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Complexity-entropy curves(CEC) is a useful tool to detect the structure of time series. It’s widely applied in many research areas since it can distinguish the chaotic system and the stochastic process well. However, the original permutation complexity-entropy curves(PCEC) based on permutation entropy(PE) has a defect for it can not take means and amplitudes of time series into consideration, which may lead to some errors when distinguishing the systems. In this paper, we propose dispersion complexity-entropy curves(DCEC) to overcome the defects of PCEC. In addition, we expand the curves from two-dimension to threedimension. We first compare DCEC with PCEC by simulated data. The result shows that DCEC are not only more distinguishable but also more robust with the change of parameters when detecting periodic series. Then, we apply our method to real-world data to illustrate its practicability. We propose a creative feature extraction method based on DCEC and combine it with MSVM to diagnose the different types of bearing fault, which obtains perfect results for the accuracy achieves 100%. We also apply DCEC to stock indices in different countries and different periods to analyze the complexity degree of financial markets. The results successfully detect American financial crisis in 2008 and the rapid development of the economy in China during 2014-2018. These demonstrate that DCEC can serve as an effective method to analyze complex systems.

In this paper, we propose a hybrid method called sliding-window amplitude-based dispersion entropy, which combines dispersion entropy with sliding-window amplitude, to characterize nonlinear time series. This hybrid method not only inherits the fast calculation speed and the ability to characterize nonlinear time series of dispersion entropy, but also has higher noise resistance than dispersion entropy. We firstly utilize three artificial data (logistic map, Hénon map, ARFIMA model) to qualify the effectiveness of the proposed method, results show that our method can correctly characterize the nonlinear time series, and has stronger robustness to noise. Next, the method is applied to analyze stock market system, the data of stock market are composed of six main indices from different countries, the result shows that the proposed method can easily distinguish the emerging markets and developed markets, and can reveal some features under the financial time series.

The ordinal pattern is an essential tool to extract the information in time series. However, little attention has been paid to the transition probability matrix of ordinal patterns, which affects the accuracy and comprehensiveness of the extracted information. In this paper, we propose transition permutation entropy (TPE) and transition dissimilarity measure (TDM) through the transition matrix. TPE can evaluate the complexity of systems. TDM measures the dissimilarity between systems through the dynamic transition and the probability distribution of ordinal patterns. The proposed methods are comprehensively evaluated by simulation experiments and vehicle dynamic response data. The results show that both TPE and TDM can distinguish complex systems and locate the rail corrugation. The combination of TDM, multidimensional scaling, and neural networks can be used for fault detection and is better than other distance calculation methods.

This paper introduces a new statistical tool: dispersion Havrda–Charvat entropy plane, which is used to analyze the complexity characteristics of time series. The Havrda–Charvat entropy with one parameter can provide flexibility in applications and provide more information about time series. The dispersion entropy algorithm is a fast and powerful algorithm for evaluating time series, which has been proposed in recent years. The statistical complexity measure defined by Jensen–Shannon divergence reflects the chaotic degrees of complex systems. The dispersion Havrda–Charvat entropy plane is constructed using the above conceptions. The performance of the dispersion entropy plane is evaluated by simulated chaotic processes and fractional Brownian motions, and then we apply the method to stock data. This demonstrates that dispersion Havrda–Charvat entropy plane can distinguish the intensive properties of time series well and is a powerful method to classify stock markets. In addition, the multiscale measure is experimented, the results show that it can eliminate the noise contained in the data and effectively extract the information contained in time series with different time scales.

Fractional calculus and entropy are two essential mathematical tools, and their conceptions support a productive interplay in the study of system dynamics and machine learning. In this article, we modify the fractional entropy and propose the cumulative permuted fractional entropy (CPFE). A theoretical analysis is provided to prove that CPFE not only meets the basic properties of the Shannon entropy but also has unique characteristics of its own. We apply it to typical discrete distributions, simulated data, and real-world data to prove its efficiency in the application. This article demonstrates that CPFE can measure the complexity and uncertainty of complex systems so that it can perform reliable and accurate classification. Finally, we introduce CPFE to support vector machines (SVMs) and get CPFE-SVM. The CPFE can be used to process data to make the irregular data linearly separable. Compared with the other five state-of-the-art algorithms, CPFE-SVM has significantly higher accuracy and less computational burden. Therefore, the CPFE-SVM is especially suitable for the classification of irregular large-scale data sets. Also, it is insensitive to noise. Implications of the results and future research directions are also presented.

