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

Authors: Jian Wang; Junseok Kim; Wei Shao
Journal: Complexity
 (2020)

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## Abstract

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 ofPM2.5 and NO2. The 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 MF-DFA, 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.


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## Body

## 1. 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]. Therefore, many researchers have started to investigate the atmospheric particulate matters. Many studies have stated that PM2.5 and NO2 are the two major pollutants among various air pollutants [2–6]. Therefore, in this research, we focused on PM2.5 and NO2 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]. Therefore, the multifractal properties of stock markets [9–13], foreign exchange markets [14–16], 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 ofPM2.5 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. Through the hourly PM2.5 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 PM2.5 and meteorological factors. Recently, Zhang et al. [28] analyzed the multifractal characteristics of the PM2.5 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. The effects of seasonal factors onPM2.5 and NO2 were analyzed, and the seasonal physical changes of PM2.5 and NO2 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.The paper is organized as follows. We briefly outline the procedure of MF-DFA in Section2. In the next section, we describe the data information. Section 4 illustrates the empirical results. Section 5 concludes the paper.


## 2. 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 seriesXi starts from X1 to XN, where N is the length of the signal, the corresponding summation sequence is constructed by the following integration:(1)Yk=∑i=1kXi−X¯,k=1,2,…,N,where X¯ is the mean value of Xi.Subsequently, the profileY is further divided into Ns 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. Thus, 2Ns windows are acquired.Next, the least squares method is used to fit the data for evaluating the local trend of each windowvv=1,2,…,2Ns, and the fitting polynomial in the vth window is denoted by yvi. The 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(2)F2s,v=1s∑i=1sYv−1s+i−yvi2,if v=1,2,…,Ns, and(3)F2s,v=1s∑i=1sYN−v−Nss+i−yvi2,if v=Ns+1,Ns+2,…,2Ns.Finally, computing the mean of2Ns windows, the q order wave function Fqs is obtained as(4)Fqs=12Ns∑v=12NsF2s,vq/21/q,if q=0; according to L’Hôpital’s rule,(5)Fqs=exp12Ns∑v=12NslnF2s,v.Through the analysis of double log plots ofFqs versus s and a varied q, the scaling behavior of the fluctuation is determined by the power-law Fqs∝shq, and from this, a family of scaling exponents hq, which are generalized Hurst components, can be obtained. Fqs is the standard DFA, if q=2. The Hurst exponents provide information on the time series, such as power-law correlated behavior, and when 0<hq<0.5, it indicates that the time series has a negative or antipersistence property. When 0.5<hq<1, then the time series has a positive persistence, and h2=0.5 indicates that the time series has an uncorrelated Brownian process.The Renyi exponentτq, which is related to the general Hurst exponent, can be expressed by(6)τq=qhq−1.In addition,(7)α=hq+qh′q,(8)fα=qα−hq+1,where α represents the Holder exponent and characterizes the singularity strength and fα is a fractal dimension of the set of points with particular α. In the plotted curve between α and fα, the shape resembles an inverted parabola and the degree of their complexity is denoted by the width of their fractal strength Δα [31].


## 3. 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. The 3 cities are located in different parts of South Korea, and there is a certain distance between any two cities. The 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.The experimental data sources for thePM2.5 μg/m3 and NO2ppm 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. Table 1 shows information of the target cities and the corresponding monitoring stations. For the multifractal analysis of API in different seasons, the samples were recorded from March 3, 2018, to February 25, 2019. We considered March to May, June to August, September to November, and December to February as the four seasons: spring, summer, autumn, and winter, respectively. Each interval was assigned a time series of 90 daily data points. For analyzing the efficiency of the implementation of the “haze special law” on February 15, 2019, we selected a time series of PM2.5 over the same period. The data were recorded from February 15, 2018, to December 16, 2018, and from February 15, 2019, to December 16, 2019, and each interval was assigned a time series of 305 daily data points. The descriptive statistics for each time series are shown in Table 2.Table 1
Information of cities in the sample.
City
Location
Monitoring station
Seoul
Northwest
Jung-gu
Busan
Southeast
Gaegeum-do
Daegu
Centrum
Guseong-dongTable 2
Descriptive statistics.
Observation
Observation number
Minimum
Maximum
Mean
Standard deviation
SpringPM2.5
90
18
195
94.289
43.470
SummerPM2.5
90
13
139
71.722
33.368
AutumnPM2.5
90
14
170
67.711
36.996
WinterPM2.5
90
35
204
102.022
33.840
SpringNO2
90
0.013
0.059
0.033
0.011
SummerNO2
90
0.008
0.039
0.024
0.007
AutumnNO2
90
0.009
0.066
0.031
0.012
WinterNO2
90
0.010
0.073
0.038
0.015
Seoul (before law)
305
13
195
79.708
39.268
Seoul (after law)
305
12
210
78.498
37.907
Busan (before law)
305
42
161
91.161
28.590
Busan (after law)
305
23
169
84.023
26.869
Daegu (before law)
305
25
158
86.138
31.332
Daegu (after law)
305
21
168
82.902
28.525


