Investigation of the Implications of “Haze Special Law” on Air Quality in South Korea

In this work, the air pollution index in three cities (Seoul, Busan, and Daegu) in South Korea was studied using multifractal detrended fluctuation analysis (MF-DFA). Hurst, Renyi, and Holder exponents were used to analyze the characteristics of the concentration time series of PM2.5 and NO2. ,e results showed that multifractality exists in each season interval and the multifractal degree of PM2.5 is stronger than that of NO2. To investigate the effects of the implementation of the “haze special law” on February 15, 2019, we analyzed the time series of PM2.5 during the time periods from February 15, 2018, to December 16, 2018, and February 15, 2019, to December 16, 2019. We found that the multifractal spectrum width after the implementation of the law was narrower than that before the law for all the cities, which shows that the enactment of the law has played a role in improving the efficiency of air pollution control in South Korea. We also conclude that the major effects of the law will be particularly visible in larger cities. To study the main causes of multifractality, the shuffled and phase-randomized series were analyzed using MFDFA, and the results demonstrated that the fat-tailed distribution resulted in the multifractality of the time series before and after the implementation of the “haze special law” in Seoul and Daegu, whereas long-range correlation resulted in multifractality of the series before and after the implementation of the law in Busan.


Introduction
With the rapid development of a social economy and the acceleration of industrialization and urbanization, air pollution has gradually become an important problem affecting people's lives and has therefore garnered considerable attention from the public and government. Air pollution is closely related to climate, ecology, and health. Studies have shown that sulfur oxides and nitrogen oxides can cause great harm to the respiratory system, even leading to respiratory failure in severe cases [1]. erefore, many researchers have started to investigate the atmospheric particulate matters. Many studies have stated that PM 2.5 and NO 2 are the two major pollutants among various air pollutants [2][3][4][5][6]. erefore, in this research, we focused on PM 2.5 and NO 2 and investigated the two pollutants based on the relationship between the air pollution index (API) and different seasons. We also checked whether the implementation of the "haze special law" on February 15, 2019, had an impact on improving the air quality in South Korea using multifractal detrended fluctuation analysis (MF-DFA).
We know that the MF-DFA model can unveil the multifractal properties hidden in nonstationary time series and that multifractality originates from temporal correlations [7,8]. erefore, the multifractal properties of stock markets [9][10][11][12][13], foreign exchange markets [14][15][16], bitcoin markets [17], atmospheric sciences [18], and phase transitions [19] have been studied previously. Recently, multifractal analysis was used to study the daily air temperature time series [20]. Moreover, MF-DFA is also an efficient method in analyzing the human heart rate time series [21]. Based on the analysis of blood pressure and heart-rate complexity, Gender Castiglioni et al. [22] used the multifractal technique to investigate cardiovascular warning signals. Moreover, MF-DFA has been used in some studies to implement specific image analysis [23,24].
In recent years, many studies on air pollutants have been conducted and the results suggested that the increased levels of PM 2.5 are associated with a higher mortality and some negative effects on the lungs [25]. Although there are many studies on API, few studies have focused on the MF-DFA of the API time series. rough the hourly PM 2.5 average concentration time series, Shi et al. [26] analyzed the multifractal nature at four air monitoring stations of Chengdu using MF-DFA. Zhang et al. [27] used the multifractal detrended cross-correlation analysis (MF-DCCA) to analyze the cross-correlations between PM 2.5 and meteorological factors. Recently, Zhang et al. [28] analyzed the multifractal characteristics of the PM 2.5 time series in Hong Kong using the empirical mode decomposition-based MF-DFA method. Multifractal property links between meteorological factors and pollutants in urban and rural areas have been verified by He [29].
Based on MF-DFA, the variations of multifractal characteristics of pollutants present in different seasons were investigated in this study. e effects of seasonal factors on PM 2.5 and NO 2 were analyzed, and the seasonal physical changes of PM 2.5 and NO 2 concentrations were confirmed using multifractality. Additionally, we conducted a multifractal comparison analysis of the effectiveness of air pollution renovation since the enactment of the "haze special law." From this, we can provide a basis for formulating a scientific and effective comprehensive air pollution control policy. e paper is organized as follows. We briefly outline the procedure of MF-DFA in Section 2. In the next section, we describe the data information. Section 4 illustrates the empirical results. Section 5 concludes the paper.