The Multiscale Entropy (MSE) is an effective measure to quantify the dynamical complexity of complex systems, which has many successful applications in physiological and physical fields. It uses different scales to mean-coarse-grain the original series, and then calculates the sample entropy for each coarse-grained series. Inspired by the MSE, we in this paper propose the Multi-Moment Multiscale Local Sample Entropy (MMMLSE), which considers both mean-coarse-grained and standard-deviation-coarse-grained characteristics of the original series for each scale, to quantify the dynamical complexity of complex systems. We use simulated data ([Formula: see text] noise, white noise and logistic map) to test the performance of our proposed method, with results showing that the MMMLSE can accurately and effectively characterize these complex systems. The ability to preserve nonlinear dynamics of the proposed method is also proved by surrogate data and nonlinearity test experiment. Furthermore, we apply the MMMLSE to analyze physiological signals, and the MMMLSE reveals that the ill individuals have lower dynamical complexity at larger scales than the healthy ones, and the elder individuals have lower dynamical complexity at larger scales than the younger ones, which are consistent with the reality.

In this paper, we propose dispersion entropy based on Hill’s diversity number (HDE) as a new method to characterize nonlinear time series such as financial time series. In order to test the performance of this new proposed method, we first apply HDE to characterize two synthetic models (logistic map, Hénon map), the results show that the proposed method can sensitively detect the changes in the state of systems, accurately distinguish different states of the system for different parameters and correctly characterize the complexity of systems. Furthermore, we apply the proposed method to analyze the financial time series obtained from the main indices from six different countries. Empirical results illustrate that the HDE can distinguish developed stock markets and emerging stock markets, and also reveal the intrinsic dynamical characteristics of financial time series.

In this paper, the symmetrical Kullback–Leibler divergence is proposed to capture the dynamical features of various complex time series more accurately. The symmetrical Kullback–Leibler divergence is modified from the Kullback–Leibler divergence to make it possess the properties of a normal distance measure. A new multidimensional scaling (MDS) method with the proposed symmetrical Kullback–Leibler divergence is established, which is an efficient clustering and visualization technique. The proposed method also includes the probability distribution of the weighted-permutation patterns, which not only provides a modified time series quantification process with more possible patterns, but also facilitates the reduction of error when extracting the hidden information from time series. The upper and lower bounds of the proposed divergence are also presented. Through simulation and reality-based data experiments, it is affirmed that the MDS based on this novel dissimilarity measure is capable of presenting more distinctive and rational clustering results by comparing it with the MDS methods based on conventional dissimilarities.

The family of visibility algorithms provides a new point of view to describe time series by transforming them into networks. In this paper, we propound a new visibility algorithm named Temporal Vector Visibility graph [Formula: see text], which maps the multivariate time series to a directed network. Computed by the [Formula: see text] and [Formula: see text] degree distributions obtained from the [Formula: see text], these statistics such as the Kullback–Leibler divergence (KLD), the normalized Shannon entropy and the statistical complexity measure, are introduced to assess the complexity of time series. Furthermore, we also apply the Multivariate Multiscale Entropy Plane (MMEP) to evaluate the complexity of multivariate time series. The experimental results of eight different types of time series verify the effectiveness of our method. Subsequently, this method is employed to explore the complexity characteristics of financial time series and classify different stock markets. Our research reveals that this method is capable of investigating the physical structures of financial time series.