## 4. Experiment Results

We first analyze the multifractal properties for two time series of each season and compare the multifractal degree ofPM2.5 with that of NO2. 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 besmin=5, and the maximum segment scale smax=15, and q is taken from −10 to 10. The 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].


### 4.1. Multifractal Analysis ofPM2.5 and NO2

From the daily API data selected in Section3, we draw the time series of PM2.5 and NO2 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.Figure 1
Time series of (a)PM2.5 concentration and (b)NO2 concentration.
(a)
(b)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 ofFqs versus time scale s of PM2.5 and NO2. In Figure 2, the continuous line represents the double plots between the calculated Fqs 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 PM2.5 and NO2 time series of all seasons have multifractal properties.Figure 2
Log-log plots of fluctuation functionFqs for (a)PM2.5 and (b)NO2. From top to bottom are spring, summer, autumn, and winter.
(a)
(b)To measure the multifractality quantitatively, we apply MF-DFA to estimate properties for two pollutants of indicesPM2.5 and NO2. First, we calculate the generalized Hurst exponent hq and then obtain the Renyi exponent τq and the Holder exponent α for both indices.Figure3 shows the relationship between hq and q. The generalized Hurst exponents for both time series in any season are not constants, which indicate that the series are not monofractal. We note that hq decreases monotonously with an increase in q, which shows that the time series of PM2.5 and NO2 are not single fractals. Figures 3(a) and 3(b) represent PM2.5 and NO2, respectively.Figure 3
Hurst exponent of (a)PM2.5 concentration and (b)NO2 concentration.
(a)
(b)As illustrated in Table3, the values of h2 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 PM2.5 time series, the persistence in spring is stronger than for other seasons, and for the NO2 time series, autumn has the strongest persistence. We compute ΔHq for each time series using ΔHq=hqmin−hqmax. From Figure 3 and Table 3, we observe that ΔHq in winter is the highest for the PM2.5 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 NO2 time series, the multifractality in spring is higher than for all other seasons.Table 3
Multifractality forPM2.5 and NO2 concentrations.
h
2
Δ
H
q
Δ
α
PM
2
.
5
Spring
1.4290
1.0297
1.2224
Summer
0.9991
1.2021
1.4787
Autumn
1.1797
1.4247
1.6851
Winter
1.1626
1.5500
2.0520
NO
2
Spring
1.0705
1.1699
1.4212
Summer
0.7468
1.1350
1.4151
Autumn
1.2635
0.4880
0.6705
Winter
0.7718
0.8884
1.0958Furthermore, 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 PM2.5 and NO2 series. Moreover, the different nonlinear behaviors of the curves of these series show that the multifractal strengths have different degrees.Figure 4
Renyi exponent of (a)PM2.5 concentration and (b)NO2 concentration.
(a)
(b)Then, we calculate the Holder exponentα and the fractal dimension fα using equation (8). The 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].Figure 5
Holder exponent of (a)PM2.5 concentration and (b)NO2 concentration.
(a)
(b)Based on Table3, 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 PM2.5 are obviously larger than those of NO2 and the multifractal degrees of PM2.5 are stronger than those of NO2.


