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.

中文翻译：

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基于水平可见性图的复杂时间序列Jensen-Shannon散度

在本文中，我们提出了基于水平可见性图（HVG）的 Jensen-Shannon 散度（JSD）来有效地测量平稳和非平稳序列的时间序列不可逆性。首先进行数值模拟以证明所提出方法的有效性，然后研究金融时间序列和交通时间序列的经验应用。可以发现，JSD 在量化时间序列不可逆性方面表现出比 Kullback-Leibler 散度（KLD）更好的鲁棒性，并正确区分了不同类型的模拟序列。对于实证分析，基于 HVG 的 JSD 能够检测股票指数的显着时间不可逆性，并揭示不同股票指数之间的关系。JSD结果显示了不同检测器的速度时间序列的时间不可逆性，并且比KLD具有更好的准确性和鲁棒性。JSD 获得的基于时间不可逆行为的层次聚类将检测器分为四组。