Estimation of entropies and dimensions by nonlinear symbolic time series analysis

Symbolic nonlinear time series analysis methods have the potential for analyzing nonlinear data efficiently with low sensitivity to noise. In symbolic nonlinear time series analysis a time series for a fixed delay is partitioned into a small number (called the alphabet size) of cells labeled by symbols, creating a symbolic time series. Symbolic methods involve computing the statistics of words made from the symbolic time series. Specifically, the Shannon entropy of the distribution of possible words for a range of word lengths is computed. The rate of increase of the entropy with word length is the metric (Kolmogorov-Sinai) entropy. Methods of computing the metric entropy for flows as well as for maps are shown. A method of computing the information dimension appropriate to symbolic analysis is proposed. In terms of this formulation, the information dimension is determined by the scaling of entropy as alphabet size is modestly increased, using the information obtained from large word length. We discuss the role of sampling time and the issue of using these methods when there may be no generating partition.     

Mixing Bandt-Pompe and Lempel-Ziv approaches: another way to analyze the complexity of continuous-state sequences

In this paper, we propose to mix the approach underlying Bandt-Pompe permutation entropy with Lempel-Ziv complexity, to design what we call Lempel-Ziv permutation complexity. The principle consists of two steps: (i) transformation of a continuous-state series that is intrinsically multivariate or arises from embedding into a sequence of permutation vectors, where the components are the positions of the components of the initial vector when re-arranged; (ii) performing the Lempel-Ziv complexity for this series of ‘symbols’, as part of a discrete finite-size alphabet. On the one hand, the permutation entropy of Bandt-Pompe aims at the study of the entropy of such a sequence; i.e., the entropy of patterns in a sequence (e.g., local increases or decreases). On the other hand, the Lempel-Ziv complexity of a discrete-state sequence aims at the study of the temporal organization of the symbols (i.e., the rate of compressibility of the sequence). Thus, the Lempel-Ziv permutation complexity aims to take advantage of both of these methods. The potential from such a combined approach – of a permutation procedure and a complexity analysis – is evaluated through the illustration of some simulated data and some real data. In both cases, we compare the individual approaches and the combined approach.     

Variance estimators for the Lempel-Ziv entropy rate estimator

Lempel and Ziv's 1976 algorithm provides an easy-to-compute way to automatically estimate the entropy rate for symbolic time series, requiring no free parameters. Here we derive an analytical variance estimate for the Lempel-Ziv entropy rate estimator that is easily computable from observations with negligible extra effort beyond the entropy rate itself, and compare to another procedure, a time-series-based bootstrap method. These provide a justified "error bar" quantifying the size of expected fluctuations on the estimate itself, given by the single time series of symbols.     

累积放电模型及其符号动力学研究

基于累积释放模型提出了一种累积放电模型.相比于累积释放模型, 累积放电模型无须变化的阈值调制, 即可出现多种状态, 例如混沌态、锁频等. 利用符号动力学对其进行研究, 发现在一定的参数条件下, 模型的输出符号序列可以被用于监测模型参数的变化, 而且与神经系统的测量相似, 都具有很高的分辨率. 计算机仿真和电路实验得到的结果也验证了上述说法. 电路实验结果显示模型的输出符号序列对输入频率的分辨率最高可以达到0.05 Hz, 对电流幅值的分辨率可达到1 μA, 并且都具有很大的动态范围. 关键词： 符号动力学 混沌 累积释放模型 非线性电路     

基于符号向量动力学的耦合映像格子参数估计

本文,将符号动力学推广到耦合映像格子中,以Logistic映射下耦合映像格子为研究对象,研究控制参数对符号向量序列动力学特性的影响.通过研究耦合映像格子逆函数,给出耦合映像格子的遍历条件.进一步,将给出系统初始向量,禁止字以及控制参数的符号向量序列描述方法,并最终给出基于符号向量动力学的耦合映像格子控制参数估计方法.实验结果表明,根据本文算法可以有效建立符号序列和耦合映像格子控制参数之间的对应关系,能够更好地刻画了实际模型的物理过程. 关键词： 符号向量动力学 耦合映像格子 参数估计 遍历性     

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.     

