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

本文建立了向量空间中带有度量的KM_2 O-Langevin方程的理论，证明了用涨落耗散定理、定态流的构造定理、定态流的延拓定理和非负定函数的延拓定理来刻画流的定态性。此外，我们还发展了向量空间中退化流的权变换理论和离散时间局部随机过程的非线性信息分析理论，解决了Masani-Wiener等人提出的一维严格平稳过程的非线性预测问题。进一步，通过对局部多维离散时间随机过程的非线性信息分析，利用KM_2 O-Langevin方程理论中的线性预测理论，建立了一种新的非线性信息分析方法， ...更多信息 作为退化流的非线性信息分析的一个应用，我们解决了多维离散随机过程的非线性预测问题和非线性滤波问题，作为多维离散局部随机过程的非线性信息分析的另一个应用，我们引入了弱因果关系的概念，弱因果关系的概念比强因果关系的概念弱。精确地说，对于给定的两个随机过程X=（X（n）; 0[小于或等于]n[小于或等于]N），Y=Y（n）; 0[小于或等于]n[小于或等于]N），如果存在一定的M_O，则X到Y存在弱因果关系（0[小于或等于]M_O[小于或等于]N）和Borel函数f_n（M_O[小于或等于]n[小于或等于]N）使得Y（n）=f_n（X（n），X（n-1），…，X（0），Y（n-1），Y（n-2），.，Y（0））（M_O[小于或等于]n[小于或等于]N）。进一步，通过建立一个判定两个时间序列数据X=（X（n）; 0[小于或等于]n[小于或等于]N），X=（X（n）; 0[小于或等于]n[小于或等于]N），我们研究了与厄尔尼诺现象有关的各种时间序列数据之间的弱因果关系和强因果关系，据说厄尔尼诺现象与全球变暖的增加有关。在1999年爱丁堡举行的ICIAM 99上，我们利用径向基函数对混沌时间序列数据的逼近定理，构造了非线性信息空间的另一个生成系统，并开发了几个软件，对混沌时间序列数据的平稳性、因果性、时间序列数据的确定性和混沌性，以及一个统一的软件，推导出时间序列数据背后的某些动力学，然后预测其未来。因此，可以在KM_2 O-Langevin方程理论的框架下，应用这些软件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