Research ArticleInvestigation of the Implications of ldquoHaze Special Lawrdquo on AirQuality in South Korea

Jian Wang 1 Junseok Kim 1 and Wei Shao 2

1Department of Mathematics Korea University Seoul 02841 Republic of Korea2Department of Economics Korea University Seoul 02841 Republic of Korea

Correspondence should be addressed to Wei Shao shaoweikoreaackr

Received 11 October 2019 Revised 24 December 2019 Accepted 19 February 2020 Published 13 March 2020

Academic Editor Bernhard C Geiger

Copyright copy 2020 Jian Wang et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this work the air pollution index in three cities (Seoul Busan and Daegu) in South Korea was studied using multifractaldetrended fluctuation analysis (MF-DFA) Hurst Renyi and Holder exponents were used to analyze the characteristics of theconcentration time series of PM25 and NO2 e results showed that multifractality exists in each season interval and themultifractal degree of PM25 is stronger than that of NO2 To investigate the effects of the implementation of the ldquohaze special lawrdquoon February 15 2019 we analyzed the time series of PM25 during the time periods from February 15 2018 to December 16 2018and February 15 2019 to December 16 2019 We found that the multifractal spectrum width after the implementation of the lawwas narrower than that before the law for all the cities which shows that the enactment of the law has played a role in improvingthe efficiency of air pollution control in South Korea We also conclude that the major effects of the law will be particularly visiblein 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 afterthe implementation of the ldquohaze special lawrdquo in Seoul and Daegu whereas long-range correlation resulted in multifractality of theseries before and after the implementation of the law in Busan

1 Introduction

With the rapid development of a social economy and theacceleration of industrialization and urbanization air pol-lution has gradually become an important problem affectingpeoplersquos lives and has therefore garnered considerable at-tention from the public and government Air pollution isclosely related to climate ecology and health Studies haveshown that sulfur oxides and nitrogen oxides can cause greatharm to the respiratory system even leading to respiratoryfailure in severe cases [1] erefore many researchers havestarted to investigate the atmospheric particulate mattersMany studies have stated that PM25 and NO2 are the twomajor pollutants among various air pollutants [2ndash6]erefore in this research we focused on PM25 and NO2and investigated the two pollutants based on the relationshipbetween the air pollution index (API) and different seasonsWe also checked whether the implementation of the ldquohaze

special lawrdquo on February 15 2019 had an impact on im-proving the air quality in South Korea using multifractaldetrended fluctuation analysis (MF-DFA)

We know that the MF-DFA model can unveil themultifractal properties hidden in nonstationary time seriesand that multifractality originates from temporal correla-tions [7 8] erefore the multifractal properties of stockmarkets [9ndash13] foreign exchange markets [14ndash16] bitcoinmarkets [17] atmospheric sciences [18] and phase transi-tions [19] have been studied previously Recently multi-fractal analysis was used to study the daily air temperaturetime series [20] Moreover MF-DFA is also an efficientmethod in analyzing the human heart rate time series [21]Based on the analysis of blood pressure and heart-ratecomplexity Gender Castiglioni et al [22] used the multi-fractal technique to investigate cardiovascular warningsignals Moreover MF-DFA has been used in some studiesto implement specific image analysis [23 24]

HindawiComplexityVolume 2020 Article ID 6193016 18 pageshttpsdoiorg10115520206193016

In recent years many studies on air pollutants have beenconducted and the results suggested that the increased levelsof PM25 are associated with a higher mortality and somenegative effects on the lungs [25] Although there are manystudies on API few studies have focused on the MF-DFA ofthe API time series rough the hourly PM25 averageconcentration time series Shi et al [26] analyzed themultifractal nature at four air monitoring stations ofChengdu using MF-DFA Zhang et al [27] used the mul-tifractal detrended cross-correlation analysis (MF-DCCA) toanalyze the cross-correlations between PM25 and meteo-rological factors Recently Zhang et al [28] analyzed themultifractal characteristics of the PM25 time series in HongKong using the empirical mode decomposition-based MF-DFA method Multifractal property links between meteo-rological factors and pollutants in urban and rural areas havebeen verified by He [29]

Based on MF-DFA the variations of multifractal char-acteristics of pollutants present in different seasons wereinvestigated in this study e effects of seasonal factors onPM25 and NO2 were analyzed and the seasonal physicalchanges of PM25 and NO2 concentrations were confirmedusing multifractality Additionally we conducted a multi-fractal comparison analysis of the effectiveness of air pol-lution renovation since the enactment of the ldquohaze speciallawrdquo From this we can provide a basis for formulating ascientific and effective comprehensive air pollution controlpolicy

e paper is organized as follows We briefly outline theprocedure of MF-DFA in Section 2 In the next section wedescribe the data information Section 4 illustrates theempirical 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 allseasons and the multifractal characteristics of the API timeseries before and after the implementation of the ldquohazespecial lawrdquo Kantelhardt summarized the technical details ofMF-DFA as follows

As a time series Xi starts from X1 to XN where N is thelength of the signal the corresponding summation sequenceis constructed by the following integration

Y(k) 1113944k

i1Xi minus X( 1113857 k 1 2 N (1)

where X is the mean value of XiSubsequently the profile Y is further divided into Ns

nonoverlapping windows of equal length s In most cases thescale s is not a multiple of the time series and a short part atthe end of series Y exists To overcome the problem ofinformation being lost in the division process the sameprocess is repeated starting from the other side of the seriesus 2Ns windows are acquired

Next the least squares method is used to fit the data forevaluating the local trend of each windowv (v 1 2 2Ns) and the fitting polynomial in the vth

window is denoted by yv(i) e variance is determined bythe detrended time series which is calculated as the dif-ference between Y and y and the resultant equation isdescribed as

F2(s v)

1s

1113944

s

i1Y[(v minus 1)s + i] minus yv(i)1113864 1113865

2 (2)

if v 1 2 Ns and

F2(s v)

1s1113944

s

i1Y N minus v minus Ns( 1113857s + i1113858 1113859 minus yv(i)1113864 1113865

2 (3)

if v Ns + 1 Ns + 2 2NsFinally computing the mean of 2Ns windows the q

order wave function Fq(s) is obtained as

Fq(s) 1

2Ns

1113944

2Ns

v1F2(s v)1113960 1113961

q2⎧⎨

⎩

⎫⎬

⎭

1q

(4)

if q 0 according to LrsquoHopitalrsquos rule

Fq(s) exp1

2Ns

1113944

2Ns

v1ln F

2(s v)1113960 1113961

⎧⎨

⎩

⎫⎬

⎭ (5)

rough the analysis of double log plots of Fq(s) versus sand a varied q the scaling behavior of the fluctuation isdetermined by the power-law Fq(s)prop sh(q) and from this afamily of scaling exponents h(q) which are generalizedHurst components can be obtained Fq(s) is the standardDFA if q 2 e Hurst exponents provide information onthe time series such as power-law correlated behavior andwhen 0lt h(q)lt 05 it indicates that the time series has anegative or antipersistence property When 05lt h(q)lt 1then the time series has a positive persistence and h(2) 05indicates that the time series has an uncorrelated Brownianprocess

e Renyi exponent τ(q) which is related to the generalHurst exponent can be expressed by

τ(q) qh(q) minus 1 (6)

In addition

α h(q) + qhprime(q) (7)

f(α) q[α minus h(q)] + 1 (8)

where α represents the Holder exponent and characterizesthe singularity strength and f(α) is a fractal dimension ofthe set of points with particular α In the plotted curvebetween α and f(α) the shape resembles an inverted pa-rabola and the degree of their complexity is denoted by thewidth of their fractal strength Δα [31]

3 Data Collection

We used the API time series of Seoul to study the multi-fractal characteristics of API sequences in different seasonsTo gauge the impact of the implementation of the ldquohazespecial lawrdquo our data set covered 3 cities in South Koreae3 cities are located in different parts of South Korea and

2 Complexity

there is a certain distance between any two cities e resultscalculated from the samples selected in this manner fullyreflect the impact and effect of the implementation of theldquohaze special lawrdquo over the entirety of South Korea

e experimental data sources for the PM25 (μgm3) andNO2(ppm) concentrations were the ldquoKorean Ministry OfEnvironmentrdquo and ldquoKorea Environment Corporationrdquo andmore specific data were provided by the monitoring stationsin South Korea For more detailed information please referto httpswwwairkoreaorkrweb and httpsaqicnorgcityseoulkr Table 1 shows information of the target cit-ies and the corresponding monitoring stations For themultifractal analysis of API in different seasons the sampleswere recorded fromMarch 3 2018 to February 25 2019 Weconsidered March to May June to August September to

November and December to February as the four seasonsspring summer autumn and winter respectively Eachinterval was assigned a time series of 90 daily data points Foranalyzing the efficiency of the implementation of the ldquohazespecial lawrdquo on February 15 2019 we selected a time series ofPM25 over the same period e data were recorded fromFebruary 15 2018 to December 16 2018 and from February15 2019 to December 16 2019 and each interval wasassigned a time series of 305 daily data points e de-scriptive statistics for each time series are shown in Table 2

4 Experiment Results

We first analyze the multifractal properties for two timeseries of each season and compare the multifractal degree of

Table 1 Information of cities in the sample

City Location Monitoring stationSeoul Northwest Jung-guBusan Southeast Gaegeum-doDaegu Centrum Guseong-dong

Table 2 Descriptive statistics

Observation Observation number Minimum Maximum Mean Standard deviationSpring (PM25) 90 18 195 94289 43470Summer (PM25) 90 13 139 71722 33368Autumn (PM25) 90 14 170 67711 36996Winter (PM25) 90 35 204 102022 33840Spring (NO2) 90 0013 0059 0033 0011Summer (NO2) 90 0008 0039 0024 0007Autumn (NO2) 90 0009 0066 0031 0012Winter (NO2) 90 0010 0073 0038 0015Seoul (before law) 305 13 195 79708 39268Seoul (after law) 305 12 210 78498 37907Busan (before law) 305 42 161 91161 28590Busan (after law) 305 23 169 84023 26869Daegu (before law) 305 25 158 86138 31332Daegu (after law) 305 21 168 82902 28525

201861 2018830 20181128 2019225201833Date

0

50

100

150

200

PM2

5

(a)

0

002

004

006

008

NO

2

201861 2018830 20181128 2019225201833Date

(b)

Figure 1 Time series of (a) PM25 concentration and (b) NO2 concentration

Complexity 3

there is a certain distance between any two cities e resultscalculated from the samples selected in this manner fullyreflect the impact and effect of the implementation of theldquohaze special lawrdquo over the entirety of South Korea

e experimental data sources for the PM25 (μgm3) andNO2(ppm) concentrations were the ldquoKorean Ministry OfEnvironmentrdquo and ldquoKorea Environment Corporationrdquo andmore specific data were provided by the monitoring stationsin South Korea For more detailed information please referto httpswwwairkoreaorkrweb and httpsaqicnorgcityseoulkr Table 1 shows information of the target cit-ies and the corresponding monitoring stations For themultifractal analysis of API in different seasons the sampleswere recorded fromMarch 3 2018 to February 25 2019 Weconsidered March to May June to August September to

November and December to February as the four seasonsspring summer autumn and winter respectively Eachinterval was assigned a time series of 90 daily data points Foranalyzing the efficiency of the implementation of the ldquohazespecial lawrdquo on February 15 2019 we selected a time series ofPM25 over the same period e data were recorded fromFebruary 15 2018 to December 16 2018 and from February15 2019 to December 16 2019 and each interval wasassigned a time series of 305 daily data points e de-scriptive statistics for each time series are shown in Table 2