In this paper, dynamic Shannon entropy (DySEn) is introduced as a novel method to detect the abnormal changes of signals. It is a combination of Shannon entropy and the permuted distribution entropy (PDE). Experiments have proved that Shannon entropy is not sensitive to local disorder, and there may be no response even if the amplitude changes significantly. PDE does not work well with chaotic sequences, unless the abnormal area and the normal one have obvious differences in periodicity. However, DySEn can deal with those problems at the same time based on both traditional statistical characteristics and dynamic characteristics. Our experiments show that it can provide an effective way to the anomaly detection for periodic signals, complex signals and the mixed signals. We also apply it to detect the rail corrugations. DySEn can effectively locate the abnormal areas, and, with the help of PDE, it can be seen that the periodicity of the abnormal areas has increased significantly, which is in line with the situation of rail corrugations.

As a practical tool, visibility graph provides a different perspective to characterize time series. In this paper, we present a new visibility algorithm called directed vector visibility graph and combine it with the Kullback–Leibler divergence to measure the irreversibility of multivariable time series. T directed vector visibility algorithm converts the time series into a directed network. Subsequently, the ingoing and outgoing degree distributions of the directed network can be got to calculate the Kullback–Leibler divergence, which will be applied to assess the level of irreversibility of the time series. This is a simple and effective method without any special symbolic process. The numerical results from various types of systems are used to validate that this method can accurately distinguish reversible time series from those irreversible ones. Finally, we employ this method to estimate the irreversibility of financial time series and the results show that our method is efficient to analyze the financial time series irreversibility.

Entropy has been an important tool for the complexity analysis of time series from various fields. Based on studying all the template mismatches, a modified sample entropy (SE) method, named as inverse sample entropy (ISE), for investigating the complexity of financial time series is proposed in this paper. Different from SE, ISE considers the far neighbors of templates; it also provides more comprehensive information combined with SE. Stock markets usually fluctuate with the economy policies; ISE allows us to detect the financial crisis by the change of complexity. By experiments on both simulated data and real-world stock data, ISE shows that the threshold $$r$$ is more flexible compared with that of SE, which allows ISE to be applied not only to limited type of data. Besides, it is more robust to high dimension $$m$$ , so ISE can be extended to the application of high dimension analysis. For studying the impact of embedding dimension $$m$$ under multiple scales on both artificial and real-world data, we made a comparison on the use of SE and ISE. Both SE and ISE are able to distinguish time series with different features and characteristics. While SE is sensitive to high dimension analysis, ISE shows robustness.

Conditional mutual information (CMI) is the basis of many coupling direction metrics and plays an important role in revealing the causal relationship between different signals. In this paper, we propose dispersion conditional mutual information (DCMI) which uses dispersion patterns to calculate the probability distribution. This method is computationally fast and can accurately extract the dynamical characteristics of signals even in the presence of noise. The effects of time lag between signals, coupling coefficient, noise, data length and sudden change in coupling direction on the performance of DCMI are evaluated by simulation experiments. Moreover, we extend DCMI to multiscale DCMI (MDCMI) through a modified multiscale method. MDCMI is adopted on the coupled chaotic model and coupled stochastic process to research the properties. The results show that both DCMI and MDCMI have excellent properties in detecting the coupling relationship and are easy to calculate. Finally, we apply the MDCMI on stock indexes of USA and the results show that there is a cross-correlation between stock price and trading volume.

Cumulative Tsallis entropy (CE) is a recently introduced entropy metric to quantify the uncertainty of time series, and its expressions of continuous random variable and discrete random variable are consistents. So far, it has proved to have a good performance in the characteristics of time series. This paper presents a new method to measure the complexity and similarity of systems—cumulative Tsallis entropy based on the dispersion entropy (DCE). It is different from the traditional PE method to simply symbolize the sequence. Instead, the complexity of the system is characterized by focusing on the amplitude information of the time series and considering the influence of past events. We applied DCE to two kinds of simulation data and six global financial time series. The results show that DCE can be used as a diagnostic model to classify global financial data according to regional characteristics, financial background and government policies. In addition, as a classical method of non-stationary time series, we combine the MSE method with DCE to observe the financial market from different time scales and obtain rich intrinsic properties.

