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For nearly a decade, quality-related fault detection algorithms have been widely used in industrial systems. However, the majority of these detection strategies rely on static assumptions of the operating environment. In this paper, taking the time series of variables into consideration, a dynamic kernel entropy component regression (DKECR) framework is proposed to address the instability of quality-related fault detection due to the existing dynamic characteristics. Compared with the typical kernel entropy component analysis method, the proposed method constructs the relationship between process states and quality states to further interpret the direct effect on the product taken by the fault. In the proposed approach, process measurements are converted to a lower-dimensional subspace with a specific angular structure that is more comprehensive than traditional subspace approaches. In addition, the angular statistics and their relevant thresholds are exploited to enhance the quality-related fault detection performance. Finally, the proposed method will be compared with three methods by means of a numerical example and two industrial scenarios to demonstrate its practicality and effectiveness. Note to Practitioners—This paper studies a quality-related fault detection problem for the dynamic nonlinear industrial system. Controlling and measuring the quality state is challenging for the high-level monitoring system of the manufacturing process due to the nonlinear dynamic feature in states. This paper proposes a new data-driven method based on the kernel entropy component analysis method to assess the correlation between the quality and fault in the industrial system, reducing unnecessary overhaul and maintenance. Based on the autoregressive moving average exogenous algorithm, the proposed method captures the dynamic interaction between the process states to decrease false alarms. In the experimental section, the DKECR method outperforms the compared approaches, which can provide stable fault detection results. Additionally, the unique angle structure of the proposed method can supply more information for engineers’ monitoring needs. [ABSTRACT FROM AUTHOR]

In this paper, we propose Jensen–Shannon divergence (JSD) based on horizontal visibility graph (HVG) to measure the time series irreversibility for both stationary and non-stationary series efficiently. Numerical simulations are first conducted to show the validity of the proposed method and then empirical applications to the financial time series and traffic time series are investigated. It can be found that JSD shows better robustness than Kullback–Leibler divergence (KLD) on quantifying time series irreversibility and correctly distinguishes the different type of simulated series. For the empirical analysis, JSD based on HVG is able to detect the significant time irreversibility of stock indices and reveal the relationship between different stock indices. JSD results show the time irreversibility of speed time series for different detectors and present better accuracy and robustness than KLD. The hierarchical clustering based on their behavior of time irreversibility obtained by JSD classifies the detectors into four groups. [ABSTRACT FROM AUTHOR]

In this paper, we develop the concept of forbidden/missing ordinal patterns into the forbidden/missing joint ordinal patterns and propose the ratio of the number of missing joint ordinal patterns (RMJPs) as a sign of interdependence. RMJP in a surrogate analysis can be used to differentiate the forbidden joint ordinal patterns from the missing joint ordinal patterns due to small sample effects. We first apply RMJP to the simulated time series: a two-component autoregressive fractionally integrated moving average process, the Hénon map, and the Rössler system using active control and discuss the effect of the length of the time series, embedding dimension, and noise contamination. RMJP has been proven to be capable of measuring the interdependence in the numerical simulation. Then, RMJP is further used on the electroencephalogram (EEG) time series for empirical analysis to explore the interdependence of brain waves. With results by RMJP obtained from a widely used open dataset of the sleep EEG time series from healthy subjects, we find that RMJP can be used to quantify the brain wave interdependence under different sleep/wake stages, reveal the overall sleep architecture, and indicate a higher level of interdependence as sleep gets deeper. The findings are consistent with existing knowledge in sleep medicine. The proposed RMJP method has shown its validity and applicability and may assist automatic sleep quantification or bring insight into the understanding of the brain activity during sleep. Furthermore, RMJP can be used on sleep EEG under various pathological conditions and in large-scale sleep studies, helping to investigate the mechanisms of the sleep process and neuron synchronization. [ABSTRACT FROM AUTHOR]

Abstract 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. Highlights • We propose MJPE method to study the synchronism between two time series. • We apply MJPE method to the simulated time series to test the validity. • The results show the necessity of multiscale and prove the effectiveness of MJPE. • MJPE method is employed to the financial and traffic time series. • MJPE results present the synchronisms from multiscale view successfully. [ABSTRACT FROM AUTHOR]

