Ordinal analysis of time series

In order to develop fast and robust methods for extracting qualitative information from non-linear time series, Bandt and Pompe have proposed to consider time series from the pure ordinal viewpoint. On the basis of counting ordinal patterns, which describe the up-and-down in a time series, they have introduced the concept of permutation entropy for quantifying the complexity of a system behind a time series. The permutation entropy only provides one detail of the ordinal structure of a time series. Here we present a method for extracting the whole ordinal information.     

A new complexity measure for time series analysis and classification

Complexity measures are used in a number of applications including extraction of information from data such as ecological time series, detection of non-random structure in biomedical signals, testing of random number generators, language recognition and authorship attribution etc. Different complexity measures proposed in the literature like Shannon entropy, Relative entropy, Lempel-Ziv, Kolmogrov and Algorithmic complexity are mostly ineffective in analyzing short sequences that are further corrupted with noise. To address this problem, we propose a new complexity measure ETC and define it as the “Effort To Compress” the input sequence by a lossless compression algorithm. Here, we employ the lossless compression algorithm known as Non-Sequential Recursive Pair Substitution (NSRPS) and define ETC as the number of iterations needed for NSRPS to transform the input sequence to a constant sequence. We demonstrate the utility of ETC in two applications. ETC is shown to have better correlation with Lyapunov exponent than Shannon entropy even with relatively short and noisy time series. The measure also has a greater rate of success in automatic identification and classification of short noisy sequences, compared to entropy and a popular measure based on Lempel-Ziv compression (implemented by Gzip).     

Event coincidence analysis for quantifying statistical interrelationships between event time series

Studying event time series is a powerful approach for analyzing the dynamics of complex dynamical systems in many fields of science. In this paper, we describe the method of event coincidence analysis to provide a framework for quantifying the strength, directionality and time lag of statistical interrelationships between event series. Event coincidence analysis allows to formulate and test null hypotheses on the origin of the observed interrelationships including tests based on Poisson processes or, more generally, stochastic point processes with a prescribed inter-event time distribution and other higher-order properties. Applying the framework to country-level observational data yields evidence that flood events have acted as triggers of epidemic outbreaks globally since the 1950s. Facing projected future changes in the statistics of climatic extreme events, statistical techniques such as event coincidence analysis will be relevant for investigating the impacts of anthropogenic climate change on human societies and ecosystems worldwide.     

气液两相流电导波动信号复杂性测度分析及其流型表征

为了考察从时间序列提取的复杂性测度与气液两相流流型变化之间的关系，本文首先讨论了三种复杂性测度(Lempel-Ziv复杂性、功率谱熵和近似熵)对周期信号、随机信号、混合随机信号和混沌信号的识别能力，然后分析了时间序列长度对复杂性计算的影响.在此基础上，从实际测量的80种垂直上升管中气液两相流电导波动信号中提取了这三种复杂性测度，结果表明：三种复杂度对两相流流型变化是敏感的，通过对三种复杂度随两相流流动参数变化规律分析，可以得到气液两相流动力学结构反演特征，为揭示气液两相流流型转化机理提供了一种有效的辅助诊断工具. 关键词： 气液两相流 Lempel和Ziv复杂性 功率谱熵 近似熵     

Measuring statistical dependences in a time series

We propose two methods to measure all (linear and nonlinear) statistical dependences in a stationary time series. Presuming ergodicity, the measures can be obtained from efficient numerical algorithms.     

混合交通流时间序列的去趋势波动分析

应用去趋势波动分析法研究交通流时间序列的复杂性,探讨了混合交通流时间序列演变行为的标度指数.根据标度指数的变化特征,进而揭示交通流时间序列所具有的长程相关性和短程相关性.通过分析发现,存在一密度ρ,当ρ₁<ρ<ρ₂时,交通流时间序列具有长程相关性;而当ρ<ρ₁或ρ>ρ₂时,交通流时间序列具有短程相关性,即密度的变化影响着标度指数的变化.另外分析了在不同慢车比率条件下时间序列的标度指数,发现慢车比率的变化 关键词： 混合交通流 去趋势波动分析 时间序列 长程相关     

Correlation between detrended fluctuation analysis and the Lempel–Ziv complexity in nonlinear time series analysis

We focus on the study of the correlation between the detrended fluctuation analysis (DFA) and the Lempel-Ziv complexity (LZC) in nonlinear time series analysis in this paper. Typical dynamical systems including logistic map and Duffing model are investigated. Moreover, the influences of the Gaussian random noise on both DFA and LZC are analyzed. The results show a high correlation between DFA and LZC, which can quantify the non-stationarity and the nonlinearity of the time series, respectively. With the enhancement of the random component, the exponent α and the normalized complexity index C show increasing trends. In addition, C is found to be more sensitive to the fluctuation in the nonlinear time series than α. Finally, the correlation between DFA and LZC is applied to the feature extraction of vibration signals for a reciprocating compressor gas valve, and an effective fault diagnosis result is obtained.     

