On the control of chaotic systems via symbolic time series analysis

Symbolic analysis of time series is extended to systems with inputs, in order to obtain input/output symbolic models to be used for control policy design. For that, the notion of symbolic word is broadened to possibly include past input values. Then, a model is derived in the form of a controlled Markov chain, i.e., transition probabilities are conditioned on the control value. The quality of alternative models with different word length and alphabet size is assessed by means of an indicator based on Shannon entropy. A control problem is formulated, with the goal of confining the system output in a smaller domain with respect to that of the uncontrolled case. Solving this problem (by means of a suitable numerical method) yields the relevant control policy, as well as an estimate of the probability distribution of the output of the controlled system. Three examples of application (based on the analysis of time series synthetically generated by the logistic map, the Lorenz system, and an epidemiological model) are presented and used to discuss the features and limitations of the method.     

Editorial

Symbolic dynamics is a powerful tool in the study of dynamical systems. The purpose of symbolic dynamics is to provide a simplified picture of complicated dynamics, that gives some insight into its complexity. To this end, the state space of the system is partitioned in a finite number of pieces, and the exact trajectories of individual points are traded off by the trajectory relative to that partition. These so-called coarse-grained trajectories turn out to be realisations of a stationary random process with a finite alphabet. In particular, the entropy of a dynamical system can be approximated by the Shannon entropy of any of its symbolic dynamics (the finer the partition, the better the approximation). Today, symbolic dynamics is an independent field of theoretical physics and applied mathematics with applications to such important disciplines as cryptology, time series analysis, and data-compression.     

Quantifying heart rate dynamics using different approaches of symbolic dynamics

The analysis of symbolic dynamics applied to physiological time series is able to retrieve information about dynamical properties of the underlying system that cannot be gained with standard methods like e.g. spectral analysis. Different approaches for the transformation of the original time series to the symbolic time series have been proposed. Yet the differences between the approaches are unknown. In this study three different transformation methods are investigated: (1) symbolization according to the deviation from the average time series, (2) symbolization according to several equidistant levels between the minimum and maximum of the time series, (3) binary symbolization of the first derivative of the time series. Furthermore, permutation entropy was used to quantify the symbolic series. Each method was applied to the cardiac interbeat interval series RR _i and its difference ΔRR _I of 17 healthy subjects obtained during head-up tilt testing. The symbolic dynamics of each method is analyzed by means of the occurrence of short sequences (“words”) of length 3. The occurrence of words is grouped according to words without variations of the symbols (0V%), words with one variation (1V%), two like variations (2LV%) and two unlike variations (2UV%). Linear regression analysis showed that for method 1 0V%, 1V%, 2LV% and 2UV% changed with increasing tilt angle. For method 2 0V%, 2LV% and 2UV% changed with increasing tilt angle and method 3 showed changes for 0V% and 1V%. Furthermore, also the permutation entropy decreased with increasing tilt angle. In conclusion, all methods are capable of reflecting changes of the cardiac autonomic nervous system during head-up tilt. All methods show that even the analysis of very short symbolic sequences is capable of tracking changes of the cardiac autonomic regulation during head-up tilt testing.     

Symbolic dynamics and topological entropy at the onset of pruning

The topological entropy and pruning rules are investigated for two-dimensional smooth maps at the onset of pruning. Typically the difference of the parameter-dependent topological entropy from its maximum value increases with a power law. Superimposed on this decrease, there are periodic or quasiperiodic oscillations on a logarithmic scale. Both, the scaling exponent and the periodicity are determined by the Lyapunov exponents of the first pruned orbit and the minimal number of letters in the alphabet of the symbolic dynamics. If, at the onset of pruning, the averaged Lyapunov exponent is sufficiently large and the first pruned orbit is homoclinic, the entropy function of area-preserving maps exhibits a series of plateaux. On the plateaux, the symbolic dynamics can be described by finitely many finite forbidden words. There is a series of plateaux which, in different systems, can be described by the same type of forbidden words.     

混沌时序相空间重构参数确定的信息论方法

根据信息论基本原理，研究了混沌时间序列相空间重构参数延迟时间和嵌入维数的选取.提出了用符号分析的方法计算互信息函数，确定出延迟时间，在此基础上，提出了一种估计嵌入维数的信息论方法，即根据重构向量条件熵随向量维数的变化关系来确定嵌入维数，通过对几种典型混沌动力学系统的数值验证，结果表明该方法能够确定出合适的相空间重构嵌入维数. 关键词： 混沌 相空间重构 互信息 条件熵 符号分析     