4 Experiment Results

We first analyze the multifractal properties for two timeseries of each season and compare the multifractal degree of

Table 1 Information of cities in the sample

City Location Monitoring stationSeoul Northwest Jung-guBusan Southeast Gaegeum-doDaegu Centrum Guseong-dong

Table 2 Descriptive statistics

Observation Observation number Minimum Maximum Mean Standard deviationSpring (PM25) 90 18 195 94289 43470Summer (PM25) 90 13 139 71722 33368Autumn (PM25) 90 14 170 67711 36996Winter (PM25) 90 35 204 102022 33840Spring (NO2) 90 0013 0059 0033 0011Summer (NO2) 90 0008 0039 0024 0007Autumn (NO2) 90 0009 0066 0031 0012Winter (NO2) 90 0010 0073 0038 0015Seoul (before law) 305 13 195 79708 39268Seoul (after law) 305 12 210 78498 37907Busan (before law) 305 42 161 91161 28590Busan (after law) 305 23 169 84023 26869Daegu (before law) 305 25 158 86138 31332Daegu (after law) 305 21 168 82902 28525

201861 2018830 20181128 2019225201833Date

0

50

100

150

200

PM2

5

(a)

0

002

004

006

008

NO

2

201861 2018830 20181128 2019225201833Date

(b)

Figure 1 Time series of (a) PM25 concentration and (b) NO2 concentration

Complexity 3

q = 10

q = 10

q = 10

q = 10

q = ndash10

q = ndash10

q = ndash10

q = ndash10

ndash1

0

1

2

3

4

ln(F

q(s)

)

2 22 24 2618ln(s)

ndash1

ndash05

0

05

1

15

2

25

ln(F

q(s)

)

2 22 24 2618ln(s)

ndash2

ndash1

0

1

2

3

ln(F

q(s)

)

2 22 24 2618ln(s)

ndash2

ndash1

0

1

2

3

4

ln(F

q(s)

)

2 22 24 2618ln(s)

(a)

q = 10

q = 10

q = 10

q = 10

q = ndash10

q = ndash10

q = ndash10

q = ndash10

ndash8

ndash7

ndash6

ndash5

ndash4

ln(F

q(s)

)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

ndash85

ndash8

ndash75

ndash7

ndash65

ndash6

ndash55

ndash5

ln(F

q(s)

)

ndash75

ndash7

ndash65

ndash6

ndash55

ndash5

ndash45

ndash4

ln(F

q(s)

)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

ndash65

ndash6

ndash55

ndash5

ndash45

ndash4

ln(F

q(s)

)

(b)

Figure 2 Log-log plots of fluctuation function Fq(s) for (a) PM25 and (b) NO2 From top to bottom are spring summer autumn andwinter

4 Complexity

05

1

15

2

25

3

h(q)

ndash5 0 5 10ndash10q

SpringSummer

AutumnWinter

(a)

ndash5 0 5 10ndash10q

05

1

15

2

h(q)

SpringSummer

AutumnWinter

(b)

Figure 3 Hurst exponent of (a) PM25 concentration and (b) NO2 concentration

Table 3 Multifractality for PM25 and NO2 concentrations

h(2) ΔH(q) Δα

PM25

Spring 14290 10297 12224Summer 09991 12021 14787Autumn 11797 14247 16851Winter 11626 15500 20520

NO2

Spring 10705 11699 14212Summer 07468 11350 14151Autumn 12635 04880 06705Winter 07718 08884 10958

ndash30

ndash20

ndash10

0

10

20

τ(q)

ndash5 0 5 10ndash10q

SpringSummer

AutumnWinter

(a)

ndash20

ndash10

0

10

20

τ(q)

ndash5 0 5 10ndash10q

SpringSummer

AutumnWinter

(b)

Figure 4 Renyi exponent of (a) PM25 concentration and (b) NO2 concentration

Complexity 5

PM25 with that of NO2 Second we conduct multifractalanalysis for two time series in the same season and confirmwhether the implementation of the ldquohaze special lawrdquo hasany implications for the air quality

In this study we choose the minimum segment scale tobe smin 5 and the maximum segment scale smax 15 andq is taken from minus10 to 10e range of qwas similarly chosenfor the multifractal analysis of stock prices and returns inLatin-American stock markets [32] Besides the range of swas selected according to suggestions in [33 34]

41 Multifractal Analysis of PM25 and NO2 From the dailyAPI data selected in Section 3 we draw the time series ofPM25 and NO2 in Figure 1 From Figure 1 we can see that apattern of high concentrations on both sides and lowconcentrations in the middle emerges which indicates thatdue to seasonal factors both air pollutants have the char-acteristics of low concentrations in summer and highconcentrations in winter In summer (June to August) theconcentration of air pollutants tends to rise while in winter(December to February of the following year) the con-centration of air pollutants tends to decrease

To check the existence of a scaling range long-rangecorrelations and multifractality of both indices in fourseasons we first show the log-log plots of Fq(s) versus timescale s of PM25 and NO2 In Figure 2 the continuous linerepresents the double plots between the calculated Fq(s) andtime scale s and the dashed line denotes the correspondingfitting line As the dashed line shows the decreasing gradientwith q increases from minus10 to 10 which indicates that PM25and NO2 time series of all seasons have multifractalproperties

To measure the multifractality quantitatively we applyMF-DFA to estimate properties for two pollutants of indicesPM25 and NO2 First we calculate the generalized Hurst

exponent h(q) and then obtain the Renyi exponent τ(q) andthe Holder exponent α for both indices

Figure 3 shows the relationship between h(q) and q egeneralized Hurst exponents for both time series in anyseason are not constants which indicate that the series arenot monofractal We note that h(q) decreases monotonouslywith an increase in q which shows that the time series ofPM25 and NO2 are not single fractals Figures 3(a) and 3(b)represent PM25 and NO2 respectively

As illustrated in Table 3 the values of h(2) for all timeseries are larger than 05 implying that the fluctuations havesignificant positive persistence Both indices of the timeseries in some seasons are even larger than 1 which isconsistent with the conclusions drawn in some previousstudies [26 35] For the PM25 time series the persistence inspring is stronger than for other seasons and for the NO2time series autumn has the strongest persistence Wecompute ΔH(q) for each time series usingΔH(q) h(qmin) minus h(qmax) From Figure 3 and Table 3 weobserve that ΔH(q) in winter is the highest for the PM25time series which implies itsrsquo multifractal characteristic isstronger in winter and themultifractalities of the other threeseasons are relatively stable However for the NO2 timeseries the multifractality in spring is higher than for all otherseasons

Furthermore the multifractality strength of a time seriescan also be measured with a nonlinear Renyi exponent curve[36] As shown in Figure 4 all curves of the Renyi exponentare nonlinear which reflects the evidence of multifractalityof the PM25 and NO2 series Moreover the differentnonlinear behaviors of the curves of these series show thatthe multifractal strengths have different degrees

en we calculate the Holder exponent α and the fractaldimension f(α) using equation (8) e width of α alsostands for the degree of the multifractality A larger Δαindicates a stronger multifractal nature In Figure 5 and

ndash2

ndash15

ndash1

ndash05

0

05

1f(α

)

1 15 2 25 3 3505α

SpringSummer

AutumnWinter

(a)

ndash1

ndash05

0

05

1

f(α)

1 15 2 2505α

SpringSummer

AutumnWinter

(b)

Figure 5 Holder exponent of (a) PM25 concentration and (b) NO2 concentration

6 Complexity

2018516 2018814 201811122018215Date

2018516 2018814 201811122018215Date

2018516 2018814 201811122018215Date

0

50

100

150

200PM

25

0

50

100

150

200

PM2

5

0

50

100

150

200

PM2

5

(a)

2019516 2019814 201911122019215Date

2019516 2019814 201911122019215Date

2019516 2019814 201911122019215Date

0

50

100

150

200

PM2

5

0

50

100

150

200

PM2

5

0

50

100

150

200

PM2

5

(b)

Figure 6 Time series of PM25 (a) before and (b) after implementation of the law From top to bottom are Seoul Busan and Daegu

Complexity 7

055

06

065

07

075

08

085

09

095h(

2)

100 150 200 250 30050Time series length

03

035

04

045

05

h(2)

05

055

06

065

07

075

08

h(2)

100 150 200 250 30050Time series length

100 150 200 250 30050Time series length

(a)

100 150 200 250 30050Time series length

05

055

06

065

07

075

08

085

h(2)

025

03

035

04

045

05

h(2)

03

035

04

045

h(2)

100 150 200 250 30050Time series length

100 150 200 250 30050Time series length

(b)

Figure 7 Confidence intervals of h(2) versus the length of PM25 time series (a) before and (b) after implementation of the law withconfidence level by 950 From top to bottom are Seoul Busan and Daegu

8 Complexity

q = 10

q = 10

q = 10

q = ndash10

q = ndash10

q = ndash10

ndash1

0

1

2

3

4

ln(F

q(s)

)

ndash1

0

1

2

3

4

ln(F

q(s)

)

ndash2

ndash1

0

1

2

3

4

ln(F

q(s)

)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

(a)

q = 10

q = 10

q = 10

q = ndash10

q = ndash10

q = ndash10

0

1

2

3

4

ln(F

q(s)

)

ndash1

0

1

2

3

4

ln(F

q(s)

)

ndash1

0

1

2

3

4

ln(F

q(s)

)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

(b)

Figure 8 Log-log plots of the fluctuation function Fq(s) for PM25 (a) before and (b) after implementation of the law From top to bottomare Seoul Busan and Daegu

Complexity 9

ndash5 0 5 10ndash10q

05

1

15

2

25

3

h(q)

Before lawAfter law

(a)

ndash5 0 5 10ndash10q

05

1

15

2

25

h(q)

Before lawAfter law

(b)

ndash5 0 5 10ndash10q

05

1

15

2

25

3

35

h(q)

Before lawAfter law

(c)

Figure 9 Hurst exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

Before lawAfter law

(a)

ndash5 0 5 10ndash10q

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

Before lawAfter law

(b)

Figure 10 Continued

10 Complexity

ndash40

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

Before lawAfter law

(c)

Figure 10 Renyi exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu

ndash15

ndash1

ndash05

0

05

1

f(α)

1 15 2 25 305α

Before lawAfter law

(a)

1 15 2 25 305α

ndash15

ndash1

ndash05

0

05

1f(α

)

Before lawAfter law

(b)

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

f(α)

1 2 3 40α

Before lawAfter law

(c)

Figure 11 Holder exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu e triangle and circle symbols represent beforeand after the law respectively

Complexity 11

Table 3 the conclusion we described above about the Hurstexponent and the Renyi exponent is discussed Moreover asshown in Figure 5 the multifractal spectra are stronglyasymmetrical and as an important indicator of fractal or-ganization the asymmetry coefficient which is used toquantify the asymmetry of the multifractal spectrum hasbeen estimated in [37]

Based on Table 3 the multifractal features can be clearlyconfirmed for all time series and it can be seen that exceptfor spring the Hurst exponents and multifractal spectrumwidths of PM25 are obviously larger than those of NO2 andthe multifractal degrees of PM25 are stronger than those ofNO2

42 Implications of the ldquoHaze Special Lawrdquo In this sectionwe focus on the enactment of the ldquohaze special lawrdquo andstudy whether the law is effective in the governance of at-mospheric pollution We select daily data from the sameperiod described in Section 3 As we studied above themultifractal degree of PM25 is stronger than that of NO2us in this section we focus on the comparison of PM25concentrations We show the PM25 time series before andafter the law was implemented in Seoul Busan and Daegu inFigure 6