Bayesian Tracking in an Uncertain Shallow Water Environment

A Bayesian source tracking approach is developed to track a moving acoustic source in an uncertain ocean environment.This approach treats the environmental parameters(e.g.,water depth,sediment and bottom parameters) at the source location and the source parameters(e.g.,source depth,range and speed) as unknown random variables that evolve as the source moves.To track a target with low signal-to-noise ratio(SNR),acoustic signals from a series of observations are treated in a simultaneous inversion.This allows real-time updating of the environment and accurate tracking of the moving source.The noise signals radiated from a surface ship target are processed and analyzed.It is found that the Bayesian source tracking method could enhance the localization accuracy in an uncertain water environment and low SNR.     

Thermodynamics of RNA/DNA hybridization in high-density oligonucleotide microarrays

We analyze a series of publicly available controlled experiments (Latin square) on Affymetrix high-density oligonucleotide microarrays using a simple physical model of the hybridization process. We plot for each gene the signal intensity vs. the hybridization free energy of RNA/DNA duplexes in solution, for perfect matching and mismatching probes. Both values tend to align on a single master curve in good agreement with Langmuir adsorption theory, provided one takes into account the decrease of the effective target concentration due to target–target hybridization in solution. We give an example of a deviation from the expected thermodynamical behavior for the probe set 1091_at due to annotation problems, i.e., the surface-bound probe is not the exact complement of the target RNA sequence, because of errors present in public databases at the time when the array was designed. We show that the parametrization of the experimental data with RNA/DNA free energy improves the quality of the fits and enhances the stability of the fitting parameters compared to previous studies.     

Polytetrahedral clusters.

By studying the structures of clusters bound by a model potential that favors polytetrahedral order, we find a previously unknown series of "magic numbers" (i.e., sizes of special stability) whose polytetrahedral structures are characterized by disclination networks that are analogous to hydrocarbons.     

Short-term forecasting of life-threatening cardiac arrhythmias based on symbolic dynamics and finite-time growth rates

Ventricular tachycardia or fibrillation (VT-VF) as fatal cardiac arrhythmias are the main factors triggering sudden cardiac death. The objective of this study is to find early signs of sustained VT-VF in patients with an implanted cardioverter-defibrillator (ICD). These devices are able to safeguard patients by returning their hearts to a normal rhythm via strong defibrillatory shocks; additionally, they store the 1000 beat-to-beat intervals immediately before the onset of a life-threatening arrhythmia. We study these 1000 beat-to-beat intervals of 17 chronic heart failure ICD patients before the onset of a life-threatening arrhythmia and at a control time, i.e., without a VT-VF event. To characterize these rather short data sets, we calculate heart rate variability parameters from the time and frequency domain, from symbolic dynamics as well as the finite-time growth rates. We find that neither the time nor the frequency domain parameters show significant differences between the VT-VF and the control time series. However, two parameters from symbolic dynamics as well as the finite-time growth rates discriminate significantly both groups. These findings could be of importance in algorithms for next generation ICD's to improve the diagnostics and therapy of VT-VF.     

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.     

Identifying parameter by identical synchronization between different systems

In this paper, parameters of a given (chaotic) dynamical system are estimated from time series by using identical synchronization between two different systems. This technique is based on the invariance principle of differential equations, i.e., a dynamical Lyapunov function involving synchronization error and the estimation error of parameters. The control used in this synchronization consists of feedback and adaptive control loop associated with the update law of estimation parameters. Our estimation process indicates that one may identify dynamically all unknown parameters of a given (chaotic) system as long as time series of the system are available. Lorenz and Rossler systems are used to illustrate the validity of this technique. The corresponding numerical results and analysis on the effect of noise are also given.     

利用复杂网络研究中国温度序列的拓扑性质

依据粗粒化方法,将中国1961—2002年逐日平均温度序列转化为由5个特征字符{R,r,e,d,D}构成的温度符号序列.以符号序列中的125种3字串组成的温度波动模态为网络的节点(即连续4d的温度波动组合),并按照时间顺序连边,构建有向加权的温度波动网络,进而将温度波动模态间的相互作用等综合信息蕴含于网络的拓扑结构之中.对随机序列和Lorenz系统的混沌序列分别构建随机和混沌波动网络.计算三种网络的度与度分布、聚类系数、最短路径长度等动 关键词： 气候变化 气候复杂网络 拓扑结构     

On a recursive method for the estimation of unknown parameters of partially observed chaotic systems

We investigate a recently proposed method for on-line parameter estimation and synchronization in chaotic systems. This novel technique has been shown effective to estimate a single unknown parameter of a primary chaotic system with known functional form that is only partially observed through a scalar time series. It works by periodically updating the parameter of interest in a secondary system, with the same functional form as the primary one but no explicit coupling between their dynamic variables, in order to minimize a suitably defined cost function. In this paper, we review the basics of the method, and investigate its robustness and new extensions. In particular, we study the performance of the novel technique in the presence of noise (either observational, i.e., an additive contamination of the observed time series, or dynamical, i.e., a random perturbation of the system dynamics) and when there is a mismatch between the primary and secondary systems. Numerical results, including comparisons with other techniques, are presented. Finally, we investigate the extension of the original method to perform the estimation of two unknown parameters and illustrate its effectiveness by means of computer simulations.     

