Denoising Nonlinear Time Series Using Singular Spectrum Analysis and Fuzzy Entropy

We present a hybrid singular spectrum analysis(SSA) and fuzzy entropy method to filter noisy nonlinear time series.With this approach,SSA decomposes the noisy time series into its constituent components including both the deterministic behavior and noise,while fuzzy entropy automatically differentiates the optimal dominant components from the noise based on the complexity of each component.We demonstrate the effectiveness of the hybrid approach in reconstructing the Lorenz and Mackey-GIass attractors,as well as improving the multi-step prediction quality of these two series in noisy environments.     

Quantifying information loss on chaotic attractors through recurrence networks

We propose an entropy measure for the analysis of chaotic attractors through recurrence networks which are un-weighted and un-directed complex networks constructed from time series of dynamical systems using specific criteria. We show that the proposed measure converges to a constant value with increase in the number of data points on the attractor (or the number of nodes on the network) and the embedding dimension used for the construction of the network, and clearly distinguishes between the recurrence network from chaotic time series and white noise. Since the measure is characteristic to the network topology, it can be used to quantify the information loss associated with the structural change of a chaotic attractor in terms of the difference in the link density of the corresponding recurrence networks. We also indicate some practical applications of the proposed measure in the recurrence analysis of chaotic attractors as well as the relevance of the proposed measure in the context of the general theory of complex networks.     

皮层脑电时间序列的相空间重构及非线性特征量的提取

对麻醉的SD大鼠在癫痫发作前后两种状态的皮层脑电(ECoG)的时间序列,用多种有效的方法和分析技术,使得大量的ECoG时间序列得以正确的分析,并得出重要的结论.首先利用延时坐标法重构吸引子;计算互信息函数,取互信息函数第一次达到最小值的延时为重构延时时间,提出将伪邻点法和Cao法相结合的方法确定最佳嵌入维数.然后采用非线性预报和替代数据法相结合的方法确定ECoG为混沌时间序列,从不同角度得出了ECoG不是低维混沌的结论.在ECoG相空间重构的基础上,计算了最大Lyapunov指数(LLE).应用了近似熵这一标量对ECoG进行刻画,计算结果表明:癫痫发作前的皮层脑电的最大Lyapunov指数和近似熵都明显地高于癫痫发作后的,这可能为理解癫痫发病机理,预报癫痫发作和治疗提供一定的思路. 关键词： 皮层脑电 互信息 伪邻点法 最大Lyapunov指数 近似熵     

A geometric theory of chaotic phase synchronization

A rigorous mathematical treatment of chaotic phase synchronization is still lacking, although it has been observed in many numerical and experimental studies. In this article we address the extension of results on phase synchronization in periodic oscillators to systems with phase coherent chaotic attractors with small phase diffusion. As models of such systems we consider special flows over diffeomorphisms in which the neutral direction is periodically perturbed. A generalization of the Averaging Theorem for periodic systems is used to extend Kuramoto's geometric theory of phase locking in periodically forced limit cycle oscillators to this class of systems. This approach results in reduced equations describing the dynamics of the phase difference between drive and response systems over long time intervals. The reduced equations are used to illustrate how the structure of a chaotic attractor is important in its response to a periodic perturbation, and to conclude that chaotic phase coherent systems may not always be treated as noisy periodic oscillators in this context. Although this approach is strictly justified for periodic perturbations affecting only the phase variable of a chaotic oscillator, we argue that these ideas are applicable much more generally.     

