Nonlinear Dynamical Time Series Analysis Methods And Its Application

Deterministic chaos that is determined by nonlinear deterministic dynamical mechanism is a kind of irregular movement widely existing in nature and society.With the development of chaos theory and research on its application,nonlinear time series analysis has become a major research domain of nonlinear signal processing,and has been widely applied to various region.The contents of this paper are arranged as follow:Progress of research of nonlinear time series analysis is reviewed in Chapter 1,the research content of this paper is also introduced.Phase space reconstruction and embedding theory of dynamical system are studied in Chapter 2,principles and algorithms of methods of principal component analysis,saturated correlation dimension and false neighbors which is three basic methods usually used to determine the embedding dimension are studied too.Chapter 3 will deeply analyze methods usually used to determine the embedding dimension,and propose the new method of determining the minimum embedding dimension based on four-order cumulant and the new method of determining the optimal embedding dimension based on nonlinear prediction.Chapter 4 will expatiate the inverse problem of dynamical system,and study theories and algorithms of global prediction,local prediction,and adaptive prediction usually used to prediction nonlinear time series. Chapter 5 is mainly concerned with local prediction methods,to simultaneously uses spatial correlation and temporal correlation,Chapter 5 proposes the improved local linear prediction method and the new local linear prediction model,and analyses the optimal embedding dimension and delay of local linear prediction model.The relation between the local prediction method and neighbors is deeply analyzed in Chapter 6, based on the information criterion,the neighbor point selection method for the local prediction method is proposed in Chapter 6,and this Chapter finally applies the nonlinear time series analysis approach to the laser data.Chapter 7 studies methods of Lyapunov exponents,surrogate data and nonlinear prediction that is usually used for the detection and analysis of nonlinear time series,and this Chapter finally applies the nonlinear time series analysis method to analyze physiological time series.Chapter 8 applies systematically the nonlinear time series analysis approach to the traffic measurements,and applies the local support vector machines prediction method to predict the traffic measurement data.Finally,the main contributions made in the paper are given in the last chapter.Results of this paper are summarized as follows:First,basic theories and general methods of nonlinear time series analysis are deeply and systematically studied.Based on the research work of basic theories including phase space reconstruction,embedding theorem,correlation dimension,local dynamics,Lyapunov exponents,surrogate data etc,based on the research work of general methods such as principal component analysis,correlation dimension GP algorithm,false neighbors method,nonlinear time series prediction,local prediction, adaptive prediction,neural network model,support vector machines regression model, prediction power,nonlinear detection,coarse-graining methodology,conditional entropy and so on,the framework of nonlinear time series analysis are constructed. Basic problems and main research areas of nonlinear time series analysis are summarized.Second,Principal component analysis is essentially a linear method based on the covariance matrix which reflects the linear dependence.Numerical experience led several researchers to express some doubts about the reliability of PCA.In this paper the matrix constructed by four-order cumulant function instead of correlation function is used to improve the method of PCA.Methods used four-order cumulant function to construct matrixes is studied and the best two methods are found.When two parameters of four-order cumulant function choose values of the diagonal direction and the off-diagonal direction of the matrix and the third parameter is zero,we can get the best matrix.Simulation results show that the improved method is fit for the small set nonlinear time series,is computationally efficient,and is stable to noise.A new method of determining the optimal embedding dimension based on nonlinear prediction is proposed to determine the optimal embedding dimension from a scalar time series.This method determines the optimal embedding dimension by optimizing the nonlinear autoregressive prediction model parameterized by the embedding dimension and the nonlinear degree.Simulation results show that this method is applicable to a short time series,stable to noise,computationally efficient,and does not contain any purposed introduced parameters.Third,based on the Bayesian information criterion,the improved local linear prediction method is proposed to predict nonlinear time series.This method simultaneously uses spatial correlation and temporal correlation.Simulation results show that the improved local linear prediction method can effectively predict nonlinear time series and the prediction performance of the improved local linear prediction method are superior to that of the traditional local linear prediction method.In the reconstructed phase space,a new local linear prediction model is proposed to predict nonlinear time series.We propose that the parameters of the local linear prediction model can be chosen values that are different to those of the state space reconstruction. We propose a criterion based on prediction power to determine the optimal parameters of the new local linear prediction model.Simulation results show that the new local linear prediction model can effectively predict nonlinear time series and the prediction performance of the new local linear prediction model is superior to that of the local linear prediction.A method of optimizing embedding dimension and delay for local linear prediction model is proposed.Simulation results show that the local linear prediction method,which has been optimized,by this method can effectively make one-step and multi-step prediction for nonlinear time series,and the one-step and multi-step prediction accuracy of the optimized local linear prediction method is superior to that of the traditional local linear prediction.Fourth,the number of nearest neighbor points is an important parameter for the local prediction method,which has an important impact on the prediction accuracy and computation complexity of the local model.Based on the information criterion,the neighbor point selection method for the local prediction method is proposed.Simulation results show that using the proposed method to select neighbor points,the one-step and multi-step prediction accuracy of the local prediction method is well,and the computation complexity is reduced.Fifth,we applied the local linear prediction method and the local support vector machines prediction method to predicting the laser measurement data.We applied the method of determining the embedding dimension based on nonlinear prediction to determining the optimal embedding dimension of this laser data,and applied the neighbor point selection method for local prediction based on information criterion to determining the nearest neighbor points.Simulation results show that the local linear prediction method and the local support vector machines prediction method whose neighbor points have been optimized can effectively predict the laser measurements data,and the one-step and multi-step prediction accuracy is well.Sixth,the heart rate variability could be explained by a low-dimensional governing mechanism.There has been increasing interest in verifying and understanding the coupling between the respiration and the heart rate.The embedding dimension of the heart rate variability is determined by the method based on nonlinear prediction.We use the nonlinear detection method to detect the nonlinear deterministic component in the physiological time series by a single variable series and by two variables series respectively,and use the conditional information entropy to analyze the correlation between the heart rate,the respiration and the blood oxygen concentration.The conclusions are that there is the nonlinear deterministic component in the heart rate data and respiration data,and the heart rate and the respiration are two variables originating from the same underlying dynamics.Seventh,this paper applied systematically the nonlinear time series analysis approach to the traffic measurements data.We demonstrated that the nonlinear time series analysis methods could be successfully used for a deeper understanding of main features of the traffic data.To reconstruct phase space of the traffic data,we applied the method of determining the embedding dimension based on nonlinear prediction to determining the optimal embedding dimension of traffic data.We applied the local support vector machines prediction method to predicting the traffic measurement data, and applied the neighbor point selection method for local prediction based on information criterion to determining the number of the nearest neighbor points. Simulation results show that the local support vector machines prediction method whose neighbor points have been optimized can effectively predict the traffic measurements data,the normalized mean squared error is very low,the time series generated by the support vector machines regression model have the same statistical properties with the real traffic data.