Financial time series analysis based on fractional and multiscale permutation entropy

Permutation entropy (PE) has been proposed to reflect the complexity of time series in recent years, and it has a wide range of applications. In this paper, we extend PE to the fractional order through the parameter a, and then obtain fractional permutation entropy (FPE). FPE can increase the sensitivity of the complex system by adjusting the fractional order. We combine FPE with fractional Jensen-Shannon divergence (FJSD) to construct the entropy plane and discover that the entropy plane can classify various time series. In particular, it can distinguish the stock indices of developing countries and developed countries. We introduce FPE and FJSD to multiscale, and gain multiscale fractional permutation entropy (MFPE) and multiscale fractional Jensen-Shannon divergence (MFJSD). We employ simulated data and financial stock data to explore the relationship of MFPE and MFJSD under different scales, and the results show that some data are similar to direct proportion relationship and others data are sharply increasing curves, which portrays the internal mechanism of the stock market excellently. (C) 2019 Elsevier B.V. All rights reserved.