Impact load identification of composite structure using genetic algorithms

For structural health monitoring of composite structure, it is important to quickly and accurately identify the impact load whenever an impact event occurs. This paper proposes a genetic algorithms (GA)-based approach for impact load identification, which can identify the impact location and reconstruct the impact force history simultaneously. In this study, impact load is represented by a set of parameters, thus the impact load identification problem in both space (impact location) and time (impact force history) domains is transformed to a parameter identification problem. A forward model characterizes the dynamic response of the structure subject to a known impact force is incorporated in the identification procedure. By minimizing the difference between the analytical responses given by the forward model and the measured ones, GA adaptively identify the impact location and force history with its global search capability. This new impact identification approach is applied to a stiffened composite panel. The stiffened composite panel is modeled as an equivalent laminate with varying properties and the forward response is obtained by using an assumed modes approach. To improve the computational efficiency, micro-GA (μGA) is employed to perform the identification task. Numerical simulation studies are conducted to demonstrate the effectiveness and applicability of the proposed method.     

基于Cholesky分解的增量式RELM及其在时间序列预测中的应用

针对应用于混沌时间序列预测的正则极端学习机(RELM)网络结构设计问题,提出一种基于Cholesky分解的增量式RELM训练算法.该算法通过逐次增加隐层神经元的方式自动确定最佳的RELM网络结构,并以Cholesky分解方式计算其输出权值,有效减小了隐层神经元递增过程的计算代价.混沌时间序列预测实例表明,该算法可有效实现最佳RELM网络结构的自动确定,且计算效率高.利用该算法训练后的RELM预测模型具有预测精度高的优点,适用于混沌时间序列预测. 关键词： 神经网络 极端学习机 混沌时间序列 时间序列预测     

Structural health monitoring and damage assessment using a novel time series analysis methodology with sensor clustering

This study presents a novel time series analysis methodology to detect, locate, and estimate the extent of the structural changes (e.g. damage). In this methodology, ARX models (Auto-Regressive models with eXogenous input) are created for different sensor clusters by using the free response of the structure. The output of each sensor in a cluster is used as an input to the ARX model to predict the output of the reference channel of that sensor cluster. Two different approaches are used for extracting Damage Features (DFs) from these ARX models. For the first approach, the coefficients of the ARX models are directly used as the DFs. It is shown with a 4 dof numerical model that damage can be identified, located and quantified for simple models and noise free data. To consider the effects of the noise and model complexity, a second approach is presented based on using the ARX model fit ratios as the DFs. The second approach is first applied to the same 4 DOF numerical model and to the numerical data coming from an international benchmark study for noisy conditions. Then, the methodology is applied to the experimental data from a large scale laboratory model. It is shown that the second approach performs successfully for different damage cases to identify and locate the damage using numerical and experimental data. Furthermore, it is observed that the DF level is a good indicator for estimating the extent of the damage for these cases. The potential and advantages of the methodology are discussed along with the analysis results. The limitations of the methodology, recommendations, and future work are also addressed.     

Noninvertibility and resonance in discrete-time neural networks for time-series processing

We present a computer-assisted study emphasizing certain elements of the dynamics of artificial neural networks (ANNs) used for discrete time-series processing and nonlinear system identification. The structure of the network gives rise to the possibility of multiple inverses of a phase point backward in time; this is not possible for the continuous-time system from which the time series are obtained. Using a two-dimensional illustrative model in an oscillatory regime, we study here the interaction of attractors predicted by the discrete-time ANN model (invariant circles and periodic points locked on them) with critical curves. These curves constitute a generalization of critical points for maps of the interval (in the sense of Julia-Fatou); their interaction with the model-predicted attractors plays a crucial role in the organization of the bifurcation structure and ultimately in determining the dynamic behavior predicted by the neural network.     

混沌时间序列的时滞判定

运用重构混沌吸引子的嵌入技术，开发了提取混沌时间序列信息的宏观信息熵方法，用以判定时滞的存在性和时滞的大小．仿真实验说明了本文方法的有效性 关键词：     

用时滞和阶跃序列组合生成网格多涡卷蔡氏混沌吸引子

提出了在规范型蔡氏电路中生成两种不同类型网格多涡卷混沌吸引子的新方法.与现有文献报道仅构造同一类型非线性函数产生多涡卷混沌吸引子的主要差别在于,这种方法能在一个蔡氏电路中同时构造时滞序列和阶跃序列,并通过其不同的组合方式来扩展相空间中指标2的鞍焦平衡点,从而生成两种不同类型的网格多涡卷混沌吸引子.理论分析、数值模拟和电路实验结果证实了该方法的可行性. 关键词： 规范型蔡氏电路 网格多涡卷混沌吸引子 时滞序列和阶跃序列 电路实现     