Scalar time series considered in most studies may be not sufficient to reconstruct the dynamics, while using multivariate time series may demonstrate great advantages over scalar time series if they are available. Multivariate time series are available in the traffic system and we intend to examine the issue for the real data in the traffic system. In this paper, we propose the multivariate predicting method and discuss the prediction performance of multivariate time series by comparison with univariate time series and K-nearest neighbor (KNN) nonparametric regression model. The three kinds of forecast accuracy measure for multivariate predicting method are smaller than those for the other two methods in all cases, which suggest the predicting results for traffic time series by multivariate predicting method are better and more accurate than those based on univariate time series and KNN model. It demonstrates that the proposed multivariate predicting method is more successful in predicting the traffic time series than univariate predicting method and KNN method. The multivariate predicting method has a broad application prospect on prediction because of its advantage on recovering the dynamics of nonlinear system. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the composite multiscale cross-sample entropy (CMCSE) which may induce undefined entropies and then further propose the refined composite multiscale cross-sample entropy (RCMCSE) which modifies CMCSE. First, we apply multiscale cross-sample entropy (MCSE), CMCSE and RCMCSE methods to three types of artificial time series in order to test the validity and accuracy of these methods. Results show that RCMCSE reduces not only standard deviation, but also the probability of inducing undefined entropy effectively, which can provide better robustness and more accurate entropies. Then, these three methods are employed to investigate financial time series including US and Chinese stock indices. For the study between stock indices in the same region, some conclusions which are consistent with previous study are drawn by the RCMCSE results. Meanwhile, it can be found that undefined entropies are induced and the numbers of inducing undefined entropy by three methods for investigation between three US stock indices and two Chinese mainland stock indices are given. Compared with MCSE and CMCSE, RCMCSE method is capable of reducing the number of undefined entropy and providing more accurate entropies. Moreover, the differences on inducing undefined entropy between results for US stock indices & two Chinese mainland stock indices and results for US stock indices & HSI demonstrate a much closer relation between US stock markets and HSI than between US stock markets and two Chinese mainland stock markets. Hence, it can be concluded that RCMCSE is more applicable for the study between US and Chinese stock markets. [ABSTRACT FROM AUTHOR]

In this paper, we introduce weighted permutation entropy (WPE) and three different symbolic approaches to investigate the complexities of stock time series containing amplitude-coded information and explore the influence of using different symbolic approaches on obtained WPE results. We employ WPE based on symbolic approaches to the US and Chinese stock markets and make a comparison between the results of US and Chinese stock markets. Three symbolic approaches are able to help the complexity containing in the stock time series by WPE method drop whatever the embedding dimension is. The similarity between these stock markets can be detected by the WPE based on Binary Δ -coding-method, while the difference between them can be revealed by the WPE based on σ -method, Max–min-method. The combinations of the symbolic approaches: σ -method and Max–min-method, and WPE method are capable of reflecting the multiscale structure of complexity by different time delay and analyze the differences between complexities of stock time series in more detail and more accurately. Furthermore, the correlations between stock markets in the same region and the similarities hidden in the S&P500 and DJI, ShangZheng and ShenCheng are uncovered by the comparison of the WPE based on Binary Δ -coding-method of six stock markets. [ABSTRACT FROM AUTHOR]

We introduce an entropic segmentation algorithm and apply it to decompose the financial sequences into compositionally homogeneous domains. To probe more about the nature of the financial time series, we investigate the statistical properties of the segment from the view of segmentation position and segment length first. We reveal some important and interesting conclusions and information hidden in these time series of stock markets. Then, we focus on the study of the intrinsic properties for each segment in the time series from two aspects: time irreversibility and correlation. The fluctuations on the time irreversibility and the scaling exponent all support that the segments present compositional heterogeneity and verify the segmentation. Meanwhile, we conclude that time irreversibility is inherent in the stock time series and verifies that stock markets are nonequilibrium systems essentially even though segmentation. Moreover, the scaling exponents for each segment point out that the traditional detrended fluctuation analysis is not applicable to measure the correlation for the whole original time series of stock market. [ABSTRACT FROM AUTHOR]

For researching multiscale behaviors from the angle of entropy, we propose a modified cross sample entropy (MCSE) and combine surrogate data analysis with it in order to compute entropy differences between original dynamics and surrogate series (MCSDiff). MCSDiff is applied to simulated signals to show accuracy and then employed to US and Chinese stock markets. We illustrate the presence of multiscale behavior in the MCSDiff results and reveal that there are synchrony containing in the original financial time series and they have some intrinsic relations, which are destroyed by surrogate data analysis. Furthermore, the multifractal behaviors of cross-correlations between these financial time series are investigated by multifractal detrended cross-correlation analysis (MF-DCCA) method, since multifractal analysis is a multiscale analysis. We explore the multifractal properties of cross-correlation between these US and Chinese markets and show the distinctiveness of NQCI and HSI among the markets in their own region. It can be concluded that the weaker cross-correlation between US markets gives the evidence for the better inner mechanism in the US stock markets than that of Chinese stock markets. To study the multiscale features and properties of financial time series can provide valuable information for understanding the inner mechanism of financial markets. [ABSTRACT FROM AUTHOR]

