## 4 Discussion and Conclusions  
In this work, we examined the descriptive statistics of PM\({}_{2.5}\) concentrations at 18 sites in Korea, and we confirmed a well-known pronounced seasonal trend in most series. Then, we selected five notable sites with lower and higher mass concentrations and then generated two groups, G1 and G2, in order to conduct comparative analyses on both. To determine the heterogeneity in underlying dynamics depending on the concentration levels, we analyzed the auto- and cross-correlation structures of both groups, i.e., G1 and G2, consisting of stations #15 and #18 and stations #5, #11, and #16, respectively. They showed a different behavior in DFA (detrended fluctuation analysis); G1 showed no crossover, while G2 showed strong evidence of crossover. This finding implies that G2 has two governing dynamics depending on the scale, while G2 has one dynamic. In terms of the impact of trans-boundary transport from China, G1 has little influence while G2 has strong influence. Furthermore, to confirm the heterogeneity between groups, we conducted DCCA (detrended cross-correlation analysis) on two series belonging to the same groups and different groups. Our finding strongly supports the heterogeneity in both. In the same group, coupling or synchronization increased as the scale increased, while, in the different group, decoupling between both series developed over increasing scale. This can be a starting point to categorize the specific region into several groups based on DCCA behaviors. Lastly, in order to reproduce the auto-correlation properties produced from real PM\({}_{2.5}\) time series, we presented the ARFIMA (auto-regressive fractionally integrated moving average) process. Although these models succeeded in reproducing crossover behaviors in the auto-correlation structure, they yielded no valid results in decoupling behavior among heterogeneous groups. Also, through the analysis of the ARFIMA process, we found slightly different behavior in the DCCA analysis when incorporating the trend term, whether exogenous or endogenous.  
In summary, the PM\({}_{2.5}\) accumulation process operates in a complicated way and shows different dynamical behavior depending on mass concentration. However, using simple DFA analysis, we could categorize all PM\({}_{2.5}\) time series into two groups with and without crossovers. Then, we confirmed that DCCA analysis works as an effective tool for checking the heterogeneity of DCCA coefficient behavior over different scales.  
Conceptualization, G.L. and S.M.; methodology, G.L. and S.M.; modeling, G.L. and S.M.; validation, S.M.; writing--original draft preparation, G.L.; writing--review and editing, S.M.; funding acquisition, S.M.  
Figure 11: DCCA fluctuation functions vs. box sizes for a pair of time series generated from the ARFIMA process with i.i.d. Gaussian variables and endogenous periodic trends of different amplitudes (A = 2 and A = 0.1). There is no complex value in fluctuation covariance, and the behaviors of DCCA coefficients are considerably different, as shown in the inset. (**a**) The cross-correlation is overwhelmed by the periodic trend, which is also shown in the scaling exponent, while (**b**) the randomness overwhelms the trend and, furthermore, its effect is accumulated in comparison to an exogenous case.  
**Funding:** This work was supported by the National Research Foundation of Korea (NRF) in a grant funded by the Korean government (MSIT) (No. 2017R1C1B5017491).  
**Conflicts of Interest:** The authors declare no conflicts of interest.