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.     

基于复杂度分析冰芯和石笋代用资料时间序列的研究

运用动态非线性时间序列分析方法——Lemper-Ziv复杂度，分别计算了古里雅冰芯资料和北 京石花洞石笋资料的复杂度，探讨了复杂度的物理内涵及其在气候变化研究中的意义；小波 多尺度分解分析表明，复杂度序列具有明显的特征周期，存在780年、380年、160年和105年 的准周期.进一步研究发现，当取不同的窗口长度时，复杂度序列的特征基本相同；并且复 杂度序列的突变和气候突变在时间尺度上有着较好的对应关系.揭示了近千年来，在380年时 间尺度上的准周期振荡，其振幅一直是在减弱的，而且其固有周期频率逐渐减小.1900年复 杂度开始持续下降，1920年之后变化很小，类似于中国6世纪中期及12世纪中期的变暖期， 因此造成20世纪气候变暖的原因还有待于深入研究. 关键词： Lemper-Ziv复杂度 小波变换 气候 自然变率     

Language time series analysis

We use the detrended fluctuation analysis (DFA) and the Grassberger–Proccacia analysis (GP) methods in order to study language characteristics. Despite that we construct our signals using only word lengths or word frequencies, excluding in this way huge amount of information from language, the application of GP analysis indicates that linguistic signals may be considered as the manifestation of a complex system of high dimensionality, different from random signals or systems of low dimensionality such as the Earth climate. The DFA method is additionally able to distinguish a natural language signal from a computer code signal. This last result may be useful in the field of cryptography.     

Permutation entropy: a natural complexity measure for time series

We introduce complexity parameters for time series based on comparison of neighboring values. The definition directly applies to arbitrary real-world data. For some well-known chaotic dynamical systems it is shown that our complexity behaves similar to Lyapunov exponents, and is particularly useful in the presence of dynamical or observational noise. The advantages of our method are its simplicity, extremely fast calculation, robustness, and invariance with respect to nonlinear monotonous transformations.     

Segmentation and classification of time series using ordinal pattern distributions

The distribution of ordinal patterns in time series has been found to reflect important qualitative features of the underlying system dynamics. Abrupt changes in the dynamics typically result in clearly visible differences between the distributions before and after the break. Recurring dynamical regimes can be discovered by classifying the distributions in different parts of the time series. This paper discusses two algorithms which exploit the relation between ordinal pattern distributions and system dynamics for the segmentation and classification of time series. The first algorithm employs a kernel-based statistic, the Maximum Mean Discrepancy of ordinal pattern distributions, to detect and locate change points in the time series. The second algorithm uses clustering of the ordinal pattern distributions to classify time series segments with similar dynamics. The methodology is applied to various real-life time series from physiology and economics.     

Autocorrelation type,timescale and statistical property in financial time series

Earlier studies have documented that three types of autocorrelations exist in financial time series: sign, volatility, and return autocorrelation. In this paper, we examine how each type of the above autocorrelations affects the statistical properties of financial time series and its role in maintaining such statistical properties. Using three different shuffling series that correspondingly destroy each type of autocorrelation upon different timescales, we find that: (1) the statistical properties of the shuffling series significantly vary from the original ones; (2) volatility and return autocorrelations show greater impacts than sign autocorrelation; (3) the effects on the statistical properties are intensified as time scale expands; (4) the nonlinear component of autocorrelation is the major drive of the effect.     

Levels of complexity in turbulent time series for weakly and high Reynolds number

We use the detrended fluctuation analysis (DFA), the detrended cross correlation analysis (DCCA) and the magnitude and sign decomposition analysis to study the fluctuations in the turbulent time series and to probe long-term nonlinear levels of complexity in weakly and high turbulent flow. The DFA analysis indicate that there is a time scaling region in the fluctuation function, segregating regimes with different scaling exponents. We discuss that this time scaling region is related to inertial range in turbulent flows. The DCCA exponent implies the presence of power-law cross correlations. In addition, we conclude its multifractality for high Reynold’s number in inertial range. Further, we find that turbulent time series exhibit complex features by magnitude and sign scaling exponents.     