As the fluctuation statistics are of asymptotic feature asufficiently long-length sequence guarantees the accurateestimation of the Hurst exponent and for a small-lengthsequence the asymptotic value in the slope of ln(Fq(s))

versus ln(s) will be lacked [38] As the total number ofobservations is selected to be 305 to eliminate the doubt ofwhether the selected time series can give representativeresults for a multifractal analysis in the investigation wedealt with this concern by computing confidence intervals ofHurst exponent h(2) to check whether there is a differencecaused by the law for all the three cities e number ofsamples increased from 50 to 300 and every additionalinterval is 10 e confidence level is selected as 950 Foreach set of sample lengths we generate the shuffled timeseries 1000 times and then the confidence interval is plottedin Figure 7

As shown in Figure 7 and as expected the domain sizedecreases with an increase in the length of the time seriesHowever when the length of the time series is larger than 200the decrease in the domain size of the confidence intervals israther slow and of a distinct asymptotic behavior Besides we

see that the confidence intervals do not overlap for Seoulwhile for Busan ([034 039] before and [032 037] after) andDaegu ([055 064] before and [058 069] after) the confi-dence intervals overlap a lot erefore we can conclude thatthere is a difference caused by the law for Seoul however thesame conclusion is inconclusive for Busan and Daegu

Now we first depict the log-log plots of the fluctuationfunction Fq(s) versus q and s for the PM25 time series Asillustrated in Figure 8 for all series the fluctuation valueincreases linearly with s indicating that a power-law be-havior and long-range correlations exist in each series paire decreasing gradient shows that all time series havemultifractal characteristics

e generalized Hurst exponent h(q) Renyi exponentτ(q) and Holder exponent α of the time series before andafter the enactment of the ldquohaze special lawrdquo for all the threecities are shown in Figures 9ndash11 respectively

In Figure 9 it is shown that the generalized Hurst ex-ponents h(q) of PM25 of all three cities decrease with variedq thereby showing the multifractal behaviors of these twoseries We present h(2) for these series in Table 4 All Hurstexponents are confirmed to be larger than 05 which in-dicates that all the time series exhibit positive persistenceBesides we observe that h(2) before the implementation ofthe law is larger than after the law implying that the fluc-tuations before the law have more significant persistenceSubsequently we calculate the range of h(q) for each timeseries and list them in Table 4 We find that for all threecities ΔH(q) after the law is larger which indicates that themultifractal characteristics are stronger and that the mul-tifractalities of the time series before the law are more stable

e Renyi exponents τ(q) plotted in Figure 10 arenonlinear along q which provides further evidence of theexistence of multifractality We notice that the curvature ofthe Renyi exponents before the law are higher which impliesthat the time series before the implementation of the lawhave stronger multifractal characteristics All of these areconsistent with the results calculated from the generalizedHurst exponents

At last the multifractal spectrums of these time series areexamined We depict the curves between α and f(α) inFigure 11 We see that the multifractal spectra are not shownas points which indicates all the time series have multi-fractality en Δα are calculated and presented in Table 4From Table 4 it can be seen that ΔH(q) and Δα of the timeseries after the enactment of law are lower than those before

Table 4 Multifractality for PM25 concentration

City Multifractality Before law After law

Seoulh(2) 12270 12085ΔH(q) 17329 11410Δα 22414 13953

Busanh(2) 12905 11041ΔH(q) 13539 09565Δα 17463 12708

Daeguh(2) 11975 10878ΔH(q) 22910 19516Δα 29610 25043

12 Complexity

05

1

15

2

25

3

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

35

h(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

0

05

1

15

2

h(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

h(q)

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 12 Hurst exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 13

ndash30

ndash20

ndash10

0

10

τ(q)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash40

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 13 Renyi exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

14 Complexity

ndash3

ndash2

ndash1

0

1

f(α)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

1 2 3 40α

05 1 15 2 25 30α

05 1 15 2 25 30α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

05 1 15 2 25 30α

05 1 15 20α

05 1 15 2 250α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 14 Holder exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 15

the implementation of the law for all three cities implyingthat the multifractality of the time series becomes morestable with the measures taken by the government and thatthe efficiency of the ldquohaze special lawrdquo in three cities isdemonstrable this conclusion can be referred to [16 39] inwhich it was pointed out that the decrease in multifractalitymeans the efficiency is improved when some measures aretaken Moreover we also notice that the efficiency of theldquohaze special lawrdquo in Seoul is most obvious when comparedwith Busan and in last place is Daegu Interestingly theorder is the same as the scale and prosperity of the citiesWe can therefore conclude that the law has a higher effecton a larger city

43 Sources of Multifractal Features e two universallyacknowledged sources of multifractality are long-rangecorrelations and a fat-tailed distribution [7 40] Now weinvestigate the major source of multifractality of the PM25concentration series To check the contribution of long-range correlations and fat-tailed distribution quantitativelywe shuffle and phase-randomize the time series

Subsequently we calculate the Hurst exponent h(q) theRenyi exponent τ(q) and the Holder exponent Δα of theoriginal shuffled and phase-randomized time series of thethree cities As shown in Figures 12ndash14 all the time series arestrongly multifractal From Table 5 all h(2) of the originalseries are larger than those of the shuffled and phase-ran-domized time series which indicates that the persistence hasbeen moved after shuffling and phase-randomizationMoreover the original series of all three cities has thegreatest spectrum width Δα while the multifractality de-creased after shuffling and phase-randomizing the seriese findings can also be confirmed in Figures 12ndash14

To examine the main source of multifractality the firstand third rows of Figure 14 show that the multifractalspectra of phase-randomized series are the narrowest ones

which means that the main source of multifractality of theseries before and after the law in Seoul and Daegu are the fat-tailed distributions e second row in Figure 14 shows thatthe multifractal spectrum of the shuffled series is the nar-rowest which implies that the long-range correlation resultsin multifractality of the series before and after imple-mentation of the law in Busan

5 Conclusions

In this study we examined the multifractal characteristics ofPM25 and NO2 time series of all seasons We determinedthat multifractality existed in each season and the multi-fractal property of PM25 was stronger than that of NO2 ineach season We also validated the effectiveness of the ldquohazespecial lawrdquo which was implemented to improve the gov-ernance of air pollution in South Korea We checkedwhether there is a difference caused by the law for these citiesand calculated confidence intervals of Hurst exponent h(2)e results showed that the confidence intervals do notoverlap for Seoul while the confidence intervals overlap a lotfor Busan and Daegu indicating that there is a differencecaused by the law for Seoul however the same conclusion isinconclusive for Busan and Daegu By comparing the Hurstexponent Renyi exponent and Holder exponent the timeseries of the PM25 concentration before the implementationof the law was found to have a higher multifractal degreewhich decreased after the enactment of the law is phe-nomenon reflected the fact that the law has played a role inimproving the efficiency of air pollution control in SouthKorea We also concluded that the effect of the law will bemore significant in a larger city To explore the major causesof multifractality we shuffled and phase-randomized theoriginal series of PM25 By analyzing the width of themultifractal spectrum Δα the results showed that the fat-tailed distribution contributed to the multifractality of bothtimes series before and after the implementation of the ldquohazespecial lawrdquo in Seoul and Daegu whereas long-range cor-relations resulted in the multifractality of the series beforeand after the implementation of the law in Busan Weconcluded that the implementation of the law was verysuccessful and effective for the improvement of PM25 levelserefore we believe that the ldquohaze special lawrdquo is a possiblecause for the change in API in South Korea

Data Availability

e data will be available upon request with the corre-sponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e first author (Jian Wang) was supported by the ChinaScholarship Council (201808260026) J S Kim expressesgratitude for the support from the BK21 PLUS programe

Table 5 Multifractality of PM25 time series

h(2) Δα LRC FTD

Seoul

BeforeOriginal 12270 22414

No YesShuffled 07232 17946Phase-randomized 06965 09574

AfterOriginal 12085 13953

No YesShuffled 03177 10978Phase-randomized 03049 08569

Busan

BeforeOriginal 12905 17463

Yes NoShuffled 03982 09999Phase-randomized 03993 13512

AfterOriginal 11041 12708

Yes NoShuffled 03678 07175Phase-randomized 03794 08673

Daegu

BeforeOriginal 11975 29610

No YesShuffled 05901 22093Phase-randomized 05868 20587

AfterOriginal 10878 25043

No YesShuffled 06756 25600Phase-randomized 06407 22981

LRC and FTD represent the long-range correlations and fat-tailed distri-bution respectively

16 Complexity

corresponding author Wei Shao thanks Professor DaraeJeong for helpful discussion

References

[1] G Shaddick and J Wakefield ldquoModelling daily multivariatepollutant data at multiple sitesrdquo Journal of the Royal StatisticalSociety Series C (Applied Statistics) vol 51 no 3 pp 351ndash3722002

[2] M OrsquoDwyer Air Quality in Ireland 2010 Key Indicators ofAmbient Air Quality EPA Wexford Ireland 2011

[3] A Challoner and L Gill ldquoIndooroutdoor air pollution re-lationships in ten commercial buildings PM25 and NO2rdquoBuilding and Environment vol 80 pp 159ndash173 2014

[4] M Hazenkamp-von Arx T Gotschi U Ackermann-Liebrichet al ldquoPM25 and NO2 assessment in 21 European studycentres of ECRHS II annual means and seasonal differencesrdquoAtmospheric Environment vol 38 no 13 pp 1943ndash19532004

[5] H K Lai L Bayer-Oglesby R Colvile et al ldquoDeterminants ofindoor air concentrations of PM25 black smoke and NO2 insix European cities (EXPOLIS study)rdquo Atmospheric Envi-ronment vol 40 no 7 pp 1299ndash1313 2006

[6] K de Hoogh J Gulliver A V Donkelaar et al ldquoDevelopmentof West-European PM25 and NO2 land use regression modelsincorporating satellite-derived and chemical transport mod-elling datardquo Environmental Research vol 151 pp 1ndash10 2016

[7] S Drozdz J Kwapien P Oswiecimka and R Rak ldquoQuan-titative features of multifractal subtleties in time seriesrdquo EPL(Europhysics Letters) vol 88 no 6 p 60003 2010

[8] R J Buonocore T Aste and T Di Matteo ldquoMeasuringmultiscaling in financial time-seriesrdquo Chaos Solitons ampFractals vol 88 pp 38ndash47 2016

[9] P Oswiecimka S Drozdz J Kwapien and A Z GorskildquoFractals log-periodicity and financial crashesrdquo Acta PhysicaPolonica A vol 117 no 4 pp 637ndash639 2010

[10] B B Mandelbrot and N Goldenfeld ldquoFractals and scaling infinance discontinuity concentration riskrdquo Physics Todayvol 51 no 8 pp 59-60 1998

[11] S Drozdz R Kowalski P Oswiecimka R Rak andR Gebarowski ldquoDynamical variety of shapes in financialmultifractalityrdquo Complexity vol 2018 Article ID 701572113 pages 2018

[12] S Lahmiri ldquoMultifractal analysis of Moroccan family businessstock returnsrdquo Physica A Statistical Mechanics and Its Ap-plications vol 486 pp 183ndash191 2017

[13] X Zhao P Shang and Q Jin ldquoMultifractal detrended cross-correlation analysis of Chinese stock markets based on timedelayrdquo Fractals vol 19 no 03 pp 329ndash338 2011

