非线性动力系统时间序列分析方法及其应用研究
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摘要
混沌现象是自然界和社会中广泛存在的一种不规则运动,是一种由确定的非线性动力系统生成的复杂行为。随着混沌理论和应用技术研究的不断深入,非线性时间序列分析已成为非线性信息处理领域中近几年来的一个重要研究热点,并在相关工程领域有着越来越重要的应用。
论文的内容大致安排为:在第一章中将综述非线性时间序列分析的研究现状,并阐述本文的选题意义和研究内容。第二章将研究动力系统吸引子的相空间重构问题和嵌入理论,研究主分量分析、关联维数GP算法以及主分量分析法、饱和关联维数法、伪邻近点法三种最常用的选取嵌入维数方法的原理和算法实现。第三章将深入分析现有选取嵌入维数方法存在的问题,提出基于高阶统计量的嵌入维数的选取方法和基于预测效果的嵌入维数的选取方法。第四章将阐述动力系统的逆问题,深入研究全局预测方法、局域预测方法、自适应预测方法等非线性时间序列常用的建模预测方法的原理和算法实现。第五章将着重研究局域预测方法,为了同时利用时间序列的时间相关性和空间相关性,提出改进的局域线性预测方法、新的局域线性预测模型,并分析局域线性预测模型的最优参数。第六章将深入分析局域预测方法与邻近点的关系,基于信息准则提出选取局域线性预测方法和局域支持向量机预测方法中邻近点个数和邻域半径的定量方法,并将应用非线性时间序列分析方法来分析实际的激光数据。第七章将深入研究Lyapunov指数、替代数据、预测效果及其相结合的检测和度量非线性时间序列及其非线性确定性程度的方法,并应用非线性时间序列分析方法来分析生物医学信号。第八章将系统地应用非线性时间序列分析方法来分析实测的网络流量序列,应用局域支持向量机预测方法来预测网路流量序列。最后,第九章总结全文的贡献之所在。
论文的主要结果为:
1.论文深入系统地研究了非线性时间序列分析的基本理论和一般方法。在对包括相空间重构、嵌入定理、关联维数、局部动力学、Lyapunov指数、替代数据、等基本理论与其物理意义的研究和讨论基础上;在对包括主分量分析、关联维数GP算法、伪邻近点法、非线性时间序列预测、局域预测、自适应预测、神经网络模型、支持向量回归模型、预测效果、非线性检测、粗粒化方法、条件熵等非线性时间序列一般分析方法的原理和算法研究基础上;构建了新的非线性时间序列分析的理论体系,归纳总结了非线性时间序列分析的基本问题和主要研究方面。
2.基于协方差矩阵的主分量分析方法,本质上是一种线性方法,其可靠性受到质疑。用能反映非线性结构的四阶累积量函数代替相关函数构造矩阵,对主分量分析方法进行改进。对比分析了用四阶累积量函数构造矩阵的多种方法,得到两种较好的构造矩阵的方法。其中当四阶累积量函数的两个变量分别在矩阵的对角线方向和偏离对角线方向取值并且第三个变量取零时,得到的矩阵的分析效果最好。实验结果表明了改进后方法适合小数据量的情况、计算效率高且对噪声稳定。基于非线性时间序列短期可预测的性质,提出了基于预测效果的嵌入维数的选取方法来从标量时间序列确定最优嵌入维数。该方法通过优化非线性自回归预测模型来确定最优嵌入维数,该模型由嵌入维数和非线性阶数两个参数来决定。仿真结果表明该方法适合小数据量的情况,对噪声的稳定性好,计算效率高,不受主观参数的影响。
3.基于Bayesian信息准则,提出了改进的局域线性预测方法来预测非线性时间序列。该方法同时利用了非线性时间序列的时间相关性和空间相关性。仿真结果表明:改进的局域线性预测方法能够有效地预测非线性时间序列,并且改进的局域线性预测方法的预测性能明显好于传统局域线性预测方法的预测性能。在重构的相空间,提出了一种新的局域线性预测模型来预测非线性时间序列。提出局域线性预测模型的参数可以取与相空间重构的参数不同的值。基于模型的预测性能,提出了确定新的局域线性预测模型的参数的方法。仿真结果表明:新的局域线性预测模型能够有效地预测非线性时间序列,并且新的局域线性预测模型的预测性能明显优于传统局域线性预测模型的预测性能。提出了一种优化局域线性预测模型参数嵌入维数与延迟时间的方法。仿真结果表明:用该方法优化后的局域线性预测方法能够有效地对非线性时间序列进行一步和多步预测,并且用该方法优化后的局域线性预测方法的一步和多步预测精度都明显好于传统局域线性预测方法的一步和多步预测精度。