In this paper, we propose multiscale detrended cross-correlation analysis (MSDCCA) to detect the long-range power-law cross-correlation of considered signals in the presence of nonstationarity. For improving the performance and getting better robustness, we further introduce the empirical mode decomposition (EMD) to eliminate the noise effects and propose MSDCCA method combined with EMD, which is called MS-EDXA method, then systematically investigate the multiscale cross-correlation structure of the real traffic signals. We apply the MSDCCA and MS-EDXA methods to study the cross-correlations in three situations: velocity and volume on one lane, velocities on the present and the next moment and velocities on the adjacent lanes, and further compare their spectrums respectively. When the difference between the spectrums of MSDCCA and MS-EDXA becomes unobvious, there is a crossover which denotes the turning point of difference. The crossover results from the competition between the noise effects in the original signals and the intrinsic fluctuation of traffic signals and divides the plot of spectrums into two regions. In all the three case, MS-EDXA method makes the average of local scaling exponents increased and the standard deviation decreased and provides a relative stable persistent scaling cross-correlated behavior which gets the analysis more precise and more robust and improves the performance after noises being removed. Applying MS-EDXA method avoids the inaccurate characteristics of multiscale cross-correlation structure at the short scale including the spectrum minimum, the range for the spectrum fluctuation and general trend, which are caused by the noise in the original signals. We get the conclusions that the traffic velocity and volume are long-range cross-correlated, which is accordant to their actual evolution, while velocities on the present and the next moment and velocities on adjacent lanes reflect the strong cross-correlations both in temporal and spatial dimensions. We also reveal the similarity and uniqueness in the cross-correlation situations between velocities. Besides, signals on one lane show stronger long-range cross-correlation than that on adjacent lanes. Thus, the multiscale cross-correlation structure acquired by MS-EDXA is more close to the intrinsic mechanism of traffic system and reflects more accurate and more abundant traffic information. [ABSTRACT FROM AUTHOR]

Abstract: We use multiscale detrended fluctuation analysis (MSDFA) and multiscale detrended cross-correlation analysis (MSDCCA) to investigate auto-correlation (AC) and cross-correlation (CC) in the US and Chinese stock markets during 1997–2012. The results show that US and Chinese stock indices differ in terms of their multiscale AC structures. Stock indices in the same region also differ with regard to their multiscale AC structures. We analyze AC and CC behaviors among indices for the same region to determine similarity among six stock indices and divide them into four groups accordingly. We choose S&P500, NQCI, HSI, and the Shanghai Composite Index as representative samples for simplicity. MSDFA and MSDCCA results and average MSDFA spectra for local scaling exponents (LSEs) for individual series are presented. We find that the MSDCCA spectrum for LSE CC between two time series generally tends to be greater than the average MSDFA LSE spectrum for individual series. We obtain detailed multiscale structures and relations for CC between the four representatives. MSDFA and MSDCCA with secant rolling windows of different sizes are then applied to reanalyze the AC and CC. Vertical and horizontal comparisons of different window sizes are made. The MSDFA and MSDCCA results for the original window size are confirmed and some new interesting characteristics and conclusions regarding multiscale correlation structures are obtained. [Copyright &y& Elsevier]

Using the first moment in coarse-graining the original time series in different time scales will lose some important information of the complexity. In this paper, we propose the generalized multivariate multiscale sample entropy (GMMSE) to analyze the complexity for multivariate time series. Before empirical analysis, numerical simulation is conducted by applying G M M S E 1 (mean), G M M S E 2 (variance), G M M S E 3 (skewness) to the autoregressive fractionally integrated moving average (ARFIMA) processes. The simulated results illustrate that G M M S E 2 is more appropriate to analyze the complexity of ARFIMA time series. Then we apply GMMSE method to traffic multivariate time series and discuss their complexities over different time scales. Comparing the G M M S E 1 , G M M S E 2 and G M M S E 3 results for different detectors, G M M S E 2 and G M M S E 3 can clearly differentiate the detectors with different complexity and reflect the characteristics and similarities of the complexity of traffic data between different locations. Besides, both the effect of traffic accident and the effect of the weekday and weekend patterns containing in traffic signals on the complexity can be detected by G M M S E 2 and G M M S E 3 methods. G M M S E 2 and G M M S E 3 are capable of providing more important information and knowledge concerning the complexity of traffic time series and will bring enhanced understanding of the properties of traffic system. • We propose the generalized multivariate multiscale sample entropy (GMMSE). • GMMSE 2 is more appropriate to analyze the complexity of ARFIMA time series. • GMMSE 2 and GMMSE 3 clearly differentiate traffic detectors with different complexity. • GMMSE 2 and GMMSE 3 can detect both the effect of traffic accident and week patterns. • GMMSE 2 and GMMSE 3 provide more information on the complexity of traffic series. [ABSTRACT FROM AUTHOR]