一种确定混沌伪随机序列复杂度的模糊关系熵测度

将模糊关系的概念引入混沌伪随机序列复杂度的测度方法之中,提出了一种新的混沌伪随机序列复杂度测度方法——模糊关系熵(fuzzy relationship entropy,简记为F-REn)测度方法,并推导了F-REn的两个基本性质.仿真结果表明,该测度方法能够有效测度混沌伪随机序列的复杂度,与近似熵(ApEn)测度方法和符号熵测度方法相比,F-REn测度具有更加好的对序列符号空间的适用性、更加小的对测量维度的敏感性和更加强的对分辨率参数的鲁棒性. 关键词： 混沌伪随机序列 模糊理论 复杂度     

Topological permutation entropy

Permutation entropy quantifies the diversity of possible ordering of the successively observed values a random or deterministic system can take, just as Shannon entropy quantifies the diversity of the values themselves. When the observable or state variable has a natural order relation, making permutation entropy possible to compute, then the asymptotic rate of growth in permutation entropy with word length forms an alternative means of describing the intrinsic entropy rate of a source. Herein, extending a previous result on metric entropy rate, we show that the topological permutation entropy rate for expansive maps equals the conventional topological entropy rate familiar from symbolic dynamics. This result is not limited to one-dimensional maps.     

Quantifying the complexity of excised larynx vibrations from high-speed imaging using spatiotemporal and nonlinear dynamic analyses

In this paper, we investigate the biomechanical applications of spatiotemporal analysis and nonlinear dynamic analysis to quantitatively describe regular and irregular vibrations of twelve excised larynges from high-speed image recordings. Regular vibrations show simple spatial symmetry, temporal periodicity, and discrete frequency spectra, while irregular vibrations show complex spatiotemporal plots, aperiodic time series, and broadband spectra. Furthermore, the global entropy and correlation length from spatiotemporal analysis and the correlation dimension from nonlinear dynamic analysis reveal a statistical difference between regular and irregular vibrations. In comparison with regular vibrations, the global entropy and correlation dimension of irregular vibrations are statistically higher, while the correlation length is significantly lower. These findings show that spatiotemporal analysis and nonlinear dynamic analysis are capable of describing the complex dynamics of vocal fold vibrations from high-speed imaging and may potentially be helpful for understanding disordered behaviors in biomedical laryngeal systems.     

How complex is the dynamics of Peptide folding?

Classical molecular dynamics simulations of the folding of alanine peptides in aqueous solution are analyzed by constructing a deterministic model of the dynamics, using methods from nonlinear time series analysis. While the dimension of the free energy landscape increases with system size, a Lyapunov analysis shows that the effective dimension of the dynamic system is rather small and even decreases with chain length. The observed reduction of phase space is a nonlinear cooperative effect that is caused by intramolecular hydrogen bonds that stabilize the secondary structure of the peptides.     

Power-law regularities in human language

Complex structure of human language enables us to exchange very complicated information. This communication system obeys some common nonlinear statistical regularities. We investigate four important long-range features of human language. We perform our calculations for adopted works of seven famous litterateurs. Zipf’s law and Heaps’ law, which imply well-known power-law behaviors, are established in human language, showing a qualitative inverse relation with each other. Furthermore, the informational content associated with the words ordering, is measured by using an entropic metric. We also calculate fractal dimension of words in the text by using box counting method. The fractal dimension of each word, that is a positive value less than or equal to one, exhibits its spatial distribution in the text. Generally, we can claim that the Human language follows the mentioned power-law regularities. Power-law relations imply the existence of long-range correlations between the word types, to convey an especial idea.     

DC-DC变换器的符号时间序列描述及模块熵分析

本文提出了一种采用符号时间序列和熵理论分析DC-DC变换器非线性行为的方法.该方法首先用离散时间序列描述非线性连续系统,然后将其转换为由简单字符构成的符号序列,再用信息学方法计算出该符号序列的模块熵,从而得到一种新的可量化的非线性动力学行为统计指标.文中以一阶电压反馈DCM和二阶电流反馈CCM Boost变换器为例进行研究.研究结果表明,模块熵这种粗粒化的统计分析方法,能够量化DC-DC变换器的倍周期分岔和混沌行为,且能够准确地确定混沌行为的发生,是一种尚未在DC-DC变换器中提出的简单、实用的分析方法. 关键词： 符号时间序列 符号动力学 模块熵 Lyapunov指数     

Symbolic dynamics of one-dimensional maps: Entropies,finite precision,and noise

In the study of nonlinear physical systems, one encounters apparently random or chaotic behavior, although the systems may be completely deterministic. Applying techniques from symbolic dynamics to maps of the interval, we compute two measures of chaotic behavior commonly employed in dynamical systems theory: the topological and metric entropies. For the quadratic logistic equation, we find that the metric entropy converges very slowly in comparison to maps which are strictly hyperbolic. The effects of finite precision arithmetric and external noise on chaotic behavior are characterized with the symbolic dynamics entropies. Finally, we discuss the relationship of these measures of chaos to algorithmic complexity, and use algorithmic information theory as a framework to discuss the construction of models for chaotic dynamics.     