### 4.2. 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 Section3. As we studied above, the multifractal degree of PM2.5 is stronger than that of NO2. Thus, in this section, we focus on the comparison of PM2.5 concentrations. We show the PM2.5 time series before and after the law was implemented in Seoul, Busan, and Daegu in Figure 6.Figure 6
Time series ofPM2.5 (a) before and (b) after implementation of the law. From top to bottom are Seoul, Busan, and Daegu.
(a)
(b)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 oflnFqs versus lns 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 h2 to check whether there is a difference caused by the law for all the three cities. The number of samples increased from 50 to 300, and every additional interval is 10. The 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.Figure 7
Confidence intervals ofh2 versus the length of PM2.5 time series (a) before and (b) after implementation of the law with confidence level by 95.0%. From top to bottom are Seoul, Busan, and Daegu.
(a)
(b)As shown in Figure7 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. Therefore, 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 functionFqs versus q and s for the PM2.5 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. The decreasing gradient shows that all time series have multifractal characteristics.Figure 8
Log-log plots of the fluctuation functionFqs for PM2.5 (a) before and (b) after implementation of the law. From top to bottom are Seoul, Busan, and Daegu.
(a)
(b)The generalized Hurst exponenthq, Renyi exponent τq, 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.Figure 9
Hurst exponent ofPM2.5 concentration of (a) Seoul, (b) Busan, and (c) Daegu.
(a)
(b)
(c)Figure 10
Renyi exponent ofPM2.5 concentration of (a) Seoul, (b) Busan, and (c) Daegu.
(a)
(b)
(c)Figure 11
Holder exponent ofPM2.5 concentration of (a) Seoul, (b) Busan, and (c) Daegu. The triangle and circle symbols represent before and after the law, respectively.
(a)
(b)
(c)In Figure9, it is shown that the generalized Hurst exponents hq of PM2.5 of all three cities decrease with varied q, thereby showing the multifractal behaviors of these two series. We present h2 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 h2 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 hq for each time series and list them in Table 4. We find that for all three cities, ΔHq 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.Table 4
Multifractality forPM2.5 concentration.
City
Multifractality
Before law
After law
Seoul
h
2
1.2270
1.2085
Δ
H
q
1.7329
1.1410
Δ
α
2.2414
1.3953
Busan
h
2
1.2905
1.1041
Δ
H
q
1.3539
0.9565
Δ
α
1.7463
1.2708
Daegu
h
2
1.1975
1.0878
Δ
H
q
2.2910
1.9516
Δ
α
2.9610
2.5043The Renyi exponentsτq 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 betweenα and fα in Figure 11. We see that the multifractal spectra are not shown as points, which indicates all the time series have multifractality. Then, Δα are calculated and presented in Table 4. From Table 4, it can be seen that ΔHq 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.


### 4.3. Sources of Multifractal Features

The 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 PM2.5 concentration series. To check the contribution of long-range correlations and fat-tailed distribution quantitatively, we shuffle and phase-randomize the time series.Subsequently, we calculate the Hurst exponenthq, the Renyi exponent τq, 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 h2 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. The findings can also be confirmed in Figures 12–14.Figure 12
Hurst exponent (a) before law and (b) after law. From top to bottom are Seoul, Busan, and Daegu.
(a)
(b)Figure 13
Renyi exponent (a) before law and (b) after law. From top to bottom are Seoul, Busan, and Daegu.
(a)
(b)Figure 14
Holder exponent (a) before law and (b) after law. From top to bottom are Seoul, Busan, and Daegu.
(a)
(b)Table 5
Multifractality ofPM2.5 time series.
h
2
Δ
α
LRC
FTD
Seoul
Before
Original
1.2270
2.2414
No
Yes
Shuffled
0.7232
1.7946
Phase-randomized
0.6965
0.9574
After
Original
1.2085
1.3953
No
Yes
Shuffled
0.3177
1.0978
Phase-randomized
0.3049
0.8569
Busan
Before
Original
1.2905
1.7463
Yes
No
Shuffled
0.3982
0.9999
Phase-randomized
0.3993
1.3512
After
Original
1.1041
1.2708
Yes
No
Shuffled
0.3678
0.7175
Phase-randomized
0.3794
0.8673
Daegu
Before
Original
1.1975
2.9610
No
Yes
Shuffled
0.5901
2.2093
Phase-randomized
0.5868
2.0587
After
Original
1.0878
2.5043
No
Yes
Shuffled
0.6756
2.5600
Phase-randomized
0.6407
2.2981
LRC and FTD represent the long-range correlations and fat-tailed distribution, respectively.To examine the main source of multifractality, the first and third rows of Figure14 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 fat-tailed distributions. The 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.


