Multiscale adaptive multifractal cross-correlation analysis of multivariate time series

In the real world, most time series generated from complex systems are nonlinear. To effectively study their fractal properties, in this work, we first generalize the adaptive fractal analysis (AFA) to the adaptive multifractal cross-correlation analysis (AMFCCA), which can be used to study the multifractal cross-correlation between two time series. Considering the complexity of time series from complex systems, we extend AMFCCA to the case of multivariate time series, namely multivariate adaptive multifractal cross-correlation analysis (MV-AMFCCA). In order to detect multiscale multifractality, we propose multiscale multivariate adaptive multifractal cross-correlation analysis (MMV-AMFCCA). By the numerical simulation on synthetic multivariate processes, our method shows the theoretical validity. Furthermore, we applied these methods to the multifractal analysis of three pollutants in urban and suburban areas in Beijing. After removing the seasonal trend, we find that the urban and suburban systems both have multifractality, especially the multifractality of the urban system are more evident than the suburban systems, and the degree of multifractality in spring and winter stronger than that in summer and autumn. Therefore, our methods are suitable for systems with multiple outputs and provide more comprehensive methods for characterizing multivariate autocorrelation and cross-correlation.