4.邻近点个数是局域预测方法的重要参数之一,它决定局域模型的预测精度和计算量。基于信息准则,提出了选取局域预测法中邻近点个数的定量方法,并用该方法分别选取局域线性预测方法和局域支持向量机预测方法的邻近点个数和领域半径。实验结果表明用该方法选取邻近点的局域线性预测方法和局域支持向量机预测方法的一步和多步预测性能较好,在满足预测精度较高的条件下,计算量较小。
5.局域预测方法是目前最常用的一种非线性时间序列预测方法。应用局域线性预测方法和局域支持向量机预测方法来预测实测激光数据。应用基于预测效果的选取嵌入维数的方法来确定该组激光数据的最优嵌入维数。应用基于信息准则的局域预测法邻近点的选取方法来确定局域线性预测法与局域支持向量机预测法的邻近点个数和邻域半径。实验结果表明邻近点优化后的局域线性预测法和局域支持向量机预测法都能够有效地预测该组激光数据,一步和多步预测性能较好,在预测精度较高的条件下,计算量较小。
6.心率变异信号可被解释为由低维非线性动力机制控制。验证呼吸和心率的耦合作用越来越受到人们的关注。应用基于预测效果的嵌入维数的选取方法来确定心率变异信号的嵌入维数。分别使用单变量时间序列和多变量时间序列,应用结合预测效果和替代数据的非线性检测方法来检测生理时间序列的非线性确定性成分。应用粗粒化方法和条件熵来分析心率信号、呼吸信号与血氧浓度信号的相互关系。实验结果表明:心率信号和呼吸信号中含有非线性确定性成分,心率信号与呼吸信号不是独立的,它们互相影响,心率信号和呼吸信号可看作起源于同一动力系统的两个变量。
7.系统地应用非线性时间序列分析方法来分析实测的网络流量序列。为了对网络流量数据相空间重构,应用基于预测效果的选取嵌入维数的方法来确定最优嵌入维数。应用局域支持向量机预测方法来预测网路流量序列。并应用基于信息准则的局域预测法邻近点的选取方法来选取局域预测的邻近点个数。实验结果表明邻近点优化后的局域支持向量机预测方法能够有效地预测网络流量序列,归一化均方误差很小;局域支持向量机回归模型生成的时间序列具有与原网络流量时间序列相一致的概率分布。
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.
论文的内容大致安排为:在第一章中将综述非线性时间序列分析的研究现状,并阐述本文的选题意义和研究内容。第二章将研究动力系统吸引子的相空间重构问题和嵌入理论,研究主分量分析、关联维数GP算法以及主分量分析法、饱和关联维数法、伪邻近点法三种最常用的选取嵌入维数方法的原理和算法实现。第三章将深入分析现有选取嵌入维数方法存在的问题,提出基于高阶统计量的嵌入维数的选取方法和基于预测效果的嵌入维数的选取方法。第四章将阐述动力系统的逆问题,深入研究全局预测方法、局域预测方法、自适应预测方法等非线性时间序列常用的建模预测方法的原理和算法实现。第五章将着重研究局域预测方法,为了同时利用时间序列的时间相关性和空间相关性,提出改进的局域线性预测方法、新的局域线性预测模型,并分析局域线性预测模型的最优参数。第六章将深入分析局域预测方法与邻近点的关系,基于信息准则提出选取局域线性预测方法和局域支持向量机预测方法中邻近点个数和邻域半径的定量方法,并将应用非线性时间序列分析方法来分析实际的激光数据。第七章将深入研究Lyapunov指数、替代数据、预测效果及其相结合的检测和度量非线性时间序列及其非线性确定性程度的方法,并应用非线性时间序列分析方法来分析生物医学信号。第八章将系统地应用非线性时间序列分析方法来分析实测的网络流量序列,应用局域支持向量机预测方法来预测网路流量序列。最后,第九章总结全文的贡献之所在。
论文的主要结果为:
1.论文深入系统地研究了非线性时间序列分析的基本理论和一般方法。在对包括相空间重构、嵌入定理、关联维数、局部动力学、Lyapunov指数、替代数据、等基本理论与其物理意义的研究和讨论基础上;在对包括主分量分析、关联维数GP算法、伪邻近点法、非线性时间序列预测、局域预测、自适应预测、神经网络模型、支持向量回归模型、预测效果、非线性检测、粗粒化方法、条件熵等非线性时间序列一般分析方法的原理和算法研究基础上;构建了新的非线性时间序列分析的理论体系,归纳总结了非线性时间序列分析的基本问题和主要研究方面。