## 4.1. Multifractal Analysis ofPM2.5 and NO2

From the daily API data selected in Section3, we draw the time series of PM2.5 and NO2 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.Figure 1
Time series of (a)PM2.5 concentration and (b)NO2 concentration.
(a)
(b)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 ofFqs versus time scale s of PM2.5 and NO2. In Figure 2, the continuous line represents the double plots between the calculated Fqs 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 PM2.5 and NO2 time series of all seasons have multifractal properties.Figure 2
Log-log plots of fluctuation functionFqs for (a)PM2.5 and (b)NO2. From top to bottom are spring, summer, autumn, and winter.
(a)
(b)To measure the multifractality quantitatively, we apply MF-DFA to estimate properties for two pollutants of indicesPM2.5 and NO2. First, we calculate the generalized Hurst exponent hq and then obtain the Renyi exponent τq and the Holder exponent α for both indices.Figure3 shows the relationship between hq and q. The generalized Hurst exponents for both time series in any season are not constants, which indicate that the series are not monofractal. We note that hq decreases monotonously with an increase in q, which shows that the time series of PM2.5 and NO2 are not single fractals. Figures 3(a) and 3(b) represent PM2.5 and NO2, respectively.Figure 3
Hurst exponent of (a)PM2.5 concentration and (b)NO2 concentration.
(a)
(b)As illustrated in Table3, the values of h2 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 PM2.5 time series, the persistence in spring is stronger than for other seasons, and for the NO2 time series, autumn has the strongest persistence. We compute ΔHq for each time series using ΔHq=hqmin−hqmax. From Figure 3 and Table 3, we observe that ΔHq in winter is the highest for the PM2.5 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 NO2 time series, the multifractality in spring is higher than for all other seasons.Table 3
Multifractality forPM2.5 and NO2 concentrations.
h
2
Δ
H
q
Δ
α
PM
2
.
5
Spring
1.4290
1.0297
1.2224
Summer
0.9991
1.2021
1.4787
Autumn
1.1797
1.4247
1.6851
Winter
1.1626
1.5500
2.0520
NO
2
Spring
1.0705
1.1699
1.4212
Summer
0.7468
1.1350
1.4151
Autumn
1.2635
0.4880
0.6705
Winter
0.7718
0.8884
1.0958Furthermore, 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 PM2.5 and NO2 series. Moreover, the different nonlinear behaviors of the curves of these series show that the multifractal strengths have different degrees.Figure 4
Renyi exponent of (a)PM2.5 concentration and (b)NO2 concentration.
(a)
(b)Then, we calculate the Holder exponentα and the fractal dimension fα using equation (8). The 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].Figure 5
Holder exponent of (a)PM2.5 concentration and (b)NO2 concentration.
(a)
(b)Based on Table3, 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 PM2.5 are obviously larger than those of NO2 and the multifractal degrees of PM2.5 are stronger than those of NO2.


