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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]

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]