## 3 Results  
### Autocorrelation Structures  
The DFA approach is a well-known method to quantify the degree of correlation properties in non-stationary time series, and it was successfully applied to various fields such as finance, meteorology, cardiac dynamics, and so on. However, there are some important problems in estimating the scaling index, such as changes in the index of scale, crossover, and scaling range. In this work, the authors firstly determined the scaling range using the surrogate dataset generated from the original series using two methods, i.e., shuffling and Fourier phase randomization, as shown in Figure 4[29]. Herein, the authors set the scaling range to the scale interval [10, 200] days. Then, the authors estimated the correlation exponent by linearly fitting the double-logarithmic plots of the detrended fluctuation functions vs. box sizes using the least-squares method. The authors confirmed that the PM\({}_{2.5}\) series showed a weak persistent behavior, as shown in Figure 5.  
Figure 4: Detrended fluctuations vs. box size plotted for two groups of surrogate data, namely, S1 and S2. S1 represents the shuffled series of an original time series, and S2 represents the shuffled series of a Fourier-phase randomized series of an original series. They have almost the same kurtosis. The authors determined the scaling range to be in the interval of [10, 200] days, which was free from the finite size effect. All other series showed a similar behavior. Figure 5: Detrended fluctuation variance vs. box size. (**a**) G1 shows no evidence of crossover, meaning no periodic trend or one with less amplitude, while (**b**) G2 exhibits pronounced evidence of crossovers, meaning the existence of periodic trends with variable amplitudes. Furthermore, crossover behavior was observed in G2 although there were variations in the degree of crossover, while G1 showed little evidence of crossovers. Considering the seasonality inherent in air pollutant series, this crossover behavior was mainly due to the seasonal trend. In order to confirm this possibility, the authors examined the crossover behaviors arising from the artificially generated time series with periodic trends. To this end, the authors used a stationary linear ARFIMA process to generate a power-law auto-correlated time series as follows [30]:  
\[y(t)=\sum_{i=1}^{\infty}a_{i}(\rho)y(t-i)+\eta(t), \tag{13}\]  
where \(\rho\) (range of \(-0.5\) and \(0.5\)) is a free parameter, \(a_{i}(\rho)\) represents weights defined by \(a_{i}(\rho)=-\Gamma(i-\rho)/[\Gamma(-\rho)\Gamma(1+i)]\), \(\Gamma(x)\) denotes the gamma function, and \(\eta(t)\) denotes an independent identically distributed (i.i.d.) Gaussian variable. The parameter \(\rho\) is related to a Hurst exponent such as \(=2+\rho\). Here, the authors added periodic trends to the ARFIMA process in two ways: one was exogenous and the other was endogenous, as shown below. Exogenous trend:  
\[y_{p}(t)=y(t)+\text{Asin}(2\pi t/T). \tag{14}\]  
Endogenous trend:  
\[y_{p}(t)=\sum_{i=1}^{\infty}a_{i}(\rho)y_{p}(t-i)+\text{Asin}\Big{(}\frac{2 \pi t}{T}\Big{)}+\eta(t). \tag{15}\]  
In the analysis, the authors set \(\rho=0.2\) in order to obtain a similar scaling exponent with that of the PM\({}_{2.5}\) series. The data size was also same as the PM\({}_{2.5}\) time series and the period \(T\) was set to 360 days. As shown in Figure 6, the authors confirmed that the crossover behavior in DFA analysis could be adjusted by the degree of periodic trends with different amplitudes. Furthermore, it is likely that a periodic trend with small amplitude existed, although no clear evidence existed in the DFA analysis. However, the authors could not determine which trend was more appropriate to describe the behavior observed in the PM\({}_{2.5}\) series. Another interesting finding was a big difference observed in the crossover behavior of stations #4 and #5, although they were located in the same city.