[14] W Shao and J Wang ldquoDoes the ldquoice-breakingrdquo of South andNorth Korea affect the South Korean financial marketrdquoChaos Solitons amp Fractals vol 132 Article ID 109564 2020

[15] D-H Wang X-W Yu and Y-Y Suo ldquoStatistical propertiesof the yuan exchange rate indexrdquo Physica A Statistical Me-chanics and Its Applications vol 391 no 12 pp 3503ndash35122012

[16] Y Ning Y Wang and C-W Su ldquoHow did Chinarsquos foreignexchange reform affect the efficiency of foreign exchangemarketrdquo Physica A Statistical Mechanics and Its Applicationsvol 483 pp 219ndash226 2017

[17] S Lahmiri and S Bekiros ldquoChaos randomness and multi-fractality in bitcoin marketrdquo Chaos Solitons amp Fractalsvol 106 pp 28ndash34 2018

[18] P Manimaran and A C Narayana ldquoMultifractal detrendedcross-correlation analysis on air pollutants of University ofHyderabad Campus Indiardquo Physica A Statistical Mechanicsand Its Applications vol 502 pp 228ndash235 2018

[19] L Zhao W Li C Yang J Han Z Su and Y Zou ldquoMul-tifractality and network analysis of phase transitionrdquo PLoSOne vol 12 Article ID e0170467 2017

[20] N Kalamaras K Philippopoulos D Deligiorgi C G Tzanisand G Karvounis ldquoMultifractal scaling properties of daily airtemperature time seriesrdquo Chaos Solitons amp Fractals vol 98pp 38ndash43 2017

[21] J Li C Chen Q Yao et al ldquoe effect of circadian rhythm onthe correlation and multifractality of heart rate signals duringexerciserdquo Physica A Statistical Mechanics and Its Applica-tions vol 509 pp 1207ndash1213 2018

[22] P Castiglioni D Lazzeroni P Coruzzi and A FainildquoMultifractal-multiscale analysis of cardiovascular signals aDFA-based characterization of blood pressure and heart-ratecomplexity by genderrdquo Complexity vol 2018 Article ID4801924 14 pages 2018

[23] W Shi R-B Zou F Wang and L Su ldquoA new image seg-mentation method based on multifractal detrended movingaverage analysisrdquo Physica A Statistical Mechanics and ItsApplications vol 432 pp 197ndash205 2015

[24] F Wang Q Fan and H E Stanley ldquoMultiscale multifractaldetrended-fluctuation analysis of two-dimensional surfacesrdquoPhysical Review E vol 93 Article ID 042213 2016

[25] X Li M Zheng J Pu et al ldquoIdentification of abnormallyexpressed lncrnas induced by PM25 in human bronchialepithelial cellsrdquo Bioscience Reports vol 38 no 5 Article IDBSR20171577 2018

[26] K Shi C Liu and Y Huang ldquoMultifractal processes and self-organized criticality of PM25 during a typical haze period inChengdu Chinardquo Aerosol and Air Quality Research vol 15no 3 pp 926ndash934 2015

[27] C Zhang Z Ni and L Ni ldquoMultifractal detrended cross-correlation analysis between PM25 and meteorological fac-torsrdquo Physica A Statistical Mechanics and Its Applicationsvol 438 pp 114ndash123 2015

[28] C Zhang X Wang S Chen L Zou X Zhang and C TangldquoA study on daily PM25 concentrations in Hong Kong usingthe EMD-based MFDFA methodrdquo Physica A StatisticalMechanics and Its Applications vol 530 p 121182 2019

[29] H-D He ldquoMultifractal analysis of interactive patterns be-tween meteorological factors and pollutants in urban andrural areasrdquo Atmospheric Environment vol 149 pp 47ndash542017

[30] J W Kantelhardt S A Zschiegner E Koscielny-BundeS Havlin A Bunde and H E Stanley ldquoMultifractaldetrended fluctuation analysis of nonstationary time seriesrdquoPhysica A Statistical Mechanics and Its Applications vol 316no 1ndash4 pp 87ndash114 2002

[31] L Zunino B M Tabak A Figliola D G PerezM Garavaglia and O A Rosso ldquoA multifractal approach forstock market inefficiencyrdquo Physica A Statistical Mechanicsand Its Applications vol 387 no 26 pp 6558ndash6566 2008

[32] L Zunino A Figliola B M Tabak D G PerezM Garavaglia and O A Rosso ldquoMultifractal structure inLatin-American market indicesrdquo Chaos Solitons amp Fractalsvol 41 no 5 pp 2331ndash2340 2009

[33] B Lashermes P Abry and P Chainais ldquoNew insights into theestimation of scaling exponentsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 2no 4 pp 497ndash523 2004

Complexity 17

[34] E A F Ihlen ldquoIntroduction to multifractal detrended fluc-tuation analysis in Matlabrdquo Frontiers in Physiology vol 3p 141 2012

[35] L Jiang X Zhao N Li F Li and Z Guo ldquoDifferent mul-tifractal scaling of the 0 cm average ground surface tem-perature of four representative weather stations over ChinardquoAdvances inMeteorology vol 2013 Article ID 341934 8 pages2013

[36] P C Ivanov L A N Amaral A L Goldberger et alldquoMultifractality in human heartbeat dynamicsrdquo Naturevol 399 no 6735 pp 461ndash465 1999

[37] S Drozdz and P Oswiecimka ldquoDetecting and interpretingdistortions in hierarchical organization of complex time se-riesrdquo Physical Review E vol 91 Article ID 030902 2015

[38] C Ibarra-Valdez J Alvarez and J Alvarez-Ramirez ldquoRan-domness confidence bands of fractal scaling exponents forfinancial price returnsrdquo Chaos Solitons amp Fractals vol 83pp 119ndash124 2016

[39] D Stosic D Stosic T Stosic and H E Stanley ldquoMultifractalanalysis of managed and independent float exchange ratesrdquoPhysica A Statistical Mechanics and its Applications vol 428pp 13ndash18 2015

[40] K Matia Y Ashkenazy and H E Stanley ldquoMultifractalproperties of price fluctuations of stocks and commoditiesrdquoEurophysics Letters (EPL) vol 61 no 3 pp 422ndash428 2003

18 Complexity

05

1

15

2

25

3

h(q)

ndash5 0 5 10ndash10q

SpringSummer

AutumnWinter

(a)

ndash5 0 5 10ndash10q

05

1

15

2

h(q)

SpringSummer

AutumnWinter

(b)

Figure 3 Hurst exponent of (a) PM25 concentration and (b) NO2 concentration

Table 3 Multifractality for PM25 and NO2 concentrations

h(2) ΔH(q) Δα

PM25

Spring 14290 10297 12224Summer 09991 12021 14787Autumn 11797 14247 16851Winter 11626 15500 20520

NO2

Spring 10705 11699 14212Summer 07468 11350 14151Autumn 12635 04880 06705Winter 07718 08884 10958

ndash30

ndash20

ndash10

0

10

20

τ(q)

ndash5 0 5 10ndash10q

SpringSummer

AutumnWinter

(a)

ndash20

ndash10

0

10

20

τ(q)

ndash5 0 5 10ndash10q

SpringSummer

AutumnWinter

(b)

Figure 4 Renyi exponent of (a) PM25 concentration and (b) NO2 concentration

Complexity 5

PM25 with that of NO2 Second we conduct multifractalanalysis for two time series in the same season and confirmwhether the implementation of the ldquohaze special lawrdquo hasany implications for the air quality

In this study we choose the minimum segment scale tobe smin 5 and the maximum segment scale smax 15 andq is taken from minus10 to 10e range of qwas similarly chosenfor the multifractal analysis of stock prices and returns inLatin-American stock markets [32] Besides the range of swas selected according to suggestions in [33 34]

41 Multifractal Analysis of PM25 and NO2 From the dailyAPI data selected in Section 3 we draw the time series ofPM25 and NO2 in Figure 1 From Figure 1 we can see that apattern of high concentrations on both sides and lowconcentrations in the middle emerges which indicates thatdue to seasonal factors both air pollutants have the char-acteristics of low concentrations in summer and highconcentrations in winter In summer (June to August) theconcentration of air pollutants tends to rise while in winter(December to February of the following year) the con-centration of air pollutants tends to decrease

To check the existence of a scaling range long-rangecorrelations and multifractality of both indices in fourseasons we first show the log-log plots of Fq(s) versus timescale s of PM25 and NO2 In Figure 2 the continuous linerepresents the double plots between the calculated Fq(s) andtime scale s and the dashed line denotes the correspondingfitting line As the dashed line shows the decreasing gradientwith q increases from minus10 to 10 which indicates that PM25and NO2 time series of all seasons have multifractalproperties

To measure the multifractality quantitatively we applyMF-DFA to estimate properties for two pollutants of indicesPM25 and NO2 First we calculate the generalized Hurst

exponent h(q) and then obtain the Renyi exponent τ(q) andthe Holder exponent α for both indices

Figure 3 shows the relationship between h(q) and q egeneralized Hurst exponents for both time series in anyseason are not constants which indicate that the series arenot monofractal We note that h(q) decreases monotonouslywith an increase in q which shows that the time series ofPM25 and NO2 are not single fractals Figures 3(a) and 3(b)represent PM25 and NO2 respectively

As illustrated in Table 3 the values of h(2) for all timeseries are larger than 05 implying that the fluctuations havesignificant positive persistence Both indices of the timeseries in some seasons are even larger than 1 which isconsistent with the conclusions drawn in some previousstudies [26 35] For the PM25 time series the persistence inspring is stronger than for other seasons and for the NO2time series autumn has the strongest persistence Wecompute ΔH(q) for each time series usingΔH(q) h(qmin) minus h(qmax) From Figure 3 and Table 3 weobserve that ΔH(q) in winter is the highest for the PM25time series which implies itsrsquo multifractal characteristic isstronger in winter and themultifractalities of the other threeseasons are relatively stable However for the NO2 timeseries the multifractality in spring is higher than for all otherseasons

Furthermore the multifractality strength of a time seriescan also be measured with a nonlinear Renyi exponent curve[36] As shown in Figure 4 all curves of the Renyi exponentare nonlinear which reflects the evidence of multifractalityof the PM25 and NO2 series Moreover the differentnonlinear behaviors of the curves of these series show thatthe multifractal strengths have different degrees

en we calculate the Holder exponent α and the fractaldimension f(α) using equation (8) e width of α alsostands for the degree of the multifractality A larger Δαindicates a stronger multifractal nature In Figure 5 and

ndash2

ndash15

ndash1

ndash05

0

05

1f(α

)

1 15 2 25 3 3505α

SpringSummer

AutumnWinter

(a)

ndash1

ndash05

0

05

1

f(α)

1 15 2 2505α

SpringSummer

AutumnWinter

(b)

Figure 5 Holder exponent of (a) PM25 concentration and (b) NO2 concentration

6 Complexity

2018516 2018814 201811122018215Date

2018516 2018814 201811122018215Date

2018516 2018814 201811122018215Date

0

50

100

150

200PM

25

0

50

100

150

200

PM2

5

0

50

100

150

200

PM2

5

(a)

2019516 2019814 201911122019215Date

2019516 2019814 201911122019215Date

2019516 2019814 201911122019215Date

0

50

100

150

200

PM2

5

0

50

100

150

200

PM2

5

0

50

100

150

200

PM2

5

(b)

Figure 6 Time series of PM25 (a) before and (b) after implementation of the law From top to bottom are Seoul Busan and Daegu