Detecting dynamical complexity changes in time series using the base-scale entropy

Timely detection of dynamical complexity changes in natural and man-made systems has deep scientific and practical meanings. We introduce a complexity measure for time series： the base-scale entropy. The definition directly applies to arbitrary real-word data. We illustrate our method on a practical speech signal and in a theoretical chaotic system. The results show that the simple and easily calculated measure of base-scale entropy can be effectively used to detect qualitative and quantitative dynamical changes.     

基于符号时间序列方法的开关变换器离散映射算法复杂度分析

开关变换器与其符号序列拓扑共轭,这使得开关变换器系统特性的研究可以归于符号序列的研究,从而得到更一般的结论.采用基于符号时间序列的算法复杂度来分析开关变换器的复杂行为,从具体内在结构得到开关变换器的本质特征.与统计复杂度不同,算法复杂度还可以定量地描述开关变换器的突变点和工作周期,为深入了解开关变换器的特性和混沌行为的利用提供理论依据.     

Fine-grained permutation entropy as a measure of natural complexity for time series

In a recent paper [2002 Phys. Rev. Lett. 88 174102], Bandt and Pompe propose permutation entropy (PE) as a natural complexity measure for arbitrary time series which may be stationary or nonstationary, deterministic or stochastic. Their method is based on a comparison of neighbouring values. This paper further develops PE, and proposes the concept of fine-grained PE (FGPE) defined by the order pattern and magnitude of the difference between neighbouring values. This measure excludes the case where vectors with a distinct appearance are mistakenly mapped onto the same permutation type, and consequently FGPE becomes more sensitive to the dynamical change of time series than does PE, according to our simulation and experimental results.     

Multifractal analysis of polyalanines time series

Multifractal properties of the energy time series of short α-helix structures, specifically from a polyalanine family, are investigated through the MF-DFA technique (multifractal detrended fluctuation analysis). Estimates for the generalized Hurst exponent h(q) and its associated multifractal exponents τ(q) are obtained for several series generated by numerical simulations of molecular dynamics in different systems from distinct initial conformations. All simulations were performed using the GROMOS force field, implemented in the program THOR. The main results have shown that all series exhibit multifractal behavior depending on the number of residues and temperature. Moreover, the multifractal spectra reveal important aspects of the time evolution of the system and suggest that the nucleation process of the secondary structures during the visits on the energy hyper-surface is an essential feature of the folding process.     

Nonlinear time series analysis of electrocardiograms

In recent years there has been an increasing number of papers in the literature, applying the methods and techniques of Nonlinear Dynamics to the time series of electrical activity in normal electrocardiograms (ECGs) of various human subjects. Most of these studies are based primarily on correlation dimension estimates, and conclude that the dynamics of the ECG signal is deterministic and occurs on a chaotic attractor, whose dimension can distinguish between healthy and severely malfunctioning cases. In this paper, we first demonstrate that correlation dimension calculations must be used with care, as they do not always yield reliable estimates of the attractor's "dimension." We then carry out a number of additional tests (time differencing, smoothing, principal component analysis, surrogate data analysis, etc.) on the ECGs of three "normal" subjects and three "heavy smokers" at rest and after mild exercising, whose cardiac rhythms look very similar. Our main conclusion is that no major dynamical differences are evident in these signals. A preliminary estimate of three to four basic variables governing the dynamics (based on correlation dimension calculations) is updated to five to six, when temporal correlations between points are removed. Finally, in almost all cases, the transition between resting and mild exercising seems to imply a small increase in the complexity of cardiac dynamics. (c) 1995 American Institute of Physics.     

两相流流型复杂网络社团结构及其统计特性

利用气液两相流电导波动信号构建了流型复杂网络. 基于K均值聚类的社团探寻算法对该网络的社团结构进行了分析,发现该网络存在分别对应于泡状流、段塞流及混状流的三个社团,并且两个社团间联系紧密的点分别对应于相应的过渡流型. 基于复杂网络理论从全新的角度探讨了两相流流型复杂网络社团结构及统计特性问题,并取得了满意的流型识别效果,与此同时,在对该网络特性进一步分析的基础上,发现了对两相流流动参数变化敏感的相关复杂网络统计量,为更好地理解两相流流型动力学特性提供了参考. 关键词： 两相流流型 复杂网络 社团探寻算法 网络统计特性