A study of identification problem for continuous model in phenomena of complex system based on the theory of Langevin equations from the viewpoint of the theory of stochastic processes

从随机过程理论的角度研究基于朗之万方程理论的复杂系统现象中的连续模型辨识问题

• 批准号: 10440026
• 负责人: OKABE Yasunori
• 金额: $ 4.29万
• 依托单位: University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B).
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-10440026/
• 关键词: KM_2O-Langevin equation; KMO-Langevin equation; weak stationarity; purely non-determinicity; fluctuation-dissipation theorem; local and global canonical representation theorem; innovation method; outer matrix function; 純非決定性,完全ランク性とテープリッツ条件; 連続系と

As a preparation for the aim to investigate weakly stationary process with continuous time by using weakly stationary process with discrete time, we developed the theory of KM_2O-Langevin equations for degenerate flows in the three directioins of the analysis of local non-linear information space, the analysis of weight transformations and the linear prediction theory. By using these results, we resolved not only 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, but also both the non-linear prediction problem and the non-linear filtering problem for multi-dimensional stochastic processes with discrete time.On the other hand, we constructed Kubo noise associated with multi-dimensional stationary flow with continuous time in a Hilbert space.Next, by taking a procedure of scaling limits of KM_2O-Langevin data that determines the local stochastic difference equation (KM_2O-Langevin … More equation) describing the time evolution of weakly stationary process with discrete time, we derived KMO-Langevin data that determines the global stochastic difference equation (KMO-Langevin equation). Conversely, for a class of weakly stationary process with continuous time, we derived KM_2O-Langevin equation from KMO-Langevin equation, by using the idea of innovation method in the filtering theory. Thus, we could obtain the algorithm calculating the discrete and global characteristics from the discrete and local characteristics and prove the representaion theorem of outer matrix function for continuous case from the one for discrete case. Therefore, we have completed the derivation of the stochastic difference equation describing the time evolution of weakly stationary process with discrete time not only for local case, but also for global case.Moreover, for a class of weakly stationary process X=(X(t) ; t∈R), we define for each positive number ∈, we define a stochastic process X_∈=(X(n∈) ; n∈Z). Then, we investigated KMO(resp.KM_2O)-Langevin data that determines the dissipation term and the fluctuation term in KMO(resp.KM_2O)-Langevin equation describing the time evolution of the weakly stationary process X and certatin ∈-dependence of KMO(resp.KM_2O)-Langevin data that determines the dissipation term and the fluctuation term in KMO (resp.KM_<>O)-Langevin equation describing the time evolution of the weakly stationary process X_∈. In particular, we could represent KMO-Langevin data associated with X as a scaling limits with respect to ∈ of KMO-Langevin data associated with X_∈. Less

为了利用离散时间的弱平稳过程研究连续时间的弱平稳过程，我们从局部非线性信息空间分析、权变换分析和线性预测理论三个方向发展了退化流的K2O-朗之万方程理论。利用这些结果，我们不仅解决了Masani-Wiener工作后长期悬而未决的一维严格平稳过程的非线性预测问题，而且还解决了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。另一方面，我们在Hilbert空间中构造了与多维连续时间平稳流相关的Kubo噪声。其次，通过对Km2O-Langevin数据的定标极限过程来确定局部随机差分方程(Km2O-Langevin…更多描述离散时间弱平稳过程的时间演化的KMO-朗之万方程，我们得到了确定全局随机差分方程(KMO-朗之万方程)的KMO-朗之万方程。相反，对于一类具有连续时间的弱平稳过程，我们利用滤波理论中的新息方法的思想，从KMO-朗之万方程出发，推导出K2O-朗之万方程。由此，我们可以得到由离散和局部特征计算离散和整体特征的算法，并从离散情况证明了连续情况下外矩阵函数的表示定理。因此，我们完成了描述离散时间弱平稳过程的时间演化的随机差分方程的推导，不仅对于局部情况，而且对于全局情况，我们定义了一类弱平稳过程X=(X(T)；t∈R)，对于每个正数∈，我们定义了一个随机过程X_∈=(X(n∈)；n∈Z)。然后，我们研究了描述弱平稳过程X随时间演化的KMO(相应KM_2O)-朗之万方程中决定耗散项和涨落项的KMO(相应KM_2O)-朗之万方程，以及确定描述弱平稳过程X_∈时间演化的KMO(相应KM_2O)-朗之万方程中耗散项和涨落项的KMO(相应KM_2O)-朗之万方程。特别地，我们可以将与X关联的KMO-朗之万数据表示为相对于与X_∈关联的KMO-朗之万数据的∈的标度极限。较少

项目成果

Y.Okabe: "On the theory of KM_2O-Langevin equations for stationary flows (II) : construction theorem"to appear in the special volume in honor of the 70th birthday of Professor Takeyuki Hida. (2001)

Y.Okabe：“关于平稳流的KM_2O-Langevin方程的理论（II）：构造定理”出现在纪念飞田武之教授70岁生日的特刊中。

N.Masuda and Y.Okabe: "Time series analysis with wavelet coefficients"Japan Journal of Industrial and Applied Mathematics. (2001)

N.Masuda 和 Y.Okabe：“利用小波系数进行时间序列分析”日本工业与应用数学杂志。

Y.Okabe and A.Kaneko: "On a non-linear prediction analysis for multi-dimensional stochastic processes with its applications to data analysis"Hokkaido Mathematical Journal. 29巻. 601-657 (2000)

Y. Okabe 和 A. Kaneko：“多维随机过程的非线性预测分析及其在数据分析中的应用”《北海道数学杂志》，第 29 卷，601-657（2000 年）。

Y.Okabe: "On a Kubo noise associated with a multidimensional stationary curve in a Hilbert space"Proceedings of the 2nd Jagna International Workshop, Mathematical Methods of Quantum Physics. (1999)

Y.Okabe：“论与希尔伯特空间中的多维平稳曲线相关的 Kubo 噪声”第二届 Jagna 国际研讨会论文集，量子物理的数学方法。

Y.Okabe and M.Matuura: "On the theory of KM_2O-Langevin equations for stationary flows (III) : extension theorem"Hokkaido Mathematical Journal. Vol.29. (2000)

Y.Okabe 和 M.Matuura：“关于平稳流的 KM_2O-Langevin 方程的理论（III）：可拓定理”北海道数学杂志。