Analysis of time series through complexity–entropy curves based on generalized fractional entropy

In this paper, we propose the complexity–entropy causality plane based on the generalized fractional entropy. When applying the proposed method into artificial time series and empirical time series, we find that both results show that the stochastic and chaotic time series are clearly distinguished. On the one hand, we could distinguish them according to the trend of the normalized generalized fractional entropy H as the parameter α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} increases. On the other hand, the stochastic and chaotic time series can be distinguished by the trend of their corresponding extreme values αC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _C$$\end{document} with the increase in embedding dimension m. However, compared with the q-complexity–entropy plane, the trend of their extreme value αC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _C$$\end{document} is irregular. Moreover, when applying the complexity–entropy causality plane into financial time series, we could obtain more accurate and clearer information on the classification of different regional financial markets.