The purpose of this study was to introduce a method in extracting and quantifying the information flow in complex system, which takes into account the temporal structure of the time series at multiple scales. It is important that the method should be able to reflect the intrinsic mechanism of information interaction faithfully. The proposed multiscale permutation mutual information (MPMI) method studies the mutual information based on permutation pattern and multiscale concept from multiscale sample entropy and is initially tested on artificially generated signals for proof of concept by comparing the MPMI results of the iterative amplitude adjusted Fourier transform surrogates and the original series. It is subsequently applied to quantify the information interaction of traffic time series. MPMI results can detect the relationship between neighboring detectors and the effect of traffic accidents on information interaction between speed and volume. MPMI method uncovers the information interaction and provides valuable insight into the underlying mechanisms in traffic system.

Vector visibility graph (VVG) is an algorithm that transforms multivariate time series into directed complex networks. However, at present, the researches of VVG mainly focus on its degree distribution. Considering the limitation of using the degree distribution of vector visibility graph alone to analyze the complexity of multivariate time series, we use the normalized Shannon entropy and the statistical complexity measure to analyze the complexity of multivariate time series based on the results of the degree distribution. We introduce the multivariate multiscale entropy plane to measure the dynamical complexity of multivariate systems. The effectiveness of the proposed method is validated by numerical simulation from several kinds of systems. In addition, we also observe that it is immune to different levels of noise in a wide range. Then, it is applied to evaluate the dynamic classification of financial time series from stock markets. Our results indicate that this method is effective to research the physical structures of stock markets.

Distribution entropy has been proved to reveal stability for short time series and to distinguish different classes of series by complexity. However, there still exists some drawbacks. For example, it does not consider the possible causality underlying the data, which may not precisely identify deterministic from stochastic processes. In addition, cumulative residual entropy can successfully solve such problems and identify randomness and complexity of time series quite clearly. We therefore combine distribution entropy with cumulative residual entropy named cumulative residual distribution entropy (CRDE), aiming at considering both distribution and values of distances in the state space. CRDE can detect the temporal and spatial structures of the series after adding multiscale analysis. Results show that the combined method can characterize series from stochastic system (white noise and 1/f noise) and deterministic system (chaotic and periodic series). Then, we apply it to physiological signals, and the result is consistent with the one that loss of complexity at larger scales is related to aging and disease.

In this work, we propose the cumulative residual entropy (CRE) plane and CRE curve based on the weighted-multiscale cumulative residual Renyi/Tsallis permutation entropy and oscillation roughness exponent to analyze complex dynamic systems. The oscillation roughness exponent method and the cumulative residual distribution theorem adopted in our proposed methods are two core theories to depict more detailed information of complex dynamic systems in a more efficient way by reducing information loss and capture the statistics of the related time series’ roughness. Trials are operated on the logistic map model as numerical experiments, and we discover that our methods are capable of discriminating different types of complex data with high accuracy. Compared with the original methods, our methods are more superior in extracting more subtle details to distinguish different dynamic systems. In the experiments with the financial stocks, our methods are still found to be more reasonable in discriminating stock indices from different parts of the world by making comparisons with original methods.

In this work, we present a comprehensive assessment of the Fisher information measure and statistical complexity measures based on Euclidean distance, Wootters distance and Jensen–Shannon divergence, regarding their abilities to (planar-) distinguish between/among (1) chaos and periodicity; (2) different degrees of periodicities; (3) different chaotic regimes; and (4) chaos and noise, and characterize delay dynamics. The Bandt–Pompe approach is used to build up the probability space to generate the entropy-complexity/information plane. The effect of embedding parameters on the evaluation is also considered. Within this framework, complexity measures based on the Wootters distance and Jensen–Shannon divergence are superior to the Fisher information measure in capturing subtle details of chaotic dynamics. The Fisher information measure shows advantages in robustness to additive noises and in planar-behavior representation of chaos and noise. Moreover, all measures are able to properly characterize the intrinsic delay dynamics of chaotic and stochastic systems. Nevertheless, the complexity measure based on the Euclidean distance is not valid by definition, thus, not applicable at any cases.

