Binary indices of time series complexity measures and entropy plane

Abstract

Complexity of time series is an important feature of dynamical systems such as financial systems. In order to bring out the complete non-linear behavior of financial time series, non-linear tools of complexity measurement like entropy measures are indispensable. Sample entropy (SampEn) and distribution entropy (DistEn) are popular methods of assessing the complexity in various fields. However, both sample entropy and distribution entropy show some limitations in detecting the complexity of stock markets. Therefore, we use two entropies as binary indices to analyze the complexity of time series and structure the entropy plane. In order to further improve the accuracy of the research results, we take the embedding dimension m as the variable, expand the entropy point into the entropy curve for further research. Furthermore, considering the shortcomings of sample entropy and distribution entropy in practical application, we generalize them by (i) replacing sample entropy and distribution entropy with multi-scale sample entropy and multi-scale distribution entropy; (ii) generalizing Shannon entropy as Tsallis entropy. Also, scale factor τ and entropic index q are taken respectively as variables to get the entropy curves. By using the artificial data, we confirm the rationality of using entropy points and entropy curves to study the complexity of time series. Finally, we apply this method to measure the complexity of real world financial time series, the results show that the entropy curves plotted by the financial time series obtained from different areas have significant differences.