A study on modeling and prediction for nonlinear phenomena from the viewpoint of the theory of stochastic processes

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We have obtained four theoretical results about the theory of KM_2 O - Langevin equations : (1) We have characterized a notion of stationarity for pair of flows by the fluctuation-dissipation theorem which holds among KM_2 O-Langevin matrix associated with the pair of flows ; (2) For any given nonnegative definite matrix function R with Toeplitz condition, we have constructed a KM _2 O-Langevin matrix satisfying the fluctuation-dissipation theorem and then a stationary pair of flows with R its correlation matrix function by solving KM _2 O-Langevin equation ; (3). We have obtained all extensions of any given stationary flow and correlation matrix function ; (4) As an application of (3) we have obtained a new prediction formulaWe have obtained six results about an application of the theory concerned to time series analysis : (1) We have made a statistical basis of both the stationary test and the causal test strong ; (2) We have proposed a deterministic test to judge whether the time evolution of time series is deterministic ; (3) We have proposed a chaotic test to judge whether the time evolution of time series is chaotic ; (4) We have studied two time series of measles and chickenbox treated in Sugihara-May 's paper (Nature, 1990). From the viewpoint of the theory concerned, we have supported their results based upon the chaotic time series analysis, by showing that both the time series behave determinately ; the former is chaotic and the latter is not so ; (5) We have obtained a theoretical method to estimate the deterministic dynamics describing the time evolution of time series after passing the deterministic test ; (6)As an application of (5), we have given an answer for the estimation problem of embeding dimension in chaotic time series analysis.