Multiscale joint permutation entropy for complex time series

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.