Complexity 7

055

06

065

07

075

08

085

09

095h(

2)

100 150 200 250 30050Time series length

03

035

04

045

05

h(2)

05

055

06

065

07

075

08

h(2)

100 150 200 250 30050Time series length

100 150 200 250 30050Time series length

(a)

100 150 200 250 30050Time series length

05

055

06

065

07

075

08

085

h(2)

025

03

035

04

045

05

h(2)

03

035

04

045

h(2)

100 150 200 250 30050Time series length

100 150 200 250 30050Time series length

(b)

Figure 7 Confidence intervals of h(2) versus the length of PM25 time series (a) before and (b) after implementation of the law withconfidence level by 950 From top to bottom are Seoul Busan and Daegu

8 Complexity

q = 10

q = 10

q = 10

q = ndash10

q = ndash10

q = ndash10

ndash1

0

1

2

3

4

ln(F

q(s)

)

ndash1

0

1

2

3

4

ln(F

q(s)

)

ndash2

ndash1

0

1

2

3

4

ln(F

q(s)

)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

(a)

q = 10

q = 10

q = 10

q = ndash10

q = ndash10

q = ndash10

0

1

2

3

4

ln(F

q(s)

)

ndash1

0

1

2

3

4

ln(F

q(s)

)

ndash1

0

1

2

3

4

ln(F

q(s)

)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

(b)

Figure 8 Log-log plots of the fluctuation function Fq(s) for PM25 (a) before and (b) after implementation of the law From top to bottomare Seoul Busan and Daegu

Complexity 9

ndash5 0 5 10ndash10q

05

1

15

2

25

3

h(q)

Before lawAfter law

(a)

ndash5 0 5 10ndash10q

05

1

15

2

25

h(q)

Before lawAfter law

(b)

ndash5 0 5 10ndash10q

05

1

15

2

25

3

35

h(q)

Before lawAfter law

(c)

Figure 9 Hurst exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

Before lawAfter law

(a)

ndash5 0 5 10ndash10q

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

Before lawAfter law

(b)

Figure 10 Continued

10 Complexity

ndash40

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

Before lawAfter law

(c)

Figure 10 Renyi exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu

ndash15

ndash1

ndash05

0

05

1

f(α)

1 15 2 25 305α

Before lawAfter law

(a)

1 15 2 25 305α

ndash15

ndash1

ndash05

0

05

1f(α

)

Before lawAfter law

(b)

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

f(α)

1 2 3 40α

Before lawAfter law

(c)

Figure 11 Holder exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu e triangle and circle symbols represent beforeand after the law respectively

Complexity 11

Table 3 the conclusion we described above about the Hurstexponent and the Renyi exponent is discussed Moreover asshown in Figure 5 the multifractal spectra are stronglyasymmetrical and as an important indicator of fractal or-ganization the asymmetry coefficient which is used toquantify the asymmetry of the multifractal spectrum hasbeen estimated in [37]

Based on Table 3 the multifractal features can be clearlyconfirmed for all time series and it can be seen that exceptfor spring the Hurst exponents and multifractal spectrumwidths of PM25 are obviously larger than those of NO2 andthe multifractal degrees of PM25 are stronger than those ofNO2

42 Implications of the ldquoHaze Special Lawrdquo In this sectionwe focus on the enactment of the ldquohaze special lawrdquo andstudy whether the law is effective in the governance of at-mospheric pollution We select daily data from the sameperiod described in Section 3 As we studied above themultifractal degree of PM25 is stronger than that of NO2us in this section we focus on the comparison of PM25concentrations We show the PM25 time series before andafter the law was implemented in Seoul Busan and Daegu inFigure 6

As the fluctuation statistics are of asymptotic feature asufficiently long-length sequence guarantees the accurateestimation of the Hurst exponent and for a small-lengthsequence the asymptotic value in the slope of ln(Fq(s))

versus ln(s) will be lacked [38] As the total number ofobservations is selected to be 305 to eliminate the doubt ofwhether the selected time series can give representativeresults for a multifractal analysis in the investigation wedealt with this concern by computing confidence intervals ofHurst exponent h(2) to check whether there is a differencecaused by the law for all the three cities e number ofsamples increased from 50 to 300 and every additionalinterval is 10 e confidence level is selected as 950 Foreach set of sample lengths we generate the shuffled timeseries 1000 times and then the confidence interval is plottedin Figure 7

As shown in Figure 7 and as expected the domain sizedecreases with an increase in the length of the time seriesHowever when the length of the time series is larger than 200the decrease in the domain size of the confidence intervals israther slow and of a distinct asymptotic behavior Besides we

see that the confidence intervals do not overlap for Seoulwhile for Busan ([034 039] before and [032 037] after) andDaegu ([055 064] before and [058 069] after) the confi-dence intervals overlap a lot erefore we can conclude thatthere is a difference caused by the law for Seoul however thesame conclusion is inconclusive for Busan and Daegu

Now we first depict the log-log plots of the fluctuationfunction Fq(s) versus q and s for the PM25 time series Asillustrated in Figure 8 for all series the fluctuation valueincreases linearly with s indicating that a power-law be-havior and long-range correlations exist in each series paire decreasing gradient shows that all time series havemultifractal characteristics

e generalized Hurst exponent h(q) Renyi exponentτ(q) and Holder exponent α of the time series before andafter the enactment of the ldquohaze special lawrdquo for all the threecities are shown in Figures 9ndash11 respectively

In Figure 9 it is shown that the generalized Hurst ex-ponents h(q) of PM25 of all three cities decrease with variedq thereby showing the multifractal behaviors of these twoseries We present h(2) for these series in Table 4 All Hurstexponents are confirmed to be larger than 05 which in-dicates that all the time series exhibit positive persistenceBesides we observe that h(2) before the implementation ofthe law is larger than after the law implying that the fluc-tuations before the law have more significant persistenceSubsequently we calculate the range of h(q) for each timeseries and list them in Table 4 We find that for all threecities ΔH(q) after the law is larger which indicates that themultifractal characteristics are stronger and that the mul-tifractalities of the time series before the law are more stable

e Renyi exponents τ(q) plotted in Figure 10 arenonlinear along q which provides further evidence of theexistence of multifractality We notice that the curvature ofthe Renyi exponents before the law are higher which impliesthat the time series before the implementation of the lawhave stronger multifractal characteristics All of these areconsistent with the results calculated from the generalizedHurst exponents

At last the multifractal spectrums of these time series areexamined We depict the curves between α and f(α) inFigure 11 We see that the multifractal spectra are not shownas points which indicates all the time series have multi-fractality en Δα are calculated and presented in Table 4From Table 4 it can be seen that ΔH(q) and Δα of the timeseries after the enactment of law are lower than those before

Table 4 Multifractality for PM25 concentration

City Multifractality Before law After law

Seoulh(2) 12270 12085ΔH(q) 17329 11410Δα 22414 13953

Busanh(2) 12905 11041ΔH(q) 13539 09565Δα 17463 12708

Daeguh(2) 11975 10878ΔH(q) 22910 19516Δα 29610 25043

12 Complexity

05

1

15

2

25

3

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

35

h(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

0

05

1

15

2

h(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

h(q)

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 12 Hurst exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 13

ndash30

ndash20

ndash10

0

10

τ(q)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash40

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 13 Renyi exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

14 Complexity

ndash3

ndash2

ndash1

0

1

f(α)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

1 2 3 40α

05 1 15 2 25 30α

05 1 15 2 25 30α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

05 1 15 2 25 30α

05 1 15 20α

05 1 15 2 250α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 14 Holder exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 15

the implementation of the law for all three cities implyingthat the multifractality of the time series becomes morestable with the measures taken by the government and thatthe efficiency of the ldquohaze special lawrdquo in three cities isdemonstrable this conclusion can be referred to [16 39] inwhich it was pointed out that the decrease in multifractalitymeans the efficiency is improved when some measures aretaken Moreover we also notice that the efficiency of theldquohaze special lawrdquo in Seoul is most obvious when comparedwith Busan and in last place is Daegu Interestingly theorder is the same as the scale and prosperity of the citiesWe can therefore conclude that the law has a higher effecton a larger city

43 Sources of Multifractal Features e two universallyacknowledged sources of multifractality are long-rangecorrelations and a fat-tailed distribution [7 40] Now weinvestigate the major source of multifractality of the PM25concentration series To check the contribution of long-range correlations and fat-tailed distribution quantitativelywe shuffle and phase-randomize the time series

Subsequently we calculate the Hurst exponent h(q) theRenyi exponent τ(q) and the Holder exponent Δα of theoriginal shuffled and phase-randomized time series of thethree cities As shown in Figures 12ndash14 all the time series arestrongly multifractal From Table 5 all h(2) of the originalseries are larger than those of the shuffled and phase-ran-domized time series which indicates that the persistence hasbeen moved after shuffling and phase-randomizationMoreover the original series of all three cities has thegreatest spectrum width Δα while the multifractality de-creased after shuffling and phase-randomizing the seriese findings can also be confirmed in Figures 12ndash14

To examine the main source of multifractality the firstand third rows of Figure 14 show that the multifractalspectra of phase-randomized series are the narrowest ones

which means that the main source of multifractality of theseries before and after the law in Seoul and Daegu are the fat-tailed distributions e second row in Figure 14 shows thatthe multifractal spectrum of the shuffled series is the nar-rowest which implies that the long-range correlation resultsin multifractality of the series before and after imple-mentation of the law in Busan

5 Conclusions

In this study we examined the multifractal characteristics ofPM25 and NO2 time series of all seasons We determinedthat multifractality existed in each season and the multi-fractal property of PM25 was stronger than that of NO2 ineach season We also validated the effectiveness of the ldquohazespecial lawrdquo which was implemented to improve the gov-ernance of air pollution in South Korea We checkedwhether there is a difference caused by the law for these citiesand calculated confidence intervals of Hurst exponent h(2)e results showed that the confidence intervals do notoverlap for Seoul while the confidence intervals overlap a lotfor Busan and Daegu indicating that there is a differencecaused by the law for Seoul however the same conclusion isinconclusive for Busan and Daegu By comparing the Hurstexponent Renyi exponent and Holder exponent the timeseries of the PM25 concentration before the implementationof the law was found to have a higher multifractal degreewhich decreased after the enactment of the law is phe-nomenon reflected the fact that the law has played a role inimproving the efficiency of air pollution control in SouthKorea We also concluded that the effect of the law will bemore significant in a larger city To explore the major causesof multifractality we shuffled and phase-randomized theoriginal series of PM25 By analyzing the width of themultifractal spectrum Δα the results showed that the fat-tailed distribution contributed to the multifractality of bothtimes series before and after the implementation of the ldquohazespecial lawrdquo in Seoul and Daegu whereas long-range cor-relations resulted in the multifractality of the series beforeand after the implementation of the law in Busan Weconcluded that the implementation of the law was verysuccessful and effective for the improvement of PM25 levelserefore we believe that the ldquohaze special lawrdquo is a possiblecause for the change in API in South Korea

Data Availability

e data will be available upon request with the corre-sponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e first author (Jian Wang) was supported by the ChinaScholarship Council (201808260026) J S Kim expressesgratitude for the support from the BK21 PLUS programe