Interdisciplinary application of nonlinear time series methods

This paper reports on the application to field measurements of time series methods developed on the basis of the theory of deterministic chaos. The major difficulties are pointed out that arise when the data cannot be assumed to be purely deterministic and the potential that remains in this situation is discussed. For signals with weakly nonlinear structure, the presence of nonlinearity in a general sense has to be inferred statistically. The paper reviews the relevant methods and discusses the implications for deterministic modeling. Most field measurements yield nonstationary time series, which poses a severe problem for their analysis. Recent progress in the detection and understanding of nonstationarity is reported. If a clear signature of approximate determinism is found, the notions of phase space, attractors, invariant manifolds, etc., provide a convenient framework for time series analysis. Although the results have to be interpreted with great care, superior performance can be achieved for typical signal processing tasks. In particular, prediction and filtering of signals are discussed, as well as the classification of system states by means of time series recordings.     

On entropy, financial markets and minority games

The paper builds upon an earlier statistical analysis of financial time series with Shannon information entropy, published in [L. Molgedey, W. Ebeling, Local order, entropy and predictability of financial time series, European Physical Journal B—Condensed Matter and Complex Systems 15/4 (2000) 733-737]. A novel generic procedure is proposed for making multistep-ahead predictions of time series by building a statistical model of entropy. The approach is first demonstrated on the chaotic Mackey-Glass time series and later applied to Japanese Yen/US dollar intraday currency data. The paper also reinterprets Minority Games [E. Moro, The minority game: An introductory guide, Advances in Condensed Matter and Statistical Physics (2004)] within the context of physical entropy, and uses models derived from minority game theory as a tool for measuring the entropy of a model in response to time series. This entropy conditional upon a model is subsequently used in place of information-theoretic entropy in the proposed multistep prediction algorithm.     

基于混沌算子网络的时间序列多步预测研究

结合相空间重构理论和时间序列分析理论,提出一种用于时间序列多步预测的网络模型.网络采用多个混沌算子加权求和的形式构成.网络各层单元采用固定权值连接,混沌算子的控制参数利用混沌优化算法进行训练调节,从而控制预测网络的动力学行为.利用已知时间序列数据构造出训练样本,训练样本在网络训练过程中仅使用一次,促使网络的动力学特性随时间的推移而变化,并逐渐逼近被预测系统的动力学特性,最终完成对未来时刻数据的预测.在对理论数据进行预测分析时,通过计算预测序列的Lyapunov指数验证了预测网络的有效性.在对实际时间序列的预测过程中,该网络表现出了良好的预测性能.仿真结果表明,该预测网络可对多种时间序列在一定的预测步长范围内实现有效的预测.     

混沌的可加性

从理论上推导了由几个混沌时间序列构成的新序列的动力学特性，提出了可加性的概念.计 算机仿真结果说明，新的时间序列具有分维吸引子，对初始值很敏感，两条邻近轨道随着时 间的增加呈现指数分离，具有短期可预测性，因此新序列也是混沌序列，即混沌具有可加性 .混沌时间序列的可加性理论不仅具有重要的理论意义，而且具有重要的应用价值，它是研 究多跳频电台的跳频码的理论基础. 关键词： 混沌时间序列 吸引子 分维 可加性     

Multi-scroll hidden attractors and multi-wing hidden attractors in a 5-dimensional memristive system

A novel 5-dimensional(5D) memristive chaotic system is proposed, in which multi-scroll hidden attractors and multiwing hidden attractors can be observed on different phase planes. The dynamical system has multiple lines of equilibria or no equilibrium when the system parameters are appropriately selected, and the multi-scroll hidden attractors and multi-wing hidden attractors have nothing to do with the system equilibria. Particularly, the numbers of multi-scroll hidden attractors and multi-wing hidden attractors are sensitive to the transient simulation time and the initial values. Dynamical properties of the system, such as phase plane, time series, frequency spectra, Lyapunov exponent, and Poincar′e map, are studied in detail. In addition, a state feedback controller is designed to select multiple hidden attractors within a long enough simulation time. Finally, an electronic circuit is realized in Pspice, and the experimental results are in agreement with the numerical ones.     

