Multivariate multiscale fractional order weighted permutation entropy of nonlinear time series

In this letter, multivariate multiscale fractional permutation entropy (MMFPE) and multivariate weighted multiscale fractional permutation entropy (MWMFPE) have been proposed to provide insights for the study of time series. When measuring the dynamics of complex systems, the MMFPE and MWMFPE methods are sensitive to the signal evolution. Meanwhile, they can provide some analysis of complexity over multiple time series as well as multiple channel signals. We perform these methods on synthetic tri-variate time series to explore some of the interesting properties, especially for negative information and information deception. It can be seen that more complex system is more likely to be deceptive. The amplitude information of time series which is taken into account in the MWMFPE can weaken this deception. The methods are also employed to the closing prices and trade volume of financial stock markets from different areas. According to the MWMFPE results, the indices can be divided into three groups: (1) CAC40, HSI, NASDAQ, S&P500, (2) N225, and (3) ShenCheng, implying that it has a capacity to distinguish these financial stock market.