This study proposes a modified recurrence quantification analysis, called global recurrence quantification analysis (GRQA). It is well known that the recurrence threshold is an important parameter in traditional recurrence quantification analysis. However, in existing researches, the selection of recurrence thresholds is often based on ‘rules of thumb.’ As many studies have shown, recurrence analysis and its quantifiers are strongly dependent on the evaluation of the vicinity threshold parameter, which indicates that a selected threshold may have an adverse effect on exploring signal inherent information and the interrelationship between different sequences. Therefore, GRQA is initialized in this paper to measure the vertical and diagonal structures of recurrence plots in a more objective way, because it considers all the information carried by all potential values of the threshold. The information described by GRQA is determined by the sequence itself and is not affected by specific thresholds. GQRA can also clearly depict the dynamical similar characteristics and recursive trajectories between sequences, which have not appeared in previous researches. We apply this method to the financial time series to find some useful information. It reveals that SZSE and SSE show similar inherent dynamic characteristics via GQRA statistics curves, and DJI and NASDAQ are similar to each other as well, while HSI is like a combination of these two groups with both of their characteristics, which is consistent with its financial background.

As a method to measure the synchronization between two different sets of signals, the multivariate synchronization index (MSI) has played an irreplaceable role in the field of frequency recognition of brain-computer interface since it was proposed. On this basis, we make a generalization of MSI by using the escort distribution to replace the original distribution. In this way, MSI can be converted from a determined value to the multivariate synchronization curve, which will vary as the parameter q of the escort distribution changes. Numerical experiments are carried out on both simulated and real-world data to confirm the effectiveness of this new method. Compared with the case of MSI (i.e., q = 1), the extended form of MSI proposed in this article can obviously capture the relationship between signals more comprehensively, implying that it is a more perfect method to describe the synchronization between them. The results reveal that this method can not only effectively extract the important information contained in different signals, but also has the potential to become a practical synchronization measurement method of multivariate signals.

In multiscale time series analysis, multiscale entropy provides a good framework to quantify the information of time series. Multiscale fractional high-order entropy based on roughness grain exponents (MFHER) is able to identify dynamical, scale dependent and oscillation information. In detail, MFHERcan be seen as a powerful tool to assess the complex characteristics of time series. A set of synthetic time series and an application of real world data financial series are researched. The results show that high order entropy performs well in distinguishing different time series. It has also been found fractional high order entropy is highly sensitive to parameter variation and thus provides a broad perspective to research the complexity of dynamic systems. This study gains an insight into the measurement of MFHER to demonstrate the wide applicability of entropy measures.Aiming at the complexity of network and the uncertainty of internal and external environment, this paper proposes MFHER to quantify the time series information on multiscale time scales. It is of great interests in identifying dynamical properties of nancial series. The results show that the volatility of the sequence is gradually stable when the scale is greater than four. High order entropy can identify the difference among the time series.

Research on complexity and uncertainty of nonlinear signals has great significance in dynamic system analysis. In order to further analyze the detailed information from the financial data to acquire deeper insight into the complex system and improve the ability of prediction, we propose the entropy of scale exponential spectrum (EOSES) as a new measure. Combined with multiscale theory, we get multiscale EOSES. The scale exponential spectrum (SES) is derived from the scale exponent of detrended fluctuation analysis. As for the entropy, we choose Renyi entropy and fractional cumulative residual entropy to compare and analyze the results. Simulated data and financial time series are used to obtain further in-depth information on the EOSES. Compared with traditional methods, we find that Renyi EOSES over moving window can provide more details of complexity which include fractal structure and scale properties. Also, it reduces the influence of degree of fitting polynomial and has higher noise immunity. In addition, through the SES and EOSES, we can better research the properties and stages of stock markets and distinguish stock markets with different characteristics.

In this paper, we propose a new method, discrete generalized past entropy based on oscillation-based grain exponent (O-DGPE), which combines the discrete generalized past entropy and O-DGPE. It is proved to be a good measure of the uncertainty and reversed hazard rate of time series, and the oscillation information inside the data can be observed by this method. Firstly, we apply O-DGPE to chaotic maps. Experiments show that O-DGPE can distinguish different states of the map, and the results are also consistent with the actual nature of the maps. Chaos has higher uncertainty for inactivity time than periodic cycle. In addition, the law of O-DGPE changing with some parameters is also revealed. After that, we further validate the effectiveness of the method through ARFIMA model. Finally, we apply the method to financial time series. Results show that O-DGPE can be used to analyze stock markets from various perspectives. Through the character of O-DGPE changing with parameters, we can get a glimpse of the internal law of the financial markets. Chinese financial market is highly uncertain, while the stock exchange market of the USA is more mature. What’s more, the application of cumulative O-DGPE turns out to be very useful in measuring information on the inactivity time of a system.