Methodology
In this section, we describe the utilization of the MF-DFA [30] to measure the multifractal behavior of haze in all seasons, and the multifractal characteristics of the API time series before and after the implementation of the "haze special law." Kantelhardt summarized the technical details of MF-DFA as follows.
As a time series X i starts from X 1 to X N , where N is the length of the signal, the corresponding summation sequence is constructed by the following integration: where X is the mean value of X i . Subsequently, the profile Y is further divided into N s nonoverlapping windows of equal length s. In most cases, the scale s is not a multiple of the time series and a short part at the end of series Y exists. To overcome the problem of information being lost in the division process, the same process is repeated starting from the other side of the series. us, 2N s windows are acquired. Next, the least squares method is used to fit the data for evaluating the local trend of each window v (v � 1, 2, . . . , 2N s ), and the fitting polynomial in the v th window is denoted by y v (i). e variance is determined by the detrended time series, which is calculated as the difference between Y and y, and the resultant equation is described as if v � 1, 2, . . . , N s , and if v � N s + 1, N s + 2, . . . , 2N s . Finally, computing the mean of 2N s windows, the q order wave function F q (s) is obtained as if q � 0; according to L'Hôpital's rule, rough the analysis of double log plots of F q (s) versus s and a varied q, the scaling behavior of the fluctuation is determined by the power-law F q (s) ∝ s h(q) , and from this, a family of scaling exponents h(q), which are generalized Hurst components, can be obtained. F q (s) is the standard DFA, if q � 2. e Hurst exponents provide information on the time series, such as power-law correlated behavior, and when 0 < h(q) < 0.5, it indicates that the time series has a negative or antipersistence property. When 0.5 < h(q) < 1, then the time series has a positive persistence, and h(2) � 0.5 indicates that the time series has an uncorrelated Brownian process.
e Renyi exponent τ(q), which is related to the general Hurst exponent, can be expressed by In addition, where α represents the Holder exponent and characterizes the singularity strength and f(α) is a fractal dimension of the set of points with particular α. In the plotted curve between α and f(α), the shape resembles an inverted parabola and the degree of their complexity is denoted by the width of their fractal strength Δα [31].

Data Collection
We used the API time series of Seoul to study the multifractal characteristics of API sequences in different seasons.
To gauge the impact of the implementation of the "haze special law," our data set covered 3 cities in South Korea. e 3 cities are located in different parts of South Korea, and 2 Complexity there is a certain distance between any two cities. e results calculated from the samples selected in this manner fully reflect the impact and effect of the implementation of the "haze special law" over the entirety of South Korea. e experimental data sources for the PM 2.5 (μg/m 3 ) and NO 2 (ppm) concentrations were the "Korean Ministry Of Environment" and "Korea Environment Corporation," and more specific data were provided by the monitoring stations in South Korea. For more detailed information, please refer to https://www.airkorea.or.kr/web and https://aqicn.org/ city/seoul/kr.

Experiment Results
We first analyze the multifractal properties for two time series of each season and compare the multifractal degree of       Complexity PM 2.5 with that of NO 2 . Second, we conduct multifractal analysis for two time series in the same season and confirm whether the implementation of the "haze special law" has any implications for the air quality.
In this study, we choose the minimum segment scale to be s min � 5, and the maximum segment scale s max � 15, and q is taken from −10 to 10. e range of q was similarly chosen for the multifractal analysis of stock prices and returns in Latin-American stock markets [32]. Besides, the range of s was selected according to suggestions in [33,34]. 2.5 and NO 2 . From the daily API data selected in Section 3, we draw the time series of PM 2.5 and NO 2 in Figure 1. From Figure 1, we can see that a pattern of high concentrations on both sides and low concentrations in the middle emerges, which indicates that due to seasonal factors, both air pollutants have the characteristics of low concentrations in summer and high concentrations in winter. In summer (June to August), the concentration of air pollutants tends to rise, while in winter (December to February of the following year), the concentration of air pollutants tends to decrease.

