Fractional multiscale phase permutation entropy for quantifying the complexity of nonlinear time series

Abstract

Permutation entropy (PE) has been regarded as a most successful measure for the complexity of the time series. To overcome the undeniable shortcomings of PE is some cases, this paper designs a novel complexity algorithm called multiscale weighted phase permutation entropy (MWPPE). The proposed MWPPE adopts phase transformation, weight influence and multiscale information to improve PE, which can help us understand the complexity of nonlinear time series in depth. The method is also further extended to fractional order to obtain fractional multiscale phase permutation entropy (FMPPE). Based on the simulation sequence, a deep and systematic discussion is carried out on the effectiveness of the proposed two complexity measure algorithms, and results show that the proposed algorithms can amplify the detection effect of dynamic changes. Aiming at the financial markets of many countries and regions, the dynamic properties of financial time series with stock index are analyzed. It is concluded that compared with the MWPPE method, the FMPPE strategy can distinguish developed country stock index and emerging country stock index more effectively.