In order to explore the complexity of multivariate time series, we propose a novel method: multiscale multivariate weighted fractional entropy (MMWFE). The research results show that MMWFE is able to measure the complexity of multivariate data correctly and reflect more information contained in the time series. In this paper, the reliability of the proposed method is supported by simulations on generated and empirical data. We analyze simulated pink noise and white noise to test the validity of this method, and the result is consistent with the fact that pink noise is more complex than white noise. Meanwhile, MMWFE shows a better robustness. MMWFE is then employed to bivariate stock return and volume to explore the complexity of stock markets. It successfully distinguishes Asia, Europe and Americas markets. Finally, dynamic MMWFE is applied to explore the evolution of complexity for mining more information containing in nonlinear time series.

In this paper, a generalized method, fractional-order approximate entropy (FOApEn), is proposed with the objective of distinguishing the complexity of time processes from statistical perspectives and characterizing differences and changes in dynamical systems. Moreover, we generalized approximate entropy (ApEn) to multiscales, which can detect complexity of time series in more scales and probe the multiscale properties containing in time series. This fractional-order approximate entropy, which provides an assessment on the multiscale complexity between measurements, is defined in terms of the FOApEn method and the multiscale method. The implementation of multiscale FOApEn is illustrated with simulated time series and financial time series. Examples taken from simulated and financial data demonstrate that tuning the fractional order allows a high sensitivity to the signal evolution and how the FOApEn for complex systems behaves on different scales and determination, which is helpful in describing the dynamics of complex systems.

In this paper, a PDF-induced distance (PID) based on permutation cross-distribution entropy is proposed to measure the dissimilarity between complex time series. It overcomes the effects of spatial distance by focusing on similar local fluctuation patterns. It also corrects the disadvantage of being insensitive to symmetric skewness distributions. We have applied PID to synthetic data and financial time series. The Euclidean distance is employed as a reference. In simulated experiments, eight signals generated from four models are detected, and results are presented by hierarchical clustering analysis via PID, which is correctly clustered and superior to other methods. Then, PID is applied to the real-world financial time series. Eight stocks in the global financial markets are employed. It reveals that they are clearly divided based on their financial backgrounds. The cophenetic correlation coefficient is used to measure the quality of solutions. It reveals that the PID is convincing and superior to the Euclidean distance. In addition, as a classic method for non-stationary time series, detrended cross-correlation analysis (DCCA) is adopted to compare the superiority of PID. Although experiments show that DCCA does perform well, it is inferior to PID. Also, we conduct additional experiments on the effects of non-Gaussian noises. Excitingly, PID can still cluster signals accurately after adding uniformly distributed noises, and the cophenetic correlation coefficient reaches 0.9788.

We modify the basic roughness grain exponent, only available for application of one single series, to complexity-invariant distance ( C I D ) for studying multifractal features between two time series. C I D is taken into consideration as a new roughness grain exponent with the large deviation spectrum to detect the similarity and correlation between different stock markets in this work. The new method is firstly applied to artificial series in order to test the scale invariance hypothesis with both the Legendre spectrum and large deviation spectrum. Results show that the large deviation spectrum is better in the detection of multifractal analysis between two time series as it exhibit plentiful scaling structures. Then we investigate the scaling behavior between financial time series so as to test whether there is scale invariance of stock markets in China and the US. Besides, from the α values responding to the maximal f α in the 5 pair series and the widths of the spectrum, a new way of measurement of the similarity and correlation has been found. To overcome the limitation of q , we then compute the surface area of the large deviation spectrum under each scale and further detect the evolution of the slope by a scaling/non-scaling criterion which helps in the measurement of scaling behavior. Moreover, the proposed method can give us more information between different markets and distinguish them from different angles.