Multifractal Analysis of PM
To check the existence of a scaling range, long-range correlations, and multifractality of both indices in four seasons, we first show the log-log plots of F q (s) versus time scale s of PM 2.5 and NO 2 . In Figure 2, the continuous line represents the double plots between the calculated F q (s) and time scale s, and the dashed line denotes the corresponding fitting line. As the dashed line shows, the decreasing gradient with q increases from −10 to 10, which indicates that PM 2.5 and NO 2 time series of all seasons have multifractal properties.
To measure the multifractality quantitatively, we apply MF-DFA to estimate properties for two pollutants of indices PM 2.5 and NO 2 . First, we calculate the generalized Hurst exponent h(q) and then obtain the Renyi exponent τ(q) and the Holder exponent α for both indices. Figure 3 shows the relationship between h(q) and q. e generalized Hurst exponents for both time series in any season are not constants, which indicate that the series are not monofractal. We note that h(q) decreases monotonously with an increase in q, which shows that the time series of PM 2.5 and NO 2 are not single fractals. Figures 3(a) and 3(b) represent PM 2.5 and NO 2 , respectively.
As illustrated in Table 3, the values of h(2) for all time series are larger than 0.5, implying that the fluctuations have significant positive persistence. Both indices of the time series in some seasons are even larger than 1, which is consistent with the conclusions drawn in some previous studies [26,35]. For the PM 2.5 time series, the persistence in spring is stronger than for other seasons, and for the NO 2 time series, autumn has the strongest persistence. We compute ΔH(q) for each time series using ΔH(q) � h(q min ) − h(q max ). From Figure 3 and Table 3, we observe that ΔH(q) in winter is the highest for the PM 2.5 time series, which implies its' multifractal characteristic is stronger in winter, and the multifractalities of the other three seasons are relatively stable. However, for the NO 2 time series, the multifractality in spring is higher than for all other seasons.
Furthermore, the multifractality strength of a time series can also be measured with a nonlinear Renyi exponent curve [36]. As shown in Figure 4, all curves of the Renyi exponent are nonlinear, which reflects the evidence of multifractality of the PM 2.5 and NO 2 series. Moreover, the different nonlinear behaviors of the curves of these series show that the multifractal strengths have different degrees. en, we calculate the Holder exponent α and the fractal dimension f(α) using equation (8).
e width of α also stands for the degree of the multifractality. A larger Δα indicates a stronger multifractal nature. In Figure 5 and           Table 3, the conclusion we described above about the Hurst exponent and the Renyi exponent is discussed. Moreover, as shown in Figure 5, the multifractal spectra are strongly asymmetrical, and as an important indicator of fractal organization, the asymmetry coefficient, which is used to quantify the asymmetry of the multifractal spectrum, has been estimated in [37].
Based on Table 3, the multifractal features can be clearly confirmed for all time series, and it can be seen that except for spring, the Hurst exponents and multifractal spectrum widths of PM 2.5 are obviously larger than those of NO 2 and the multifractal degrees of PM 2.5 are stronger than those of NO 2 .

Implications of the "Haze Special Law".
In this section, we focus on the enactment of the "haze special law" and study whether the law is effective in the governance of atmospheric pollution. We select daily data from the same period described in Section 3. As we studied above, the multifractal degree of PM 2.5 is stronger than that of NO 2 .
us, in this section, we focus on the comparison of PM 2.5 concentrations. We show the PM 2.5 time series before and after the law was implemented in Seoul, Busan, and Daegu in Figure 6.
As the fluctuation statistics are of asymptotic feature, a sufficiently long-length sequence guarantees the accurate estimation of the Hurst exponent, and for a small-length sequence, the asymptotic value in the slope of ln(F q (s)) versus ln(s) will be lacked [38]. As the total number of observations is selected to be 305, to eliminate the doubt of whether the selected time series can give representative results for a multifractal analysis in the investigation, we dealt with this concern by computing confidence intervals of Hurst exponent h (2) to check whether there is a difference caused by the law for all the three cities. e number of samples increased from 50 to 300, and every additional interval is 10. e confidence level is selected as 95.0%. For each set of sample lengths, we generate the shuffled time series 1000 times, and then, the confidence interval is plotted in Figure 7.
As shown in Figure 7 and as expected, the domain size decreases with an increase in the length of the time series. However, when the length of the time series is larger than 200, the decrease in the domain size of the confidence intervals is rather slow, and of a distinct asymptotic behavior. Besides, we see that the confidence intervals do not overlap for Seoul, while for Busan ([0.34, 0.39] before and [0.32, 0.37] after) and Daegu ([0.55, 0.64] before and [0.58, 0.69] after), the confidence intervals overlap a lot. erefore, we can conclude that there is a difference caused by the law for Seoul; however, the same conclusion is inconclusive for Busan and Daegu. Now, we first depict the log-log plots of the fluctuation function F q (s) versus q and s for the PM 2.5 time series. As illustrated in Figure 8, for all series, the fluctuation value increases linearly with s, indicating that a power-law behavior and long-range correlations exist in each series pair. e decreasing gradient shows that all time series have multifractal characteristics. e generalized Hurst exponent h(q), Renyi exponent τ(q), and Holder exponent α of the time series before and after the enactment of the "haze special law" for all the three cities are shown in Figures 9-11, respectively.
In Figure 9, it is shown that the generalized Hurst exponents h(q) of PM 2.5 of all three cities decrease with varied q, thereby showing the multifractal behaviors of these two series. We present h(2) for these series in Table 4. All Hurst exponents are confirmed to be larger than 0.5, which indicates that all the time series exhibit positive persistence. Besides, we observe that h(2) before the implementation of the law is larger than after the law, implying that the fluctuations before the law have more significant persistence. Subsequently, we calculate the range of h(q) for each time series and list them in Table 4. We find that for all three cities, ΔH(q) after the law is larger, which indicates that the multifractal characteristics are stronger, and that the multifractalities of the time series before the law are more stable.
e Renyi exponents τ(q) plotted in Figure 10 are nonlinear along q, which provides further evidence of the existence of multifractality. We notice that the curvature of the Renyi exponents before the law are higher, which implies that the time series before the implementation of the law have stronger multifractal characteristics. All of these are consistent with the results calculated from the generalized Hurst exponents.
At last, the multifractal spectrums of these time series are examined. We depict the curves between α and f(α) in Figure 11. We see that the multifractal spectra are not shown as points, which indicates all the time series have multifractality. en, Δα are calculated and presented in Table 4. From Table 4, it can be seen that ΔH(q) and Δα of the time series after the enactment of law are lower than those before  the implementation of the law for all three cities, implying that the multifractality of the time series becomes more stable with the measures taken by the government and that the efficiency of the "haze special law" in three cities is demonstrable; this conclusion can be referred to [16,39], in which it was pointed out that the decrease in multifractality means the efficiency is improved when some measures are taken. Moreover, we also notice that the efficiency of the "haze special law" in Seoul is most obvious, when compared with Busan, and in last place is Daegu. Interestingly, the order is the same as the scale and prosperity of the cities. We can therefore conclude that the law has a higher effect on a larger city.

