The dependence index based on martingale difference correlation: An efficient tool to distinguish different complex systems

Abstract

For the study of complex systems, the existing correlation measures suffer from deficiencies and may lead to the loss of information and inefficient characterizations of complex systems. We aim to capture detailed dynamical characteristics from complex systems effectively, including periodical features, chaotic features, and mixed features. Our objectives also include discriminating different complex systems and performing clustering of complex data with higher efficiency. So in this paper, we propose the dependence index (DI) based on the martingale difference correlation (MDC), the synchronization index, and the phase space reconstruction. Through simulation experiments, we demonstrate that our proposed method is effective in detecting different and subtle dynamical features, including periodic, chaotic, and mixed behaviors. For applications, we apply our method to analyze real-world data, including heartbeat signals, stock indices, bearing fault data, image outline data, CBF data, and food spectrum data. To achieve our objectives and demonstrate the applicability of our method, the DI is applied to multidimensional scaling (MDS) to cluster reality-based data with better performance when compared with the MDS based on other measures. We affirm that our method can distinguish different complex systems more effectively and obtain detailed information.