Discussing the complexity of time series has been a long-standing problem, including the use of information entropy to determine the complexity of the sequence. Permutation entropy (PE) has been regarded as a learning process to investigate the complexity of time series, such as financial time series and physiological time series. The permutation entropy, which is based on the Shannon entropy (SE), has undeniable shortcomings in dealing with some specific problems. Hence in this paper, we propose the multiscale Tsallis permutation entropy (MTPE) as an improved measuring tool for assessing the hidden temporal correlations in time series. The modified method not only presents a different way showing clear characteristics but also provides more significant results compared with the SE. Robustness of the hypothesis is proved by the correlation between Tsallis permutation entropy (TPE) and HURST exponent obtained from the sequences which are generated from fractional brownian motion stochastic model. Experimental results of autoregressive sequences (AR) and electroencephalograph time series (EEG) make the advantages of Tsallis permutation entropy more obvious, which also well validate the efficiency and integrity of principle of maximum entropy (PME). Multiscale analysis provides us with another perspective to analyze permutation entropy, which also allows us to better analyze the complexity of the sequence.

An important feature of the time series in the real world is that its distribution has different degrees of asymmetry, which is what we call irreversibility. In this paper, we propose a new method named permutation pattern (PP) to calculate the Kullback–Leibler divergence ( $${D}_{\mathrm{KL}}$$ ) and the Jensen–Shannon divergence ( $${D}_{\mathrm{JS}}$$ ) to explore the irreversibility of time series. Meanwhile, we improve $${D}_{\mathrm{JS}}$$ and obtain a complete mean divergence ( $${D}_{\mathrm{m}}$$ ) through averaging $${D}_{\mathrm{KL}}$$ of a time series and its inverse time series. The variation trend of $${D}_{\mathrm{m}}$$ is similar to $${D}_{\mathrm{JS}}$$ , but the value of $${D}_{\mathrm{m}}$$ is slightly larger and the description of irreversibility is more intuitive. Furthermore, we compare $${D}_{\mathrm{JS}}$$ and $${D}_{\mathrm{m}}$$ calculated by PP with those calculated by the horizontal visibility graph, and discuss their respective characteristics. Then, we investigate the advantages of $$\mathrm{D}_{\mathrm{JS}}$$ and $${D}_{\mathrm{m}}$$ through length variation, dynamic time variation, multiscale and so on. It is worth mentioning that we introduce Score and variance to analyze the practical significance of stock irreversibility.

The multivariate multiscale complexity-entropy causality plane (MMCECP) is introduced for evaluating the dynamical complexity and long-range correlations of multivariate nonlinear systems. Numerical simulations from different classes of systems are applied to confirm the effectiveness of the proposed measure. We observe that the MMCECP not only can characterize the deterministic properties of the systems, but also can distinguish Gaussian and non-Gaussian processes. Moreover, it is immune to varying degrees of noises at large scales. Then we apply it to financial time series analysis, mainly investigating the classification of stock market dynamics. Empirical results illustrate that the MMCECP is robust and valid to detect the physical structures of stock markets.

Entropy is an accessible way to work as a measure of the irregularity and the uncertainty between the predicting knowledge and the given time series. Statistical complexity measure (SCM) combining Shannon entropy and the extensive Jensen–Shannon divergence provides important additional information regarding the peculiarities of the underlying probability distribution, not already detected by the entropy. In this paper, we extend the traditional complexity-entropy causality plane, which applies the diagram of SCM versus normalized Shannon entropy, to two generalized complexity-entropy plane based on Permutation entropy (PE) and Permuted distribution entropy (PEDisEn). Moreover, as the important extension of the Shannon entropy, the Tsallis entropy and Renyi entropy are used to construct the plane. We discuss the parameter selection for the PE plane and PEDisEn plane respectively. Outlier detection is recently a heated point focusing on discovering patterns that occur infrequently in the time series in data mining. However, there exists few entropy plane based methods in outlier detection. We apply the proposed procedure to the real world data for outlier detection. It turns out that the generalized entropy plane is robust to the type of original series and is efficient for detecting outliers.