Sources of Multifractal
Features. e two universally acknowledged sources of multifractality are long-range correlations and a fat-tailed distribution [7,40]. Now, we investigate the major source of multifractality of the PM 2.5 concentration series. To check the contribution of longrange correlations and fat-tailed distribution quantitatively, we shuffle and phase-randomize the time series.
Subsequently, we calculate the Hurst exponent h(q), the Renyi exponent τ(q), and the Holder exponent Δα of the original, shuffled, and phase-randomized time series of the three cities. As shown in Figures 12-14, all the time series are strongly multifractal. From Table 5, all h(2) of the original series are larger than those of the shuffled and phase-randomized time series, which indicates that the persistence has been moved after shuffling and phase-randomization. Moreover, the original series of all three cities has the greatest spectrum width Δα, while the multifractality decreased after shuffling and phase-randomizing the series. e findings can also be confirmed in Figures 12-14.
To examine the main source of multifractality, the first and third rows of Figure 14 show that the multifractal spectra of phase-randomized series are the narrowest ones, which means that the main source of multifractality of the series before and after the law in Seoul and Daegu are the fattailed distributions. e second row in Figure 14 shows that the multifractal spectrum of the shuffled series is the narrowest, which implies that the long-range correlation results in multifractality of the series before and after implementation of the law in Busan.

Conclusions
In this study, we examined the multifractal characteristics of PM 2.5 and NO 2 time series of all seasons. We determined that multifractality existed in each season, and the multifractal property of PM 2.5 was stronger than that of NO 2 in each season. We also validated the effectiveness of the "haze special law," which was implemented to improve the governance of air pollution in South Korea. We checked whether there is a difference caused by the law for these cities and calculated confidence intervals of Hurst exponent h (2). e results showed that the confidence intervals do not overlap for Seoul, while the confidence intervals overlap a lot for Busan and Daegu, indicating that there is a difference caused by the law for Seoul; however, the same conclusion is inconclusive for Busan and Daegu. By comparing the Hurst exponent, Renyi exponent, and Holder exponent, the time series of the PM 2.5 concentration before the implementation of the law was found to have a higher multifractal degree, which decreased after the enactment of the law. is phenomenon reflected the fact that the law has played a role in improving the efficiency of air pollution control in South Korea. We also concluded that the effect of the law will be more significant in a larger city. To explore the major causes of multifractality, we shuffled and phase-randomized the original series of PM 2.5 . By analyzing the width of the multifractal spectrum Δα, the results showed that the fattailed distribution contributed to the multifractality of both times series before and after the implementation of the "haze special law" in Seoul and Daegu, whereas long-range correlations resulted in the multifractality of the series before and after the implementation of the law in Busan. We concluded that the implementation of the law was very successful and effective for the improvement of PM 2.5 levels.
erefore, we believe that the "haze special law" is a possible cause for the change in API in South Korea.
Data Availability e data will be available upon request with the corresponding author.   LRC and FTD represent the long-range correlations and fat-tailed distribution, respectively.