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

构造了具有度量的向量空间中流动的km_20 - langevin方程理论，证明了用涨落耗散定理、稳态流动的构造定理、稳态流动的可拓定理和非负定函数的可拓定理来表征流动的平稳性。此外，我们还发展了具有度量的向量空间中退化流的权变换理论和具有离散时间的局部随机过程的非线性信息分析理论。利用这些结果，我们解决了Masani-Wiener之后长期未解决的一维严格平稳过程的非线性预测问题。在此基础上，对具有离散时间的局部多维随机过程进行了非线性信息分析，利用退化流的km_20 - langevin方程理论中的线性预测理论，解决了具有离散时间的多维随机过程的非线性预测问题和非线性滤波问题。作为非线性信息分析在离散时间局部多维随机过程中的另一个应用，我们引入了弱因果关系的概念，它比目前所研究的强因果关系的概念更弱。精确地说，对于给定的两个随机过程，X=(X(n)；0【小于等于】n【小于等于】n)， Y=Y(n)；0（小于等于）n（小于等于）n)，如果存在某种M_O（0（小于等于）M_O（小于等于）n）和Borel函数f_n（M_O（小于等于）n（小于等于）n）使得Y(n)=f_n(X（n)， X(n-1)，…），则存在从X到Y的弱因果关系， X(0)， Y(n-1)， Y(n-2)，…， Y(0))（m0【小于等于】n【小于等于】n）。进一步，通过建立检验标准（CS），确定给定两个时间序列数据X=(X(n))从X到Y存在弱因果关系；0【小于等于】n【小于等于】n)， X=(X(n)；0【小于或等于】n【小于或等于】n)，我们研究了与地球大气温度升高有关的E1 Nino现象的各种时间序列数据之间的弱因果关系和强因果关系，并在1999年在爱丁堡举行的ICIAM99上公布了他们的结果。利用混沌时间序列数据中的径向基函数证明了一个近似定理，构造了另一个非线性信息空间的生成系统。此外，我们还开发了几个软件来确定时间序列数据的平稳性、因果性、确定性和混沌性，并开发了一个统一的软件来导出时间序列数据背后的某种动态并预测其未来。因此，我们可以利用这些软件在km_20 - langevin方程理论的框架下对时间序列数据进行因果分析、模型分析和预测分析。少

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