Table 5 Multifractality of PM25 time series

h(2) Δα LRC FTD

Seoul

BeforeOriginal 12270 22414

No YesShuffled 07232 17946Phase-randomized 06965 09574

AfterOriginal 12085 13953

No YesShuffled 03177 10978Phase-randomized 03049 08569

Busan

BeforeOriginal 12905 17463

Yes NoShuffled 03982 09999Phase-randomized 03993 13512

AfterOriginal 11041 12708

Yes NoShuffled 03678 07175Phase-randomized 03794 08673

Daegu

BeforeOriginal 11975 29610

No YesShuffled 05901 22093Phase-randomized 05868 20587

AfterOriginal 10878 25043

No YesShuffled 06756 25600Phase-randomized 06407 22981

LRC and FTD represent the long-range correlations and fat-tailed distri-bution respectively

16 Complexity

corresponding author Wei Shao thanks Professor DaraeJeong for helpful discussion

References

[1] G Shaddick and J Wakefield ldquoModelling daily multivariatepollutant data at multiple sitesrdquo Journal of the Royal StatisticalSociety Series C (Applied Statistics) vol 51 no 3 pp 351ndash3722002

[2] M OrsquoDwyer Air Quality in Ireland 2010 Key Indicators ofAmbient Air Quality EPA Wexford Ireland 2011

[3] A Challoner and L Gill ldquoIndooroutdoor air pollution re-lationships in ten commercial buildings PM25 and NO2rdquoBuilding and Environment vol 80 pp 159ndash173 2014

[4] M Hazenkamp-von Arx T Gotschi U Ackermann-Liebrichet al ldquoPM25 and NO2 assessment in 21 European studycentres of ECRHS II annual means and seasonal differencesrdquoAtmospheric Environment vol 38 no 13 pp 1943ndash19532004

[5] H K Lai L Bayer-Oglesby R Colvile et al ldquoDeterminants ofindoor air concentrations of PM25 black smoke and NO2 insix European cities (EXPOLIS study)rdquo Atmospheric Envi-ronment vol 40 no 7 pp 1299ndash1313 2006

[6] K de Hoogh J Gulliver A V Donkelaar et al ldquoDevelopmentof West-European PM25 and NO2 land use regression modelsincorporating satellite-derived and chemical transport mod-elling datardquo Environmental Research vol 151 pp 1ndash10 2016

[7] S Drozdz J Kwapien P Oswiecimka and R Rak ldquoQuan-titative features of multifractal subtleties in time seriesrdquo EPL(Europhysics Letters) vol 88 no 6 p 60003 2010

[8] R J Buonocore T Aste and T Di Matteo ldquoMeasuringmultiscaling in financial time-seriesrdquo Chaos Solitons ampFractals vol 88 pp 38ndash47 2016

[9] P Oswiecimka S Drozdz J Kwapien and A Z GorskildquoFractals log-periodicity and financial crashesrdquo Acta PhysicaPolonica A vol 117 no 4 pp 637ndash639 2010

[10] B B Mandelbrot and N Goldenfeld ldquoFractals and scaling infinance discontinuity concentration riskrdquo Physics Todayvol 51 no 8 pp 59-60 1998

[11] S Drozdz R Kowalski P Oswiecimka R Rak andR Gebarowski ldquoDynamical variety of shapes in financialmultifractalityrdquo Complexity vol 2018 Article ID 701572113 pages 2018

[12] S Lahmiri ldquoMultifractal analysis of Moroccan family businessstock returnsrdquo Physica A Statistical Mechanics and Its Ap-plications vol 486 pp 183ndash191 2017

[13] X Zhao P Shang and Q Jin ldquoMultifractal detrended cross-correlation analysis of Chinese stock markets based on timedelayrdquo Fractals vol 19 no 03 pp 329ndash338 2011

[14] W Shao and J Wang ldquoDoes the ldquoice-breakingrdquo of South andNorth Korea affect the South Korean financial marketrdquoChaos Solitons amp Fractals vol 132 Article ID 109564 2020

[15] D-H Wang X-W Yu and Y-Y Suo ldquoStatistical propertiesof the yuan exchange rate indexrdquo Physica A Statistical Me-chanics and Its Applications vol 391 no 12 pp 3503ndash35122012

[16] Y Ning Y Wang and C-W Su ldquoHow did Chinarsquos foreignexchange reform affect the efficiency of foreign exchangemarketrdquo Physica A Statistical Mechanics and Its Applicationsvol 483 pp 219ndash226 2017

[17] S Lahmiri and S Bekiros ldquoChaos randomness and multi-fractality in bitcoin marketrdquo Chaos Solitons amp Fractalsvol 106 pp 28ndash34 2018

[18] P Manimaran and A C Narayana ldquoMultifractal detrendedcross-correlation analysis on air pollutants of University ofHyderabad Campus Indiardquo Physica A Statistical Mechanicsand Its Applications vol 502 pp 228ndash235 2018

[19] L Zhao W Li C Yang J Han Z Su and Y Zou ldquoMul-tifractality and network analysis of phase transitionrdquo PLoSOne vol 12 Article ID e0170467 2017

[20] N Kalamaras K Philippopoulos D Deligiorgi C G Tzanisand G Karvounis ldquoMultifractal scaling properties of daily airtemperature time seriesrdquo Chaos Solitons amp Fractals vol 98pp 38ndash43 2017

[21] J Li C Chen Q Yao et al ldquoe effect of circadian rhythm onthe correlation and multifractality of heart rate signals duringexerciserdquo Physica A Statistical Mechanics and Its Applica-tions vol 509 pp 1207ndash1213 2018

[22] P Castiglioni D Lazzeroni P Coruzzi and A FainildquoMultifractal-multiscale analysis of cardiovascular signals aDFA-based characterization of blood pressure and heart-ratecomplexity by genderrdquo Complexity vol 2018 Article ID4801924 14 pages 2018

[23] W Shi R-B Zou F Wang and L Su ldquoA new image seg-mentation method based on multifractal detrended movingaverage analysisrdquo Physica A Statistical Mechanics and ItsApplications vol 432 pp 197ndash205 2015

[24] F Wang Q Fan and H E Stanley ldquoMultiscale multifractaldetrended-fluctuation analysis of two-dimensional surfacesrdquoPhysical Review E vol 93 Article ID 042213 2016

[25] X Li M Zheng J Pu et al ldquoIdentification of abnormallyexpressed lncrnas induced by PM25 in human bronchialepithelial cellsrdquo Bioscience Reports vol 38 no 5 Article IDBSR20171577 2018

[26] K Shi C Liu and Y Huang ldquoMultifractal processes and self-organized criticality of PM25 during a typical haze period inChengdu Chinardquo Aerosol and Air Quality Research vol 15no 3 pp 926ndash934 2015

[27] C Zhang Z Ni and L Ni ldquoMultifractal detrended cross-correlation analysis between PM25 and meteorological fac-torsrdquo Physica A Statistical Mechanics and Its Applicationsvol 438 pp 114ndash123 2015

[28] C Zhang X Wang S Chen L Zou X Zhang and C TangldquoA study on daily PM25 concentrations in Hong Kong usingthe EMD-based MFDFA methodrdquo Physica A StatisticalMechanics and Its Applications vol 530 p 121182 2019

[29] H-D He ldquoMultifractal analysis of interactive patterns be-tween meteorological factors and pollutants in urban andrural areasrdquo Atmospheric Environment vol 149 pp 47ndash542017

[30] J W Kantelhardt S A Zschiegner E Koscielny-BundeS Havlin A Bunde and H E Stanley ldquoMultifractaldetrended fluctuation analysis of nonstationary time seriesrdquoPhysica A Statistical Mechanics and Its Applications vol 316no 1ndash4 pp 87ndash114 2002

[31] L Zunino B M Tabak A Figliola D G PerezM Garavaglia and O A Rosso ldquoA multifractal approach forstock market inefficiencyrdquo Physica A Statistical Mechanicsand Its Applications vol 387 no 26 pp 6558ndash6566 2008

[32] L Zunino A Figliola B M Tabak D G PerezM Garavaglia and O A Rosso ldquoMultifractal structure inLatin-American market indicesrdquo Chaos Solitons amp Fractalsvol 41 no 5 pp 2331ndash2340 2009

[33] B Lashermes P Abry and P Chainais ldquoNew insights into theestimation of scaling exponentsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 2no 4 pp 497ndash523 2004

Complexity 17

[34] E A F Ihlen ldquoIntroduction to multifractal detrended fluc-tuation analysis in Matlabrdquo Frontiers in Physiology vol 3p 141 2012

[35] L Jiang X Zhao N Li F Li and Z Guo ldquoDifferent mul-tifractal scaling of the 0 cm average ground surface tem-perature of four representative weather stations over ChinardquoAdvances inMeteorology vol 2013 Article ID 341934 8 pages2013

[36] P C Ivanov L A N Amaral A L Goldberger et alldquoMultifractality in human heartbeat dynamicsrdquo Naturevol 399 no 6735 pp 461ndash465 1999

[37] S Drozdz and P Oswiecimka ldquoDetecting and interpretingdistortions in hierarchical organization of complex time se-riesrdquo Physical Review E vol 91 Article ID 030902 2015

[38] C Ibarra-Valdez J Alvarez and J Alvarez-Ramirez ldquoRan-domness confidence bands of fractal scaling exponents forfinancial price returnsrdquo Chaos Solitons amp Fractals vol 83pp 119ndash124 2016

[39] D Stosic D Stosic T Stosic and H E Stanley ldquoMultifractalanalysis of managed and independent float exchange ratesrdquoPhysica A Statistical Mechanics and its Applications vol 428pp 13ndash18 2015

[40] K Matia Y Ashkenazy and H E Stanley ldquoMultifractalproperties of price fluctuations of stocks and commoditiesrdquoEurophysics Letters (EPL) vol 61 no 3 pp 422ndash428 2003

18 Complexity

PM25 with that of NO2 Second we conduct multifractalanalysis for two time series in the same season and confirmwhether the implementation of the ldquohaze special lawrdquo hasany implications for the air quality

In this study we choose the minimum segment scale tobe smin 5 and the maximum segment scale smax 15 andq is taken from minus10 to 10e range of qwas similarly chosenfor the multifractal analysis of stock prices and returns inLatin-American stock markets [32] Besides the range of swas selected according to suggestions in [33 34]

41 Multifractal Analysis of PM25 and NO2 From the dailyAPI data selected in Section 3 we draw the time series ofPM25 and NO2 in Figure 1 From Figure 1 we can see that apattern of high concentrations on both sides and lowconcentrations in the middle emerges which indicates thatdue to seasonal factors both air pollutants have the char-acteristics of low concentrations in summer and highconcentrations in winter In summer (June to August) theconcentration of air pollutants tends to rise while in winter(December to February of the following year) the con-centration of air pollutants tends to decrease

To check the existence of a scaling range long-rangecorrelations and multifractality of both indices in fourseasons we first show the log-log plots of Fq(s) versus timescale s of PM25 and NO2 In Figure 2 the continuous linerepresents the double plots between the calculated Fq(s) andtime scale s and the dashed line denotes the correspondingfitting line As the dashed line shows the decreasing gradientwith q increases from minus10 to 10 which indicates that PM25and NO2 time series of all seasons have multifractalproperties

To measure the multifractality quantitatively we applyMF-DFA to estimate properties for two pollutants of indicesPM25 and NO2 First we calculate the generalized Hurst

exponent h(q) and then obtain the Renyi exponent τ(q) andthe Holder exponent α for both indices

Figure 3 shows the relationship between h(q) and q egeneralized Hurst exponents for both time series in anyseason are not constants which indicate that the series arenot monofractal We note that h(q) decreases monotonouslywith an increase in q which shows that the time series ofPM25 and NO2 are not single fractals Figures 3(a) and 3(b)represent PM25 and NO2 respectively

