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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]

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]