Multiscale reconstruction of time series

A new method is proposed which allows a reconstruction of time series based on higher order multiscale statistics given by a hierarchical process. This method is able to model the time series not only on a specific scale but for a range of scales. It is possible to generate complete new time series, or to model the next steps for a given sequence of data. The method itself is based on the joint probability density which can be extracted directly from given data, thus no estimation of parameters is necessary. The results of this approach are shown for a real world dataset, namely for turbulence. The unconditional and conditional probability densities of the original and reconstructed time series are compared and the ability to reproduce both is demonstrated. Therefore in the case of Markov properties the method proposed here is able to generate artificial time series with correct n-point statistics.     

Relationships between (1)H NMR relaxation data and some technological parameters of meat: a chemometric approach

In this paper chemometrics (ANOVA and PCR) is used to measure unbiased correlations between NMR spin-echo decays of pork M. Longissimus dorsi obtained through Carr-Purcell-Meiboom-Gill (CPMG) experiments at low frequency (20 MHz) and the values of 14 technological parameters commonly used to assess pork meat quality. On the basis of the ANOVA results, it is also found that the CPMG decays of meat cannot be best interpreted with a "discrete" model (i.e., by expanding the decays in a series of a discrete number of exponential components, each with a different transverse relaxation time), but rather with a "continuous" model, by which a continuous distribution of T(2)'s is allowed. The latter model also agrees with literature histological results.     

Anomalous diffusion of inertial, weakly damped particles

The anomalous (i.e., non-Gaussian) dynamics of particles subject to a deterministic acceleration and a series of "random kicks" is studied. Based on an extension of the concept of continuous time random walks to position-velocity space, a new fractional equation of the Kramers-Fokker-Planck type is derived. The associated collision operator necessarily involves a fractional substantial derivative, representing important nonlocal couplings in time and space. For the force-free case, a closed solution is found and discussed.     

Entropy and optimal partition for data analysis

The concept of symbolic dynamics, entropy and complexity measures has been widely utilized for the analysis of measured time series. However, little attention as been devoted to investigate the effects of choosing different partitions to obtain the coarse-grained symbolic sequences. Because the theoretical concepts of generating partitions mostly fail in the case of empirical data, one commonly introduces a homogeneous partition which ensures roughly equidistributed symbols. We will show that such a choice may lead to spurious results for the estimated entropy and will not fully reveal the randomness of the sequence. Received 1st September 2000     

A long-range memory stochastic model of the return in financial markets

We present a nonlinear stochastic differential equation (SDE) which mimics the probability density function (PDF) of the return and the power spectrum of the absolute return in financial markets. Absolute return as a measure of market volatility is considered in the proposed model as a long-range memory stochastic variable. The SDE is obtained from the analogy with an earlier proposed model of trading activity in the financial markets and generalized within the nonextensive statistical mechanics framework. The proposed stochastic model generates time series of the return with two power law statistics, i.e., the PDF and the power spectral density, reproducing the empirical data for the one-minute trading return in the NYSE.     

A relation between multipath group velocity, mode number, and ray cycle distance

Weston's ray invariant or "characteristic time" in a range-dependent environment is exactly equivalent to the Wentzel-Kramers-Brillouin phase integral for ducted normal modes. By considering a ray element it is shown that the ray invariant can also be written in terms of ray cycle distance and cycle time. This leads to a useful formula for group velocity in terms of cycle distance and mode number. Drawing a distinction between the ray and wave interpretation, the Airy phase (i.e., the existence of a group velocity minimum) can be included in this approach. Favorable comparisons are made with group velocities derived from a normal mode model. The relationship is valid for variable sound speed and variable bathymetry, and this is demonstrated numerically. The formula is applicable to active sonar, multipath pulse shape, target signatures, reverberation, tomography, and underwater communications.