Early warning in high-dimensional complex systems requires indicators that can characterize multivariate covariance changes across different interaction scales. Localized instabilities may concentrate covariance amplification within a small subset of variables, whereas diffuse stress may produce weaker but broadly distributed correlation growth. This article introduces the lambda-spectrum of dynamical network markers, a multiscale extension of covariance-based DNM analysis based on a family of generalized covariance means. Low aggregation orders emphasize distributed weak coherence, while high aggregation orders emphasize localized covariance hotspots. Therefore, the spectral shape provides interpretable information about how instability-related covariance is organized within the system. Numerical simulations on networked harvesting systems show that the lambda-spectrum reflects different instability localization mechanisms: localized community failure and core-driven collapse generate strong high-lambda amplification, whereas diffuse and peripheral stress produce more compressed spectral responses. Sensitivity analyses further show that the qualitative spectral structure remains stable across a broad range of DNM sizes. For engineering systems in which strict bifurcation assumptions cannot always be verified, the proposed method is further interpreted as a multiscale covariance fingerprinting framework. Experiments on railway bogie data, C-MAPSS turbofan degradation data, and the secure water treatment industrial process dataset demonstrate that the lambda-spectrum captures state-dependent covariance fingerprints, reveals systematic degradation evolution, and provides competitive anomaly detection with earlier warning than conventional scalar indicators.

Entropy is an effective tool for assessing the irregularity and complexity of nonlinear time series and complex systems. In recent years, Dempster-Shafer evidence theory has gained increasing attention in time series analysis and has been integrated with traditional entropy measures, leading to the development of belief entropy, which offers a novel perspective for characterizing nonlinear time series. However, most existing belief entropy approaches remain limited to simple combinations of existing methods and rely on static patterns, which may overlook important dynamical information embedded in the time series. In this paper, we extend belief entropy into a statistical complexity measure framework. Moreover, by incorporating transition network methods, we propose link belief entropy and its corresponding statistical complexity measure—along with complexity-entropy causality diagram analysis—to capture the dynamic structures and properties of multivariate time series. Comparative experiments on both simulated signals and real-world datasets confirm the effectiveness of the proposed methods. The results demonstrate that the belief entropy and link belief entropy frameworks, along with their complexity-entropy diagrams, can successfully distinguish complex systems with different characteristics, offering a new perspective for nonlinear multivariate time series analysis.

With the rapid development of artificial intelligence, nonlinear time series analysis and mining have become an indispensable part of many disciplines. How to analyze and identify the properties of complex data with higher dimensionality and complexity is still the key problem to be solved in complex system analysis accurately and effectively. For the purpose of helping address this issue, in this paper, we propose a novel nonlinear time series clustering method based on modified stochastic neighbor embedding and improved information dissimilarity measure. This method can effectively reveal the local structure of high dimensional data, so that similar data points can be kept close in low dimensional space. Because it uses probability distribution to calculate the similarity between data points, it can handle nonlinear relations and has strong ability to capture them. Secondly, the method can visualize high-dimensional data effectively, especially when the distribution of data points has a complex structure. It can map high-dimensional data into two-dimensional or three-dimensional space, which is easy to observe and analyze the distribution and clustering of data. In addition, this technique does not need to specify the intrinsic dimensions of the data in advance, and it can automatically learn the intrinsic dimensions of the data, which makes it more flexible in the face of different types of high-dimensional data. In real data applications, similar data of the same class will be clustered together in a low-dimensional space. This makes it excellent in processing high-dimensional data with complex nonlinear structures.

In this paper, we propose a novel Euclidean-distance-based coefficient, named differential distance correlation, to measure the strength of dependence between a random variable Y is an element of R and a random vector X is an element of Rp. The coefficient has a concise expression and is invariant to arbitrary orthogonal transformations of the random vector. Moreover, the coefficient is a strongly consistent estimator of a simple and interpretable dependent measure, which is 0 if and only if X and Y are independent and equal to 1 if and only if Y determines X almost surely. An alternative approach is also proposed to address the limitation that the coefficient is non-robust to outliers. Furthermore, the coefficient exhibits asymptotic normality with a simple variance under the independent hypothesis, facilitating fast and accurate estimation of p-value for testing independence. Three simulation experiments show that the proposed coefficient is more computationally efficient for independence testing and more effective in detecting oscillatory relationships than several competing methods. We also apply our method to analyze a real data example.