2.基于协方差矩阵的主分量分析方法,本质上是一种线性方法,其可靠性受到质疑。用能反映非线性结构的四阶累积量函数代替相关函数构造矩阵,对主分量分析方法进行改进。对比分析了用四阶累积量函数构造矩阵的多种方法,得到两种较好的构造矩阵的方法。其中当四阶累积量函数的两个变量分别在矩阵的对角线方向和偏离对角线方向取值并且第三个变量取零时,得到的矩阵的分析效果最好。实验结果表明了改进后方法适合小数据量的情况、计算效率高且对噪声稳定。基于非线性时间序列短期可预测的性质,提出了基于预测效果的嵌入维数的选取方法来从标量时间序列确定最优嵌入维数。该方法通过优化非线性自回归预测模型来确定最优嵌入维数,该模型由嵌入维数和非线性阶数两个参数来决定。仿真结果表明该方法适合小数据量的情况,对噪声的稳定性好,计算效率高,不受主观参数的影响。
3.基于Bayesian信息准则,提出了改进的局域线性预测方法来预测非线性时间序列。该方法同时利用了非线性时间序列的时间相关性和空间相关性。仿真结果表明:改进的局域线性预测方法能够有效地预测非线性时间序列,并且改进的局域线性预测方法的预测性能明显好于传统局域线性预测方法的预测性能。在重构的相空间,提出了一种新的局域线性预测模型来预测非线性时间序列。提出局域线性预测模型的参数可以取与相空间重构的参数不同的值。基于模型的预测性能,提出了确定新的局域线性预测模型的参数的方法。仿真结果表明:新的局域线性预测模型能够有效地预测非线性时间序列,并且新的局域线性预测模型的预测性能明显优于传统局域线性预测模型的预测性能。提出了一种优化局域线性预测模型参数嵌入维数与延迟时间的方法。仿真结果表明:用该方法优化后的局域线性预测方法能够有效地对非线性时间序列进行一步和多步预测,并且用该方法优化后的局域线性预测方法的一步和多步预测精度都明显好于传统局域线性预测方法的一步和多步预测精度。
4.邻近点个数是局域预测方法的重要参数之一,它决定局域模型的预测精度和计算量。基于信息准则,提出了选取局域预测法中邻近点个数的定量方法,并用该方法分别选取局域线性预测方法和局域支持向量机预测方法的邻近点个数和领域半径。实验结果表明用该方法选取邻近点的局域线性预测方法和局域支持向量机预测方法的一步和多步预测性能较好,在满足预测精度较高的条件下,计算量较小。
5.局域预测方法是目前最常用的一种非线性时间序列预测方法。应用局域线性预测方法和局域支持向量机预测方法来预测实测激光数据。应用基于预测效果的选取嵌入维数的方法来确定该组激光数据的最优嵌入维数。应用基于信息准则的局域预测法邻近点的选取方法来确定局域线性预测法与局域支持向量机预测法的邻近点个数和邻域半径。实验结果表明邻近点优化后的局域线性预测法和局域支持向量机预测法都能够有效地预测该组激光数据,一步和多步预测性能较好,在预测精度较高的条件下,计算量较小。
6.心率变异信号可被解释为由低维非线性动力机制控制。验证呼吸和心率的耦合作用越来越受到人们的关注。应用基于预测效果的嵌入维数的选取方法来确定心率变异信号的嵌入维数。分别使用单变量时间序列和多变量时间序列,应用结合预测效果和替代数据的非线性检测方法来检测生理时间序列的非线性确定性成分。应用粗粒化方法和条件熵来分析心率信号、呼吸信号与血氧浓度信号的相互关系。实验结果表明:心率信号和呼吸信号中含有非线性确定性成分,心率信号与呼吸信号不是独立的,它们互相影响,心率信号和呼吸信号可看作起源于同一动力系统的两个变量。
7.系统地应用非线性时间序列分析方法来分析实测的网络流量序列。为了对网络流量数据相空间重构,应用基于预测效果的选取嵌入维数的方法来确定最优嵌入维数。应用局域支持向量机预测方法来预测网路流量序列。并应用基于信息准则的局域预测法邻近点的选取方法来选取局域预测的邻近点个数。实验结果表明邻近点优化后的局域支持向量机预测方法能够有效地预测网络流量序列,归一化均方误差很小;局域支持向量机回归模型生成的时间序列具有与原网络流量时间序列相一致的概率分布。
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.
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