As illustrated in Table 3 the values of h(2) for all timeseries are larger than 05 implying that the fluctuations havesignificant positive persistence Both indices of the timeseries in some seasons are even larger than 1 which isconsistent with the conclusions drawn in some previousstudies [26 35] For the PM25 time series the persistence inspring is stronger than for other seasons and for the NO2time series autumn has the strongest persistence Wecompute ΔH(q) for each time series usingΔH(q) h(qmin) minus h(qmax) From Figure 3 and Table 3 weobserve that ΔH(q) in winter is the highest for the PM25time series which implies itsrsquo multifractal characteristic isstronger in winter and themultifractalities of the other threeseasons are relatively stable However for the NO2 timeseries the multifractality in spring is higher than for all otherseasons

Furthermore the multifractality strength of a time seriescan also be measured with a nonlinear Renyi exponent curve[36] As shown in Figure 4 all curves of the Renyi exponentare nonlinear which reflects the evidence of multifractalityof the PM25 and NO2 series Moreover the differentnonlinear behaviors of the curves of these series show thatthe multifractal strengths have different degrees

en we calculate the Holder exponent α and the fractaldimension f(α) using equation (8) e width of α alsostands for the degree of the multifractality A larger Δαindicates a stronger multifractal nature In Figure 5 and

ndash2

ndash15

ndash1

ndash05

0

05

1f(α

)

1 15 2 25 3 3505α

SpringSummer

AutumnWinter

(a)

ndash1

ndash05

0

05

1

f(α)

1 15 2 2505α

SpringSummer

AutumnWinter

(b)

Figure 5 Holder exponent of (a) PM25 concentration and (b) NO2 concentration

6 Complexity

2018516 2018814 201811122018215Date

2018516 2018814 201811122018215Date

2018516 2018814 201811122018215Date

0

50

100

150

200PM

25

0

50

100

150

200

PM2

5

0

50

100

150

200

PM2

5

(a)

2019516 2019814 201911122019215Date

2019516 2019814 201911122019215Date

2019516 2019814 201911122019215Date

0

50

100

150

200

PM2

5

0

50

100

150

200

PM2

5

0

50

100

150

200

PM2

5

(b)

Figure 6 Time series of PM25 (a) before and (b) after implementation of the law From top to bottom are Seoul Busan and Daegu

Complexity 7

055

06

065

07

075

08

085

09

095h(

2)

100 150 200 250 30050Time series length

03

035

04

045

05

h(2)

05

055

06

065

07

075

08

h(2)

100 150 200 250 30050Time series length

100 150 200 250 30050Time series length

(a)

100 150 200 250 30050Time series length

05

055

06

065

07

075

08

085

h(2)

025

03

035

04

045

05

h(2)

03

035

04

045

h(2)

100 150 200 250 30050Time series length

100 150 200 250 30050Time series length

(b)

Figure 7 Confidence intervals of h(2) versus the length of PM25 time series (a) before and (b) after implementation of the law withconfidence level by 950 From top to bottom are Seoul Busan and Daegu

8 Complexity

q = 10

q = 10

q = 10

q = ndash10

q = ndash10

q = ndash10

ndash1

0

1

2

3

4

ln(F

q(s)

)

ndash1

0

1

2

3

4

ln(F

q(s)

)

ndash2

ndash1

0

1

2

3

4

ln(F

q(s)

)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

(a)

q = 10

q = 10

q = 10

q = ndash10

q = ndash10

q = ndash10

0

1

2

3

4

ln(F

q(s)

)

ndash1

0

1

2

3

4

ln(F

q(s)

)

ndash1

0

1

2

3

4

ln(F

q(s)

)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

(b)

Figure 8 Log-log plots of the fluctuation function Fq(s) for PM25 (a) before and (b) after implementation of the law From top to bottomare Seoul Busan and Daegu

Complexity 9

ndash5 0 5 10ndash10q

05

1

15

2

25

3

h(q)

Before lawAfter law

(a)

ndash5 0 5 10ndash10q

05

1

15

2

25

h(q)

Before lawAfter law

(b)

ndash5 0 5 10ndash10q

05

1

15

2

25

3

35

h(q)

Before lawAfter law

(c)

Figure 9 Hurst exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

Before lawAfter law

(a)

ndash5 0 5 10ndash10q

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

Before lawAfter law

(b)

Figure 10 Continued

10 Complexity

ndash40

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

Before lawAfter law

(c)

Figure 10 Renyi exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu

ndash15

ndash1

ndash05

0

05

1

f(α)

1 15 2 25 305α

Before lawAfter law

(a)

1 15 2 25 305α

ndash15

ndash1

ndash05

0

05

1f(α

)

Before lawAfter law

(b)

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

f(α)

1 2 3 40α

Before lawAfter law

(c)

Figure 11 Holder exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu e triangle and circle symbols represent beforeand after the law respectively

Complexity 11

Table 3 the conclusion we described above about the Hurstexponent and the Renyi exponent is discussed Moreover asshown in Figure 5 the multifractal spectra are stronglyasymmetrical and as an important indicator of fractal or-ganization the asymmetry coefficient which is used toquantify the asymmetry of the multifractal spectrum hasbeen estimated in [37]

Based on Table 3 the multifractal features can be clearlyconfirmed for all time series and it can be seen that exceptfor spring the Hurst exponents and multifractal spectrumwidths of PM25 are obviously larger than those of NO2 andthe multifractal degrees of PM25 are stronger than those ofNO2

42 Implications of the ldquoHaze Special Lawrdquo In this sectionwe focus on the enactment of the ldquohaze special lawrdquo andstudy whether the law is effective in the governance of at-mospheric pollution We select daily data from the sameperiod described in Section 3 As we studied above themultifractal degree of PM25 is stronger than that of NO2us in this section we focus on the comparison of PM25concentrations We show the PM25 time series before andafter the law was implemented in Seoul Busan and Daegu inFigure 6

As the fluctuation statistics are of asymptotic feature asufficiently long-length sequence guarantees the accurateestimation of the Hurst exponent and for a small-lengthsequence the asymptotic value in the slope of ln(Fq(s))

versus ln(s) will be lacked [38] As the total number ofobservations is selected to be 305 to eliminate the doubt ofwhether the selected time series can give representativeresults for a multifractal analysis in the investigation wedealt with this concern by computing confidence intervals ofHurst exponent h(2) to check whether there is a differencecaused by the law for all the three cities e number ofsamples increased from 50 to 300 and every additionalinterval is 10 e confidence level is selected as 950 Foreach set of sample lengths we generate the shuffled timeseries 1000 times and then the confidence interval is plottedin Figure 7

As shown in Figure 7 and as expected the domain sizedecreases with an increase in the length of the time seriesHowever when the length of the time series is larger than 200the decrease in the domain size of the confidence intervals israther slow and of a distinct asymptotic behavior Besides we

see that the confidence intervals do not overlap for Seoulwhile for Busan ([034 039] before and [032 037] after) andDaegu ([055 064] before and [058 069] after) the confi-dence intervals overlap a lot erefore we can conclude thatthere is a difference caused by the law for Seoul however thesame conclusion is inconclusive for Busan and Daegu

Now we first depict the log-log plots of the fluctuationfunction Fq(s) versus q and s for the PM25 time series Asillustrated in Figure 8 for all series the fluctuation valueincreases linearly with s indicating that a power-law be-havior and long-range correlations exist in each series paire decreasing gradient shows that all time series havemultifractal characteristics

e generalized Hurst exponent h(q) Renyi exponentτ(q) and Holder exponent α of the time series before andafter the enactment of the ldquohaze special lawrdquo for all the threecities are shown in Figures 9ndash11 respectively

In Figure 9 it is shown that the generalized Hurst ex-ponents h(q) of PM25 of all three cities decrease with variedq thereby showing the multifractal behaviors of these twoseries We present h(2) for these series in Table 4 All Hurstexponents are confirmed to be larger than 05 which in-dicates that all the time series exhibit positive persistenceBesides we observe that h(2) before the implementation ofthe law is larger than after the law implying that the fluc-tuations before the law have more significant persistenceSubsequently we calculate the range of h(q) for each timeseries and list them in Table 4 We find that for all threecities ΔH(q) after the law is larger which indicates that themultifractal characteristics are stronger and that the mul-tifractalities of the time series before the law are more stable

e Renyi exponents τ(q) plotted in Figure 10 arenonlinear along q which provides further evidence of theexistence of multifractality We notice that the curvature ofthe Renyi exponents before the law are higher which impliesthat the time series before the implementation of the lawhave stronger multifractal characteristics All of these areconsistent with the results calculated from the generalizedHurst exponents

At last the multifractal spectrums of these time series areexamined We depict the curves between α and f(α) inFigure 11 We see that the multifractal spectra are not shownas points which indicates all the time series have multi-fractality en Δα are calculated and presented in Table 4From Table 4 it can be seen that ΔH(q) and Δα of the timeseries after the enactment of law are lower than those before

Table 4 Multifractality for PM25 concentration

City Multifractality Before law After law

Seoulh(2) 12270 12085ΔH(q) 17329 11410Δα 22414 13953

Busanh(2) 12905 11041ΔH(q) 13539 09565Δα 17463 12708

Daeguh(2) 11975 10878ΔH(q) 22910 19516Δα 29610 25043

12 Complexity

05

1

15

2

25

3

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

35

h(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

0

05

1

15

2

h(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

h(q)

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 12 Hurst exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 13

ndash30

ndash20

ndash10

0

10

τ(q)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash40

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 13 Renyi exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

14 Complexity

ndash3

ndash2

ndash1

0

1

f(α)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

1 2 3 40α

05 1 15 2 25 30α

05 1 15 2 25 30α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

05 1 15 2 25 30α

05 1 15 20α

05 1 15 2 250α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 14 Holder exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 15

the implementation of the law for all three cities implyingthat the multifractality of the time series becomes morestable with the measures taken by the government and thatthe efficiency of the ldquohaze special lawrdquo in three cities isdemonstrable this conclusion can be referred to [16 39] inwhich it was pointed out that the decrease in multifractalitymeans the efficiency is improved when some measures aretaken Moreover we also notice that the efficiency of theldquohaze special lawrdquo in Seoul is most obvious when comparedwith Busan and in last place is Daegu Interestingly theorder is the same as the scale and prosperity of the citiesWe can therefore conclude that the law has a higher effecton a larger city

43 Sources of Multifractal Features e two universallyacknowledged sources of multifractality are long-rangecorrelations and a fat-tailed distribution [7 40] Now weinvestigate the major source of multifractality of the PM25concentration series To check the contribution of long-range correlations and fat-tailed distribution quantitativelywe shuffle and phase-randomize the time series

Subsequently we calculate the Hurst exponent h(q) theRenyi exponent τ(q) and the Holder exponent Δα of theoriginal shuffled and phase-randomized time series of thethree cities As shown in Figures 12ndash14 all the time series arestrongly multifractal From Table 5 all h(2) of the originalseries are larger than those of the shuffled and phase-ran-domized time series which indicates that the persistence hasbeen moved after shuffling and phase-randomizationMoreover the original series of all three cities has thegreatest spectrum width Δα while the multifractality de-creased after shuffling and phase-randomizing the seriese findings can also be confirmed in Figures 12ndash14