The use of complex network for nonlinear analysis of time series has attracted increasing attention, among which, ordinal networks have been widely studied for their simplicity and computational efficiency. However, existing ordinal network methodologies exhibit two limitations: 1) inadequate handling of equal values in time series discretization, 2) neglect of some amplitude information. They potentially compromising the characterization of complex systems. To address these limitations, this paper proposes a dispersion transition network framework, which includes node-wise entropy and Wasserstein entropy curve based on the dynamic transition information and Wasserstein distance, to identify the properties of systems. The introduction of Wasserstein distance improves the stability and comprehensiveness of the information extracted from the probability distribution. The node-wise entropy and entropy curve are local metric and global metric, respectively. Numerical results show that the framework have a strong ability to distinguish signals with different dynamics and can show the variation of system properties with parameters. In the empirical application, the proposed methods can be applied to financial data classification without complex preprocessing. We also apply these methods to railway corrugation detection and compare them with other nine dimensionality reduction methods. The most satisfactory classification results are obtained by the proposed methods. The entropy curve can also be combined with multidimensional scaling for physiological time series classification.

This paper introduces a novel methodology for analyzing the complexity of time series using the fractional order dispersion entropy (qDE) and fractional order reverse dispersion entropy (qRDE) plane. Building on the theoretical foundations of dispersion entropy and its transformation, this paper introduces new entropy measures, namely, qDE and qRDE, detailing the computation processes of these entropies and constructing an entropy plane. This new approach offers fresh perspectives and tools for time series complexity analysis. Through theoretical derivations and numerical experiments, including analysis of both simulated and real stock market data, this method has demonstrated strong effectiveness in distinguishing time series of varying complexities, particularly in capturing the trends of financial market outputs over different periods. The results demonstrate that the qDE-qRDE plane can distinctly differentiate time series with varying dynamics and effectively reveal the complexities and cyclical changes in financial markets, showcasing the potential and practical value of this method in time series analysis.

The evaluation of node centrality remains a critical challenge in the field of complex network research. This paper proposes a novel method, the JSD method, which integrates local and global information to measure node centrality. The method employs Shannon entropy to quantify local centrality and Jensen-Shannon (JS) divergence to compute inter-community distances, thereby assessing topological differences between communities and measuring global centrality. Experiments conducted on both real and random networks evaluated the impact of central nodes identified by the JSD method on network efficiency. Simulation results demonstrate that in networks with distinct community structures, the JSD method provides more accurate node centrality measurements compared to the BC, CCI, CBC, COMM, and CI methods. Additionally, the repetition frequency of central ranking nodes indicates that the JSD method effectively distinguishes the centrality of different nodes. Finally, the paper discusses the impact of different combination methods on measurement performance, revealing that incorporating additional community information further enhances the method's effectiveness.

Lempel-Ziv complexity (LZC) is a key measure for detecting the irregularity and complexity of nonlinear time series and has seen various improvements in recent decades. However, existing LZC-based metrics, such as Permutation Lempel-Ziv complexity (PLZC) and Dispersion-Entropy based Lempel-Ziv complexity (DELZC), focus mainly on patterns of independent embedding vectors, often overlooking the transition patterns within the time series. To address this gap, this paper introduces a novel LZC-based method called Bidirectional Transition Dispersion Entropy-based Lempel-Ziv complexity (BT-DELZC). Leveraging Markov chain theory, this method integrates a bidirectional transition network framework with DELZC to better capture dynamic signal information. Additionally, an improved hierarchical decomposition algorithm is used to extract features from various frequency components of the time series. The proposed BT-DELZC method is first evaluated through four simulated experiments, demonstrating its robustness and effectiveness in characterizing nonlinear time series. Additionally, two fault-bearing diagnosis experiments are conducted by combining the hierarchical BT-DELZC method with various classifiers from the machine learning domain. The results indicate that BT-DELZC achieves the highest accuracy across both datasets, significantly outperforming existing methods such as LZC, PLZC, and DELZC in extracting features related to fault bearings.

Information theory and Shannon entropy are essential for quantifying irregularity in complex systems or signals. Recently, two-dimensional entropy methods, such as two-dimensional sample entropy, distribution entropy, and permutation entropy, have been proposed for analyzing 2D texture or image data. This paper introduces Gradient entropy (GradEn), an extension of slope entropy to 2D, which considers both symbolic patterns and amplitude information, enabling better feature extraction from image data. We evaluate GradEn with simulated data, including 2D colored noise, 2D mixed processes, and the logistic map. Results show the ability of GradEn to distinguish images with various characteristics while maintaining low computational cost. Real-world datasets, consist of texture, fault gear, and railway corrugation signals, demonstrate the superior performance of GradEn in classification tasks compared to other 2D entropy methods. In conclusion, GradEn is an effective tool for image characterization, offering a novel approach for image processing and recognition.