Empirical mode decomposition (EMD) is a self-adaptive signal processing method that can be applied to nonlinear and non-stationary processes perfectly. In view of this good ability of EMD, in this paper, we propose a new method—EMD energy entropy plane—which combines two different tools—EMD energy entropy and complexity-entropy causality plane—to analyze time series. Firstly, we apply EMD energy entropy plane to synthetic data, such as logistic map, Henon map, ARFIMA model and so on, finding that the EMD energy entropy plane presents different trends and distributions when the map is in periodic cycles and chaos. Then we demonstrate the application of EMD energy entropy plane in stock markets. Results show that it is an effective tool of distinguishing two kinds of financial markets. In addition, the introduction of multi-scale reveals the variation law of EMD energy entropy plane at different scales.

In this paper, we propose the complexity–entropy causality plane based on the generalized fractional entropy. When applying the proposed method into artificial time series and empirical time series, we find that both results show that the stochastic and chaotic time series are clearly distinguished. On the one hand, we could distinguish them according to the trend of the normalized generalized fractional entropy H as the parameter $$\alpha $$ increases. On the other hand, the stochastic and chaotic time series can be distinguished by the trend of their corresponding extreme values $$\alpha _C$$ with the increase in embedding dimension m. However, compared with the q-complexity–entropy plane, the trend of their extreme value $$\alpha _C$$ is irregular. Moreover, when applying the complexity–entropy causality plane into financial time series, we could obtain more accurate and clearer information on the classification of different regional financial markets.

In this paper, we propose the multiscale joint permutation entropy (MJPE) to study the synchronism between two complex time series from the view of ordinal pattern and multiple scales. First, we use the Rossler system using active control, two-component ARFIMA processes to test the effectiveness of MJPE and also add some noise to the ARFIMA time series and apply MJPE to find the effect of noise. The results show the necessity of investigating the synchronism on the multiple scales, prove the effectiveness of MJPE method and show the sensitiveness of MJPE method to noise. Then MJPE method is employed to financial time series and traffic time series to validate the applicability of the proposed MJPE method for the complex time series in the real world. The conclusion from these MJPE results for financial time series is consistent with the actual situation of the synchronism and correlation between stock indices. Meanwhile, the results for traffic time series suggest the need for study the synchronism from the perspective of multiple scales and point out the different synchronisms for traffic time series of weekdays and weekends. MJPE method has a broad application prospect on the investigation of synchronism on the complex time series from different fields.

The complexity of time series has become necessary to understand the dynamics of the control system. In this paper, the multivariate multiscale distribution entropy (MMSDE) is introduced as a new method to assess the complexity of dynamical systems such as financial systems, physiological systems, etc. Distribution entropy (DE) takes full advantage of the information hidden in the state–space by the estimation of the probability density of distances among vectors. Based on this, MMSDE can quantify the complexity of multivariate time series from multiple time scales. We test the performance of this method with simulated data. Results show that MMSDE has less dependence on parameters and the test of short time series is very effective. As for the real data, we explore the high dimensional series that are composed of opening price, closing price, volume of stocks data including US stock indices and Chinese stock indices. MMSDE can quantify the change in the complexity of the stock market data. In addition, we get richer information from MMSDE and gain some features about the difference between the U.S. and Chinese stock indexes.

In this letter, multivariate multiscale fractional permutation entropy (MMFPE) and multivariate weighted multiscale fractional permutation entropy (MWMFPE) have been proposed to provide insights for the study of time series. When measuring the dynamics of complex systems, the MMFPE and MWMFPE methods are sensitive to the signal evolution. Meanwhile, they can provide some analysis of complexity over multiple time series as well as multiple channel signals. We perform these methods on synthetic tri-variate time series to explore some of the interesting properties, especially for negative information and information deception. It can be seen that more complex system is more likely to be deceptive. The amplitude information of time series which is taken into account in the MWMFPE can weaken this deception. The methods are also employed to the closing prices and trade volume of financial stock markets from different areas. According to the MWMFPE results, the indices can be divided into three groups: (1) CAC40, HSI, NASDAQ, S&P500, (2) N225, and (3) ShenCheng, implying that it has a capacity to distinguish these financial stock market.