## 4.2. 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 Section3. As we studied above, the multifractal degree of PM2.5 is stronger than that of NO2. Thus, in this section, we focus on the comparison of PM2.5 concentrations. We show the PM2.5 time series before and after the law was implemented in Seoul, Busan, and Daegu in Figure 6.Figure 6
Time series ofPM2.5 (a) before and (b) after implementation of the law. From top to bottom are Seoul, Busan, and Daegu.
(a)
(b)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 oflnFqs versus lns 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 h2 to check whether there is a difference caused by the law for all the three cities. The number of samples increased from 50 to 300, and every additional interval is 10. The 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.Figure 7
Confidence intervals ofh2 versus the length of PM2.5 time series (a) before and (b) after implementation of the law with confidence level by 95.0%. From top to bottom are Seoul, Busan, and Daegu.
(a)
(b)As shown in Figure7 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. Therefore, 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 functionFqs versus q and s for the PM2.5 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. The decreasing gradient shows that all time series have multifractal characteristics.Figure 8
Log-log plots of the fluctuation functionFqs for PM2.5 (a) before and (b) after implementation of the law. From top to bottom are Seoul, Busan, and Daegu.
(a)
(b)The generalized Hurst exponenthq, Renyi exponent τq, 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.Figure 9
Hurst exponent ofPM2.5 concentration of (a) Seoul, (b) Busan, and (c) Daegu.
(a)
(b)
(c)Figure 10
Renyi exponent ofPM2.5 concentration of (a) Seoul, (b) Busan, and (c) Daegu.
(a)
(b)
(c)Figure 11
Holder exponent ofPM2.5 concentration of (a) Seoul, (b) Busan, and (c) Daegu. The triangle and circle symbols represent before and after the law, respectively.
(a)
(b)
(c)In Figure9, it is shown that the generalized Hurst exponents hq of PM2.5 of all three cities decrease with varied q, thereby showing the multifractal behaviors of these two series. We present h2 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 h2 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 hq for each time series and list them in Table 4. We find that for all three cities, ΔHq 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.Table 4
Multifractality forPM2.5 concentration.
City
Multifractality
Before law
After law
Seoul
h
2
1.2270
1.2085
Δ
H
q
1.7329
1.1410
Δ
α
2.2414
1.3953
Busan
h
2
1.2905
1.1041
Δ
H
q
1.3539
0.9565
Δ
α
1.7463
1.2708
Daegu
h
2
1.1975
1.0878
Δ
H
q
2.2910
1.9516
Δ
α
2.9610
2.5043The Renyi exponentsτq 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 betweenα and fα in Figure 11. We see that the multifractal spectra are not shown as points, which indicates all the time series have multifractality. Then, Δα are calculated and presented in Table 4. From Table 4, it can be seen that ΔHq 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.


## 4.3. Sources of Multifractal Features

The 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 PM2.5 concentration series. To check the contribution of long-range correlations and fat-tailed distribution quantitatively, we shuffle and phase-randomize the time series.Subsequently, we calculate the Hurst exponenthq, the Renyi exponent τq, 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 h2 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. The findings can also be confirmed in Figures 12–14.Figure 12
Hurst exponent (a) before law and (b) after law. From top to bottom are Seoul, Busan, and Daegu.
(a)
(b)Figure 13
Renyi exponent (a) before law and (b) after law. From top to bottom are Seoul, Busan, and Daegu.
(a)
(b)Figure 14
Holder exponent (a) before law and (b) after law. From top to bottom are Seoul, Busan, and Daegu.
(a)
(b)Table 5
Multifractality ofPM2.5 time series.
h
2
Δ
α
LRC
FTD
Seoul
Before
Original
1.2270
2.2414
No
Yes
Shuffled
0.7232
1.7946
Phase-randomized
0.6965
0.9574
After
Original
1.2085
1.3953
No
Yes
Shuffled
0.3177
1.0978
Phase-randomized
0.3049
0.8569
Busan
Before
Original
1.2905
1.7463
Yes
No
Shuffled
0.3982
0.9999
Phase-randomized
0.3993
1.3512
After
Original
1.1041
1.2708
Yes
No
Shuffled
0.3678
0.7175
Phase-randomized
0.3794
0.8673
Daegu
Before
Original
1.1975
2.9610
No
Yes
Shuffled
0.5901
2.2093
Phase-randomized
0.5868
2.0587
After
Original
1.0878
2.5043
No
Yes
Shuffled
0.6756
2.5600
Phase-randomized
0.6407
2.2981
LRC and FTD represent the long-range correlations and fat-tailed distribution, respectively.To examine the main source of multifractality, the first and third rows of Figure14 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 fat-tailed distributions. The 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.


## 5. Conclusions

In this study, we examined the multifractal characteristics ofPM2.5 and NO2 time series of all seasons. We determined that multifractality existed in each season, and the multifractal property of PM2.5 was stronger than that of NO2 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 h2. The 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 PM2.5 concentration before the implementation of the law was found to have a higher multifractal degree, which decreased after the enactment of the law. This 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 PM2.5. By analyzing the width of the multifractal spectrum Δα, the results showed that the fat-tailed 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 PM2.5 levels. Therefore, we believe that the “haze special law” is a possible cause for the change in API in South Korea.



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*Source: 6193016-2020-03-13.xml*