Fourier transform and entropy are two essential mathematical tools, and they have a fruitful role in system dynamics and machine learning research. In this paper, we propose a generalized composite multiscale amplitude dispersion entropy (GCMADE) and time-frequency dispersion entropy plane. GCMADE measures the complexity of the frequency domain of a time series and can approach the zero complexity of periodic sequences, unlike most other entropy methods. The time-frequency dispersion entropy plane further extracts time domain and frequency domain features of complex signals simultaneously through entropy. Its ability to measure the uncertainty of complex systems is analyzed by simulated data, and the results show that it can effectively distinguish between periodic sequences, chaotic sequences and stochastic processes. Finally, we introduce support vector machine (SVM) to perform mechanical fault diagnosis on five datasets. Compared with the other six algorithms, our method has significantly higher accuracy.

Topological permutation entropy is based on ordinal partition networks (OPNs) to approximate topological entropy of low-dimensional chaotic systems. But the ordinal patterns of OPN ignore the magnitude of the amplitude value. To solve the problem, we propose topological dispersion entropy (TDE) and weighted topological dispersion entropy (WTDE) based on dispersion patterns to characterize the complexity of a system. Furthermore, WTDE strengthens the topological structure analysis of complex networks by weighting the adjacency matrix of the improved ordinal partition networks, thereby more accurately capturing the dynamic evolution of time series. The proposed methods are comprehensively evaluated by numerical experiments. The results show that the performance of both TDE and WTDE is significantly better than TPE. Especially, WTDE has good stability to parameters, data length, and noise. Finally, combining support vector machines and K-Nearest Neighbor, TDE and WTDE are applied to the classification of physiological data and mechanical failure data.

Detecting dependence between variables is a crucial issue in statistical science. In this paper, we propose a novel metric called label projection correlation to measure the dependence between numerical and categorical variables. The proposed correlation does not require any conditions on numerical variables, and it is equal to zero if and only if the two variables are independent. When the numerical variable is one-dimensional, we demonstrate that the computational cost of the correlation estimation can be reduced to 𝒪(n log n), where n is the sample size. Additionally, if the one-dimensional variable is continuous, the correlation can be simplified to a concise rank-based expression. The asymptotic theorems of the estimation are also established. Two simulated experiments are presented to demonstrate the effectiveness of the proposed correlation in feature selection. Furthermore, the metric is applied to feature selection in drivers' facial images and cancer mass-spectrometric data.

Entropy serves as an effective method for quantifying the irregularity and complexity of nonlinear time series or complex signals. Recently, a novel entropy measure, attention entropy (AE), has been introduced for detecting interbeat interval time series. However, the original AE focuses solely on peak points, potentially overlooking crucial information embedded in signals. In this paper, we present the global ordinal pattern attention entropy (GOPAE), a novel measure that integrates AE with the principles of phase space reconstruction (PSR). Additionally, the connections between GOPAE and state-of-the-art time series network methods, including ordinal pattern transition network (OPTN) and recurrence quantification analysis (RQA), are elucidated to showcase its proficiency in extracting dynamic information from complex signals. Comparative experiments, both qualitative and quantitative, are conducted, using both simulated data and real-world signals. The results of the experiments suggest that GOPAE can effectively distinguishing complex signals in real application scenarios.

Analysis of correlation between time series is an essential step for complex system studies and dynamical characteristics extractions. Martingale difference correlation (MDC) theory is mainly concerned with the correlation of conditional mean values between response variables and predictor variables. It is the generalization and deepening of the Pearson correlation coefficient, Spearman correlation coefficient, Kendall correlation coefficient, and other statistics. In this paper, on the basis of phase space reconstruction, the generalized dependence index (GDI) is proposed by using MDC and martingale difference divergence matrix theories, which can measure the degree of dependence between time series more effectively. Moreover, motivated by the theoretical framework of the refined distance correlation method, the corresponding dependence measure (DE) is employed in this paper to construct the DE-GDI plane, so as to comprehensively and intuitively distinguish different types of data and deeply explore the operating mechanism behind the relevant time series and complex systems. According to the performances tested by the different simulated and real-world data, our proposed method performs relatively reasonably and reliably in dependence measuring and data distinguishing. The proposal of this complex data clustering method can not only recognize the features of complex systems but also distinguish them effectively so as to acquire more relevant detailed information.

