## 3 Results
### Cross-Correlation Structures  
In order for a power-law correlation to exist between two time series, two conditions must be satisfied; one is a common error term in both series, and the other involves sharing the same periodic term overwhelming random errors. As shown in Figure 9, PM\({}_{2.5}\) series showed a considerably robust behavior in power-law cross-correlation scaling although there were different behaviors in the DCCA coefficients, i.e., decoupling in different groups and coupling in the same groups. These results imply that all PM\({}_{2.5}\) series in the Korean peninsula, which are small compared with China and India, have at least one common error term and share a similar seasonal trend.  
Figure 8: Multifractal fluctuation functions (\(q=+5\)) vs. box size. As a reference, the authors added the plots of both groups of the surrogate dataset. For (**a**) G1 and (**b**) G2, the overall behavior is very similar to the detrended fluctuation analysis (DFA) functions. Also, the scaling exponents are not different from those of DFA.  
Figure 7: Multifractal fluctuation functions (\(q=-5\)) vs. box size. In order to determine the scaling range, the authors added the plots of both groups of the surrogate dataset. (**a**) there is no sign of stable scaling range for stations #3 and #5. (**b**) for stations #2 and #4, a noticeable crossover appears in the large scale.  
According to the simulation results based on the ARFIMA process with periodic trends, both time series of the same periodic trend with independent identically distributed (i.i.d.) Gaussian variables showed an incomplete behavior in DCCA scaling, i.e., with complex-valued covariance in small scales, as shown in Figure 10. Depending on the characteristics of a periodic trend, i.e., exogenous or endogenous, there are two distinguishable tendencies. When the amplitude of a trend function is large enough to overwhelm the randomness, the endogenous factor promotes coupling instead of the exogenous one. On the other hand, decoupling lasts longer in terms of DCCA coefficients in the endogenous case than the exogenous case, as shown in Figures 10b and 11b. This may be due to the accumulation effect in the endogenous process because a trend is incorporated into the underlying dynamics. Also, an incomplete covariance appeared in cases where both time series had a common error term but periodic terms with different periods. Therefore, the authors concluded that the PM\({}_{2.5}\) series shared more than one factor governing the dynamics, and that they were governed by a similar seasonal trend with variable amplitudes.  
Figure 10: (**a**) DCCA fluctuation functions plotted with respect to box size for a pair of time series generated from the ARFIMA process with independent identically distributed (i.i.d.) Gaussian variables and exogenous periodic trends of different amplitudes (A = 2 and A = 0.1). (**b**) the complex values in fluctuation covariance increase in size as the randomness increases due to a difference in amplitudes, and the correlation coefficient increases slowly.  
Figure 9: Detrended cross-correlation analysis (DCCA) fluctuation functions are plotted with respect to box size for a pair of time series from the same group and different groups. Different behaviors in the DCCA coefficients can be observed, as shown in insets: coupling in (**a**) and decoupling in (**b**). Stable cross-correlated power-law behaviors remain in both, stressing that at least one common factor governs their dynamics.