In this work, we propose the discrete general cumulative residual Kullback-Leibler information(DGCKL)to measure the complexity of nonlinear time series. Theoretically, the novel method benefits from the core concept of distribution entropy (DistEn)and possesses the advantages of the general cumulative Kullback-Leibler information(GCKL), presenting significant superiorities in quantifying the complexity of time series. Furthermore, comparative experiments conducted on simulation data confirm thatDGCKLis robust to noise and inherits the excellent properties ofDistEn to extract the features of time series. In practical application, DGCKLis provided with the ability to detect the abnormal behaviors of the rail dynamic time series and identify the significant events in specific periods of stock indices. Supported by theoretical analysis, simulated research and experimental verifications, this work establishes DGCKLas a meaningful and practical model for quantifying the complexity of time series.

This article presents a volatility quantification method based on the Wasserstein–Fourier (WF) distance to detect dynamic changes and identify abnormal behaviors of time series. Specifically, if one of the time series in WF distance is determined as having known statistical properties, then the focus of the research is naturally shifted to explore the properties of the other time series it contains. Theoretically, the proposed method named constrained WF (CWF) combines the natures of WF distance with the statistical characteristics of the fixed time series in a good way, and obtains some unique properties. In terms of application, experiments conducted on both simulated and real-world datasets demonstrate the effectiveness of this new proposed volatility quantification method. Furthermore, the comparison with other methods of the same type gives promising results, highlighting its advantages in applicability and recognition accuracy. Based on theoretical analysis, simulated research and experimental verifications, this work establishes CWF as a meaningful and practical model for quantifying the volatility of time series.

In this paper, a generalized form of Shannon–Fisher​ (SF) index is introduced. The generalized SF index (GSF) is in view of the original SF index, which is provided with the ability to quantify the instability of time series. The key of this proposed method is to replace the original distribution with its escort distribution, whose advantage lies in its capability of scanning the properties of original distribution. Besides, since the effectiveness of visibility graph (VG) used in the original SF index is limited to univariate time series, we utilize the vector visibility graph (VVG) which is suitable for multivariate time series to broaden the adaptability of our new method. Trials for both simulated and real-world time series are carried out for providing the contrastive research. Compared with the original SF index (i.e. q=1), the new parameter q introduced by employing the escort distribution gives us the opportunity to explore more hidden information in the time series, implying that this new method is expected to be regarded as a practical method to quantify the instability of multivariate time series.

Time irreversibility (TIR) is one of the basic properties of nonlinear systems, and it can be used as an effective way to explore the nonlinear complexity of dynamic systems. In this paper, we propose the TIR measure algorithm based on Riemannian geometry and apply this algorithm to abnormality detection tasks. The core idea is to calculate the positive definite covariance matrices of the forward sequence and its inverse sequence based on the theories of sliding window and phase space reconstruction, and compute the affine invariant Riemannian metric (AIRM) between them to characterize the TIR of the time series. The advantage of this algorithm is that the framework based on Riemannian geometry can more effectively use the spatial information of time series. In addition, compared with the traditional methods, the new method is suitable for nonstationary complex systems, which introduces a sliding window to make subsequences satisfy weak stationary conditions to measure local time irreversibility. The simulated and surrogate data are used to verify the effectiveness of the algorithm. Finally, we apply the algorithm to empirical analysis. The results show that, the algorithm detects the financial crisis period and stable period of the stock markets in the analysis of the financial system, and the loss degree of TIR of physiologic signals of different types of cardiac patients is detected in the study of physiologic signals. In the research of Parkinson’s gait, we verify that the walker is an effective tool to improve the gait of early Parkinson’s patients through statistical tests.

In this paper, we propose a reverse form of Shannon-Fisher (SF) index. The new method is based on the original SF index, which is capable of quantifying the instability of time series. The core of this method mainly includes two points. Firstly, considering that the visibility graph (VG) is only suitable for univariate time series, we replace it with the vector visibility graph (VVG) applicable for multivariate time series. Secondly, in order to provide a new perspective to explore the information contained in time series, we change the original distribution applied in the SF index to the negation of this original distribution. Compared with the original SF index, our new method can not only obtain the information contained in simulated time series from a complementary point of view, but also improve the limitation of data length in the original SF index. In the process of experimenting with the stock data, it can be realized that our method is able to identify those special years of different regions, which suggests that it is provided with the possibility of becoming a new effective method for instability quantification of multivariate time series.

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.

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.

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.

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.

Complexity of time series is an important feature of dynamical systems such as financial systems. In order to bring out the complete non-linear behavior of financial time series, non-linear tools of complexity measurement like entropy measures are indispensable. Sample entropy (SampEn) and distribution entropy (DistEn) are popular methods of assessing the complexity in various fields. However, both sample entropy and distribution entropy show some limitations in detecting the complexity of stock markets. Therefore, we use two entropies as binary indices to analyze the complexity of time series and structure the entropy plane. In order to further improve the accuracy of the research results, we take the embedding dimension m as the variable, expand the entropy point into the entropy curve for further research. Furthermore, considering the shortcomings of sample entropy and distribution entropy in practical application, we generalize them by (i) replacing sample entropy and distribution entropy with multi-scale sample entropy and multi-scale distribution entropy; (ii) generalizing Shannon entropy as Tsallis entropy. Also, scale factor τ and entropic index q are taken respectively as variables to get the entropy curves. By using the artificial data, we confirm the rationality of using entropy points and entropy curves to study the complexity of time series. Finally, we apply this method to measure the complexity of real world financial time series, the results show that the entropy curves plotted by the financial time series obtained from different areas have significant differences.