Volatility clustering, widely observed in daily equity market returns, hasn’t been analyzed for high-resolution intraday and overnight returns, nor has its time scale dependency been systematically explored. This paper examines the volatility clustering of intraday and overnight returns in 15 global equity markets, both developed and emerging. Findings reveal universal volatility clustering in intraday and overnight returns across various time scales, from daily to monthly and beyond. It appears that the volatility clustering of overnight returns is even more pronounced than intraday returns. However, the cross clustering between two volatility series is generally weak within each market. These observations suggest both short- and long-term investment risks, providing meaningful insights for equity market investors’ risk management.

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.

In this paper, we introduce detrended cross-correlation analysis (DCCA) method into visibility graph (VG) algorithm and propose VGDCCA method to reflect the time series irreversibility from a new perspective. The validity and reliability of the proposed VGDCCA method is supported by numerical simulations on synthesized short-term correlated chaotic systems and long-term correlated fractal processes, and by the empirical analysis on stock indices and traffic parameter. The VGDCCA planes suggest that autoregressive fractionally integrated moving average (ARFIMA) and fractional Gaussian noise (FGN) series show time reversible behavior, whatever the long-term correlation of the series is or however strong the persistence or anti-persistence of the series is. Meanwhile, Logistic map with a=3.3 similar to 3.6 and H & eacute;non map show more time irreversible behaviors than those of ARFIMA, FGN series and other Logistic maps. It can be found that the relationship between cross-correlation of ingoing degree sequence and outgoing degree sequence for the simulated series with different parameters is consistent with the complexity and autocorrelation behavior in the corresponding definition of the time series. For the empirical analysis, VGDCCA method declares the similarity and dissimilarity between stock indices on time series irreversibility and captures the time irreversibility of traffic time series recorded by the detectors in different locations.

We propose a distance metric to quantify the dissimilarity between time series from the perspective of the frequency domain, especially non-stationary time series, called the Wasserstein Hilbert marginal spectral distance (WHMS). It calculates the Wasserstein distance between the Hilbert marginal spectral (HMS) of the time series, where HMS is the integral of Hilbert spectrum. Hilbert spectrum is obtained by the Hilbert transform performed on the intrinsic mode functions (IMFs) generated after the Empirical Mode Decomposition (EMD) operation. On this basis, we propose an IMF adaptive selection algorithm to improve the feature extraction accuracy of the HMS, free from a large number of prior experiments to set the threshold. We demonstrate that WHMS distance can be used as a general measure of time series from both theoretical and practical aspects. Combined with multidimensional scaling (MDS), we comprehensively evaluate the proposed method through simulation experiments and empirical data. The results confirm the good applicability of the WHMS distance combined with the IMF adaptive selection algorithm, which can distinguish different types of complex systems, is robust to noise, and is superior to Wasserstein-Fourier (WF) distance, Wasserstein-Fourier distance with short time Fourier transform(STFT), Euclidean distance and Chebyshev distance.

With the explosive growth of data quantity and rapid development of nonlinear dynamics as well as the growing demand for complex data classification in the field of artificial intelligence and machine learning, the research of complex time series, generated by complex systems, has attracted enormous interests. However, how to simultaneously distinguish different types of time series data and extract more accurate and detailed information from them in the light of localized and global scale perspectives remains significant and needs to be tackled. Thus, in this paper, we propose the fractional DisEn–Fisher plane, which is innovatively constructed by the fractional form of dispersion entropy and Fisher information measure. These are both effective tools to diagnose the essential properties of complex time series, to analyze the complexity of systems and to depict the contained statistical information with higher accuracy and effectiveness. Several classical entropy plane methods are selected as a comparison to design simulation experiments by simulated data and three real-world datasets. Comparative experimental results show that this method is a feasible and reliable improvement method, which will help to provide more additional information in time series recognition and dynamic characterization. It may not only provide new insights for further improvement of complex time series analysis, but also have important implications for developing complex data clustering methods.

In this paper, the cumulative residual Tsallis singular entropy (CRTSE) is introduced to measure the complex characteristics of nonlinear signals. Firstly, we do singular value decomposition on time series, which can reduce the interference of noise on information extraction, and the singular values represent the information characteristics of the signal. Then the statistical distribution of the signal is described by the cumulative residual function of singular values, and the Tsallis entropy is calculated to quantify the complexity. We verify the effectiveness and robustness of CRTSE through simulation experiments. Finally, we propose a grey wolf optimized support vector machine based on CRTSE called CRTSE-GWOSVM to intelligently diagnose complex systems. The results show that CRTSE can effectively measure the complex characteristics of time series, GWOSVM is superior to SVM and particle swarm optimized support vector machine (PSOSVM) in data identification, and CRTSE-GWOSVM model can identify complex systems more effectively and accurately.