Generalized multivariate multiscale sample entropy for detecting the complexity in complex systems

Abstract

Using the first moment in coarse-graining the original time series in different time scales will lose some important information of the complexity. In this paper, we propose the generalized multivariate multiscale sample entropy (GMMSE) to analyze the complexity for multivariate time series. Before empirical analysis, numerical simulation is conducted by applying GMMSE1 (mean), GMMSE2 (variance), GMMSE3 (skewness) to the autoregressive fractionally integrated moving average (ARFIMA) processes. The simulated results illustrate that GMMSE2 is more appropriate to analyze the complexity of ARFIMA time series. Then we apply GMMSE method to traffic multivariate time series and discuss their complexities over different time scales. Comparing the GMMSE1, GMMSE2 and GMMSE3 results for different detectors, GMMSE2 and GMMSE3 can clearly differentiate the detectors with different complexity and reflect the characteristics and similarities of the complexity of traffic data between different locations. Besides, both the effect of traffic accident and the effect of the weekday and weekend patterns containing in traffic signals on the complexity can be detected by GMMSE2 and GMMSE3 methods. GMMSE2 and GMMSE3 are capable of providing more important information and knowledge concerning the complexity of traffic time series and will bring enhanced understanding of the properties of traffic system.