Development of unified system with non-linear analytic technique for time series data based upon the fluctuation-dissipation theorem

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We constructed a theory of KM_2O-Langevin equations for flows in a vector space with a metric in which a characterization theorem that characterizes stationarity of flow in terms of the fluctuation-dissipation theorem, a construction theorem of stationary flow, an extension theorem of stationary flow and an extension theorem of non-negative definite functions were proved. Moreover, we developed both the theory of weight transformation for degenerate flows in a vector space with a metric and the theory of non-linear information analysis for local stochastic processes with discrete time.By using these results, we resolved the non-linear prediction problem for one-dimensional strictly stationary processes which had remained to be solved for a long time after Masani-Wiener's work. Furthermore, by developing a non-linear information analysis for local multi-dimensional stochastic processes with discrete time and then using the theory of linear prediction in the theory of KM_2O-Langevin equa … More tions for degenerate flows, we solved both the non-linear prediction problem and the non-linear filtering problem for multi-dimensional stochastic processes with discrete time.As another application of the non-linear information analysis for local multi-dimensional stochastic processes with discrete time, we introduced the concept of weak causality that is weaker than yhe one of strong causality investigated so far. Precisely speaking, for given two stochastic proceses X=(X(n) ; 0【less than or equal】n【less than or equal】N), Y=Y(n) ; 0【less than or equal】n【less than or equal】N), we say that there exists a weak causality from X to Y if there exist certain M_O (0【less than or equal】M_O【less than or equal】N) and Borel functions f_n (M_O【less than or equal】n【less than or equal】N) such that Y(n)=f_n(X(n), X(n-1), ..., X(0), Y(n-1), Y(n-2), ..., Y(0))(M_O【less than or equal】n【less than or equal】N). Furthermore, by establishing a criteria of Test (CS) that determines the existence of weak causality from X to Y for given two time series data X=(X(n) ; 0【less than or equal】n【less than or equal】N), X=(X(n) ; 0【less than or equal】n【less than or equal】N), we investigated both th weak causality and the strong causality among the various time series data that are related to the E1 Nino phenomena that is said to be related to the increase of earth atmosphere temperature and announced their results at ICIAM99 that was held in Edinburg in 1999.We constructed another genarating system of the non-linear information space by proving an approximation theorem by the radial basic functions that are used in the chaotic time series data.Moreover, we developed several softs that deterimine the stationarity, causality, determinicity and chaotic property of time series data and a unified soft that derives cetain dynamics behind time series data and then predicts its future. Thus, we can apply these softs to carry out causal analysis, model analysis and prediction analysis for time series data with the framework of the theory of KM_2O-Langevin equations. Less