The dependence measurements based on martingale difference correlation and distance correlation: Efficient tools to distinguish different complex systems

The study of the relationships between time series is the basis of nonlinear dynamics and especially plays an important role in revealing the dynamical relationships between different systems. So in this work, we use the volatility martingale difference divergence matrix (VMDDM) to construct the dependence index (DI) from the perspective of the martingale difference correlation (MDC) and the phase space reconstruc-tion. The dependence measure (DM) is also structured based on the phase space reconstruction and the refined version of distance correlation (RDC), which involves a modified distance dependence statistic. Through simulation experiments, it is demonstrated that our proposed methods are efficient in discover-ing various subtle dynamical characteristics. For applications, data from the real world are used, contain-ing human heartbeat data, stocks indices series, data of complex image outlines, artificial CBF data, and data of food (coffee) spectrum. We are sure that the proposed approaches can obtain information with more specifics when compared with the state-of-the-art cross-sample entropy (CSE) and the frequently used synchrony measuring technique, the synchronization index (SI). Moreover, for complex systems, we construct the DM-tr(VMDDM) plane to achieve distinct and reasonable results of clustering.(c) 2022 Elsevier Ltd. All rights reserved.