Detecting asynchrony of two series using multiscale cross-trend sample entropy

We develop a new cross-sample entropy, namely the multiscale cross-trend sample entropy (MCTSE), to investigate the synchronism of dynamical structure regarding two series with potential trends. It incorporates symbolic representation and polynomial fitting trend. Numerical tests illustrated that the newly proposed MCTSE can address the series with various trends well and detect asynchrony between two series more sensitively. The MCTSE is used to analyze the asynchrony between air quality index (AQI) series of Beijing, Changsha and Zhuhai and the series of six air quality impact factors (PM2.5, PM10, \(\hbox {SO}_2\), CO, \(\hbox {NO}_2\) and \(\hbox {O}_3\)). We find that the consistency between the AQI series and the two fine particulates series of PM2.5 and PM10 is obviously higher than other four factors series. Comparing the entropy values of these pollution factors between every two cities, we also found that the pollutants in Changsha and Zhuhai are similar in quality, but they are quite different from those in Beijing. These findings help to further explore the similarities and differences in dynamic structure of the AQI series and identify the main pollutant. This has certain reference value for the reduction and control of similar pollutants.