Generalized Multivariate Singular Spectrum Analysis for Nonlinear Time Series De-Noising and Prediction

Singular spectrum analysis and its multivariate or multichannel singular spectrum analysis(MSSA)variant are effective methods for time series representation,denoising and prediction,with broad application in many fields.However,a key element in MSSA is singular value decomposition of a high-dimensional matrix stack of component matrices,where the spatial(structural)information among multivariate time series is lost or distorted.This vector-space model also leads to difficulties including high dimensionality,small sample size,and numerical instability when applied to multi-dimensional time series.We present a generalized multivariate singular spectrum analysis(GMSSA)method to simultaneously decompose multivariate time series into constituent components,which can overcome the limitations of conventional multivariate singular spectrum analysis.In addition,we propose a Samp En-based method to determine the dominant components in GMSSA.We demonstrate the effectiveness and efficiency of GMSSA to simultaneously de-noise multivariate time series for attractor reconstruction,and to predict both simulated and real-world multivariate noisy time series.     

基于Bernstein多项式的自适应混沌时间序列预测算法

提出了利用Bernstein多项式对混沌时间序列的动力学方程进行建模的方法,并将该方法与递推最小二乘(RLS)算法相结合,从而可以自适应地逼近混沌时间序列的动力学特性,以达到预测的目的.理论分析和仿真实验表明该方法对一些常见的混沌时间序列具有较高的预测精度和较理想的准确预测率.由于RLS算法的收敛速度较快,因此该方法比较适合于对短混沌时间序列进行实时预测. 关键词： 混沌 预测 Bernstein多项式 RLS算法     

Testing statistical significance of multivariate time series analysis techniques for epileptic seizure prediction

Nonlinear time series analysis techniques have been proposed to detect changes in the electroencephalography dynamics prior to epileptic seizures. Their applicability in practice to predict seizure onsets is hampered by the present lack of generally accepted standards to assess their performance. We propose an analytic approach to judge the prediction performance of multivariate seizure prediction methods. Statistical tests are introduced to assess patient individual results, taking into account that prediction methods are applied to multiple time series and several seizures. Their performance is illustrated utilizing a bivariate seizure prediction method based on synchronization theory.     

Joining of components of complex structures for improved dynamic response

The goal of this work is to provide a method for choosing joining (e.g., bolt) locations for attaching structural reinforcements onto complex structures. The joining locations affect structural performance criteria such as the frequency response and the static compliance of the modified structure. One approach to finding improved/optimal joining locations is to place the joints such that the total amount of energy input into the structure (from external forces) is lowered/minimized, thus ensuring that the performance of the structure is least affected by the structural modifications. However, such an approach does not account for the stresses in the joints. Therefore, in this work, the amount of strain energy concentrated in the joints is also considered. The cost function for this optimization problem is then composed of two energies. These energies are different for the undamped and damped cases. Herein, the focus is on the (more realistic) damped case. The cost function is minimized by a modified optimality criteria method. This process is time consuming because it requires the calculation of sensitivities of the joint strain energy, which in turn requires the calculation of the displacements of all candidate joint locations by using the system-level mass and stiffness matrices and force vector (at each frequency in the range of interest). To address this issue, a series of complex algebraic manipulations and approximations are used to significantly reduce the computational cost. In addition, for the case where structural and geometrical variations are necessary, parametric reduced-order models are used to compute the cost function with further significant gains in computational speed. Numerical results for improved/optimal joining are presented for representative complex structures with structural variabilities.