A new complexity measure: Modified discrete generalized past entropy based on grain exponent.

• We propose a new complexity measure of time series called modified discrete generalized past entropy based on grain exponent. • We validate the method with logistic map and Hénon map, it can characterize the feature of both map effectively, distinguish the periodic and chaotic state of system accurately and sensitively. • We compare the method with the discrete generalized past entropy based on oscillation-based grain exponent (O-DGPE), our method can characterize the complexity of a system more accurately. • Stock markets of different area are distinguished well by our method, and it can get the result that the US market and Hong Kong market are more mature than the Chinese mainland market, which is consistent with the reality. In this paper, we propose the modified discrete generalized past entropy based on grain exponent (GE-MDGPE), to analyze complex dynamical systems. Gao et al. proposed discrete generalized past entropy based on oscillation-based grain exponent (O-DGPE) method in 2019, which has been proved to be a good measure of uncertainty of time series. Whereas, it still has some drawbacks, such as the effectiveness of O-DGPE is not good when characterizing some special systems. In order to solve these drawbacks, we therefore generalize O-DGPE method to put forward GE-MDGPE which can better characterize complex systems. While using two artificial model (logistic map, Hénon map) to qualify the proposed method, we find that the method can characterize the system more accurately than O-DPGE, and can distinguish the periodic system and chaotic system effectively and sensitively. Moreover, we discuss the influence of parameters β and j on the proposed method. At last, we apply the proposed method to analyze the financial series which are extracting from six indices: three U.S. stock indices and three Chinese stock indices. The results show that the method can clearly distinguish the stock markets of different levels of development, and the U.S. market and the Hong Kong market are more mature than the Chinese mainland market. [ABSTRACT FROM AUTHOR]