区域之间的一个新的定性性质，即分离性。本文通过处理全局时间演化受[α，β，γ]-Langevin方程控制的连续时间平稳过程X，导出了描述随机过程X局部时间演化的连续时间KM_2 O-Langevin方程，以及描述该方程中系数的方程组。

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.