Estimating good discrete partitions from observed data: symbolic false nearest neighbors

A symbolic analysis of observed time series requires a discrete partition of a continuous state space containing the dynamics. A particular kind of partition, called "generating," preserves all deterministic dynamical information in the symbolic representation, but such partitions are not obvious beyond one dimension. Existing methods to find them require significant knowledge of the dynamical evolution operator. We introduce a statistic and algorithm to refine empirical partitions for symbolic state reconstruction. This method optimizes an essential property of a generating partition, avoiding topological degeneracies, by minimizing the number of "symbolic false nearest neighbors." It requires only the observed time series and is sensible even in the presence of noise when no truly generating partition is possible.     

变尺度概率净化法的优化

变尺度概率净化法是一种混沌时间序列非线性动力学降噪方法，该方法需对整个相空间点列做联合处理，因此算法的计算量和所需内存量会随着每个轨道点的修正参考点数和嵌入维数的增加而呈指数增长.根据变尺度概率净化法的特点，对前向概率和转移概率的估计方法作了一些改进，使算法的运算量减小到了原来的0.27左右，而降噪性能并没有下降，并提出了数据较长情况下的算法实现结构，大大降低了算法运行所需内存. 关键词： 混沌 非线性动力学 降噪 最大后验概率估计 变尺度概率净化     

基于时间序列符号化模式表征的有向加权复杂网络

时间序列复杂网络分析近些年已发展成为非线性信号分析领域的一个国际热点课题.为了能更有效地挖掘时间序列(特别是非线性时间序列)中的结构特征,同时简化时间序列分析的复杂度,提出了一种新的基于时间序列符号化结合滑窗技术模式表征的有向加权复杂网络建网方法.该方法首先按照等概率区段划分的方式将时间序列做符号化处理,结合滑窗技术确定不同时刻的符号化模式作为网络的节点;然后将待分析时间序列符号化模式的转换频次和方向作为网络连边的权重和方向,从而建立时间序列有向加权复杂网络.通过对Logistic系统不同参数设置对应的时间序列复杂网络建网测试结果表明,相比经典的可视图建网方法,本文方法的网络拓扑能更简洁、直观地展示时间序列的结构特征.进而,将本文方法应用于规则排列采集的自然风场信号分析,其网络特性指标能较准确地预测采集信号的排布规律,而可视图建网方法的网络特性指标没有任何规律性的结果.     

评价奇怪吸引子分形特征的Grassberger-Procaccia算法

基于Lorenz,Rssler和H啨ｎｏｎ三种典型的奇怪吸引子,全面分析了GrassbergerProcaccia(缩写GP)算法,详细讨论了采样数据量、延迟时间、重构相空间维数和线性区长度等参数对计算关联维数和Kolmogorov熵的影响,结果表明这些关键参数是相互关联的.通过分析关联积分谱的变化趋势,发现延迟时间与重构相空间维数对连续动力系统和离散动力系统的作用效果是不同的,且选择最佳延迟时间对计算关联维数的意义不大.指出了实际中应用GP算法应注意的问题 关键词： 奇怪吸引子 GrassbergerProcaccia算法 关联维数 Kolmogorov熵     

Measuring persistence in a stationary time series using the complex network theory

A growing interest exists currently in the analysis of time series by the complex network theory. Here we present a simple and quick way for mapping time series to complex networks. Using a simple rule allows us to transform time series into a textual sequence then we divide it into words with fixed size. Distinct words are nodes of the network, and we have complete control on the network scale by adjusting the word size. Two nodes are linked if their associated words co-occur in sequence. We show that the network topological measures quantify the persistence and the long range correlations in fractional Brownian processes. For a particular word size we assume some relations between the topological measures and the Hurst exponent which characterised the persistence in fractional Brownian processes.     

基于改进的符号转移熵的心脑电信号耦合研究

试图探究动力系统中的耦合关系一直以来都是国内外众多学者关注的热点,传统的时间序列符号化分析方法会使研究结果受序列非平稳性的严重影响,本文在原有转移熵的研究基础上,应用粗粒化提取,经过理论与实验的分析,发现心脑电信号耦合研究中的转移熵值在不同提取情况下对应不同的分布趋势,并选择效果最好的信号数据提取方法用在其后的应用分析中. 此外,对时间序列符号化方法提出改进,采用动态的自适应分割方法. 实验结果表明,无论清醒期还是睡眠期,改进的符号转移熵算法观测分析到的心脑电信号耦合作用更显著,能更好的捕捉到信号中的动态信息、系统动力学复杂性的改变,更利于医学临床实践应用中的检测,在分析非平稳的时间序列上具有更好的效果. 关键词： 心脑电信号 粗粒化 符号转移熵 基本尺度