A novel and effective method for characterizing time series correlations based on martingale difference correlation

Analysis of correlation between time series is an essential step for complex system studies and dynamical characteristics extractions. Martingale difference correlation (MDC) theory is mainly concerned with the correlation of conditional mean values between response variables and predictor variables. It is the generalization and deepening of the Pearson correlation coefficient, Spearman correlation coefficient, Kendall correlation coefficient, and other statistics. In this paper, on the basis of phase space reconstruction, the generalized dependence index (GDI) is proposed by using MDC and martingale difference divergence matrix theories, which can measure the degree of dependence between time series more effectively. Moreover, motivated by the theoretical framework of the refined distance correlation method, the corresponding dependence measure (DE) is employed in this paper to construct the DE-GDI plane, so as to comprehensively and intuitively distinguish different types of data and deeply explore the operating mechanism behind the relevant time series and complex systems. According to the performances tested by the different simulated and real-world data, our proposed method performs relatively reasonably and reliably in dependence measuring and data distinguishing. The proposal of this complex data clustering method can not only recognize the features of complex systems but also distinguish them effectively so as to acquire more relevant detailed information.