To examine the main source of multifractality the firstand third rows of Figure 14 show that the multifractalspectra of phase-randomized series are the narrowest ones

which means that the main source of multifractality of theseries before and after the law in Seoul and Daegu are the fat-tailed distributions e second row in Figure 14 shows thatthe multifractal spectrum of the shuffled series is the nar-rowest which implies that the long-range correlation resultsin multifractality of the series before and after imple-mentation of the law in Busan

5 Conclusions

In this study we examined the multifractal characteristics ofPM25 and NO2 time series of all seasons We determinedthat multifractality existed in each season and the multi-fractal property of PM25 was stronger than that of NO2 ineach season We also validated the effectiveness of the ldquohazespecial lawrdquo which was implemented to improve the gov-ernance of air pollution in South Korea We checkedwhether there is a difference caused by the law for these citiesand calculated confidence intervals of Hurst exponent h(2)e results showed that the confidence intervals do notoverlap for Seoul while the confidence intervals overlap a lotfor Busan and Daegu indicating that there is a differencecaused by the law for Seoul however the same conclusion isinconclusive for Busan and Daegu By comparing the Hurstexponent Renyi exponent and Holder exponent the timeseries of the PM25 concentration before the implementationof the law was found to have a higher multifractal degreewhich decreased after the enactment of the law is phe-nomenon reflected the fact that the law has played a role inimproving the efficiency of air pollution control in SouthKorea We also concluded that the effect of the law will bemore significant in a larger city To explore the major causesof multifractality we shuffled and phase-randomized theoriginal series of PM25 By analyzing the width of themultifractal spectrum Δα the results showed that the fat-tailed distribution contributed to the multifractality of bothtimes series before and after the implementation of the ldquohazespecial lawrdquo in Seoul and Daegu whereas long-range cor-relations resulted in the multifractality of the series beforeand after the implementation of the law in Busan Weconcluded that the implementation of the law was verysuccessful and effective for the improvement of PM25 levelserefore we believe that the ldquohaze special lawrdquo is a possiblecause for the change in API in South Korea

Data Availability

e data will be available upon request with the corre-sponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e first author (Jian Wang) was supported by the ChinaScholarship Council (201808260026) J S Kim expressesgratitude for the support from the BK21 PLUS programe

Table 5 Multifractality of PM25 time series

h(2) Δα LRC FTD

Seoul

BeforeOriginal 12270 22414

No YesShuffled 07232 17946Phase-randomized 06965 09574

AfterOriginal 12085 13953

No YesShuffled 03177 10978Phase-randomized 03049 08569

Busan

BeforeOriginal 12905 17463

Yes NoShuffled 03982 09999Phase-randomized 03993 13512

AfterOriginal 11041 12708

Yes NoShuffled 03678 07175Phase-randomized 03794 08673

Daegu

BeforeOriginal 11975 29610

No YesShuffled 05901 22093Phase-randomized 05868 20587

AfterOriginal 10878 25043

No YesShuffled 06756 25600Phase-randomized 06407 22981

LRC and FTD represent the long-range correlations and fat-tailed distri-bution respectively

16 Complexity

corresponding author Wei Shao thanks Professor DaraeJeong for helpful discussion

References

[1] G Shaddick and J Wakefield ldquoModelling daily multivariatepollutant data at multiple sitesrdquo Journal of the Royal StatisticalSociety Series C (Applied Statistics) vol 51 no 3 pp 351ndash3722002

[2] M OrsquoDwyer Air Quality in Ireland 2010 Key Indicators ofAmbient Air Quality EPA Wexford Ireland 2011

[3] A Challoner and L Gill ldquoIndooroutdoor air pollution re-lationships in ten commercial buildings PM25 and NO2rdquoBuilding and Environment vol 80 pp 159ndash173 2014

[4] M Hazenkamp-von Arx T Gotschi U Ackermann-Liebrichet al ldquoPM25 and NO2 assessment in 21 European studycentres of ECRHS II annual means and seasonal differencesrdquoAtmospheric Environment vol 38 no 13 pp 1943ndash19532004

[5] H K Lai L Bayer-Oglesby R Colvile et al ldquoDeterminants ofindoor air concentrations of PM25 black smoke and NO2 insix European cities (EXPOLIS study)rdquo Atmospheric Envi-ronment vol 40 no 7 pp 1299ndash1313 2006

[6] K de Hoogh J Gulliver A V Donkelaar et al ldquoDevelopmentof West-European PM25 and NO2 land use regression modelsincorporating satellite-derived and chemical transport mod-elling datardquo Environmental Research vol 151 pp 1ndash10 2016

[7] S Drozdz J Kwapien P Oswiecimka and R Rak ldquoQuan-titative features of multifractal subtleties in time seriesrdquo EPL(Europhysics Letters) vol 88 no 6 p 60003 2010

[8] R J Buonocore T Aste and T Di Matteo ldquoMeasuringmultiscaling in financial time-seriesrdquo Chaos Solitons ampFractals vol 88 pp 38ndash47 2016

[9] P Oswiecimka S Drozdz J Kwapien and A Z GorskildquoFractals log-periodicity and financial crashesrdquo Acta PhysicaPolonica A vol 117 no 4 pp 637ndash639 2010

[10] B B Mandelbrot and N Goldenfeld ldquoFractals and scaling infinance discontinuity concentration riskrdquo Physics Todayvol 51 no 8 pp 59-60 1998

[11] S Drozdz R Kowalski P Oswiecimka R Rak andR Gebarowski ldquoDynamical variety of shapes in financialmultifractalityrdquo Complexity vol 2018 Article ID 701572113 pages 2018

[12] S Lahmiri ldquoMultifractal analysis of Moroccan family businessstock returnsrdquo Physica A Statistical Mechanics and Its Ap-plications vol 486 pp 183ndash191 2017

[13] X Zhao P Shang and Q Jin ldquoMultifractal detrended cross-correlation analysis of Chinese stock markets based on timedelayrdquo Fractals vol 19 no 03 pp 329ndash338 2011

[14] W Shao and J Wang ldquoDoes the ldquoice-breakingrdquo of South andNorth Korea affect the South Korean financial marketrdquoChaos Solitons amp Fractals vol 132 Article ID 109564 2020

[15] D-H Wang X-W Yu and Y-Y Suo ldquoStatistical propertiesof the yuan exchange rate indexrdquo Physica A Statistical Me-chanics and Its Applications vol 391 no 12 pp 3503ndash35122012

[16] Y Ning Y Wang and C-W Su ldquoHow did Chinarsquos foreignexchange reform affect the efficiency of foreign exchangemarketrdquo Physica A Statistical Mechanics and Its Applicationsvol 483 pp 219ndash226 2017

[17] S Lahmiri and S Bekiros ldquoChaos randomness and multi-fractality in bitcoin marketrdquo Chaos Solitons amp Fractalsvol 106 pp 28ndash34 2018

[18] P Manimaran and A C Narayana ldquoMultifractal detrendedcross-correlation analysis on air pollutants of University ofHyderabad Campus Indiardquo Physica A Statistical Mechanicsand Its Applications vol 502 pp 228ndash235 2018

[19] L Zhao W Li C Yang J Han Z Su and Y Zou ldquoMul-tifractality and network analysis of phase transitionrdquo PLoSOne vol 12 Article ID e0170467 2017

[20] N Kalamaras K Philippopoulos D Deligiorgi C G Tzanisand G Karvounis ldquoMultifractal scaling properties of daily airtemperature time seriesrdquo Chaos Solitons amp Fractals vol 98pp 38ndash43 2017

[21] J Li C Chen Q Yao et al ldquoe effect of circadian rhythm onthe correlation and multifractality of heart rate signals duringexerciserdquo Physica A Statistical Mechanics and Its Applica-tions vol 509 pp 1207ndash1213 2018

[22] P Castiglioni D Lazzeroni P Coruzzi and A FainildquoMultifractal-multiscale analysis of cardiovascular signals aDFA-based characterization of blood pressure and heart-ratecomplexity by genderrdquo Complexity vol 2018 Article ID4801924 14 pages 2018

[23] W Shi R-B Zou F Wang and L Su ldquoA new image seg-mentation method based on multifractal detrended movingaverage analysisrdquo Physica A Statistical Mechanics and ItsApplications vol 432 pp 197ndash205 2015

[24] F Wang Q Fan and H E Stanley ldquoMultiscale multifractaldetrended-fluctuation analysis of two-dimensional surfacesrdquoPhysical Review E vol 93 Article ID 042213 2016

[25] X Li M Zheng J Pu et al ldquoIdentification of abnormallyexpressed lncrnas induced by PM25 in human bronchialepithelial cellsrdquo Bioscience Reports vol 38 no 5 Article IDBSR20171577 2018

[26] K Shi C Liu and Y Huang ldquoMultifractal processes and self-organized criticality of PM25 during a typical haze period inChengdu Chinardquo Aerosol and Air Quality Research vol 15no 3 pp 926ndash934 2015

[27] C Zhang Z Ni and L Ni ldquoMultifractal detrended cross-correlation analysis between PM25 and meteorological fac-torsrdquo Physica A Statistical Mechanics and Its Applicationsvol 438 pp 114ndash123 2015

[28] C Zhang X Wang S Chen L Zou X Zhang and C TangldquoA study on daily PM25 concentrations in Hong Kong usingthe EMD-based MFDFA methodrdquo Physica A StatisticalMechanics and Its Applications vol 530 p 121182 2019

[29] H-D He ldquoMultifractal analysis of interactive patterns be-tween meteorological factors and pollutants in urban andrural areasrdquo Atmospheric Environment vol 149 pp 47ndash542017

[30] J W Kantelhardt S A Zschiegner E Koscielny-BundeS Havlin A Bunde and H E Stanley ldquoMultifractaldetrended fluctuation analysis of nonstationary time seriesrdquoPhysica A Statistical Mechanics and Its Applications vol 316no 1ndash4 pp 87ndash114 2002

[31] L Zunino B M Tabak A Figliola D G PerezM Garavaglia and O A Rosso ldquoA multifractal approach forstock market inefficiencyrdquo Physica A Statistical Mechanicsand Its Applications vol 387 no 26 pp 6558ndash6566 2008

[32] L Zunino A Figliola B M Tabak D G PerezM Garavaglia and O A Rosso ldquoMultifractal structure inLatin-American market indicesrdquo Chaos Solitons amp Fractalsvol 41 no 5 pp 2331ndash2340 2009

[33] B Lashermes P Abry and P Chainais ldquoNew insights into theestimation of scaling exponentsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 2no 4 pp 497ndash523 2004

Complexity 17

[34] E A F Ihlen ldquoIntroduction to multifractal detrended fluc-tuation analysis in Matlabrdquo Frontiers in Physiology vol 3p 141 2012

[35] L Jiang X Zhao N Li F Li and Z Guo ldquoDifferent mul-tifractal scaling of the 0 cm average ground surface tem-perature of four representative weather stations over ChinardquoAdvances inMeteorology vol 2013 Article ID 341934 8 pages2013

[36] P C Ivanov L A N Amaral A L Goldberger et alldquoMultifractality in human heartbeat dynamicsrdquo Naturevol 399 no 6735 pp 461ndash465 1999

[37] S Drozdz and P Oswiecimka ldquoDetecting and interpretingdistortions in hierarchical organization of complex time se-riesrdquo Physical Review E vol 91 Article ID 030902 2015

[38] C Ibarra-Valdez J Alvarez and J Alvarez-Ramirez ldquoRan-domness confidence bands of fractal scaling exponents forfinancial price returnsrdquo Chaos Solitons amp Fractals vol 83pp 119ndash124 2016

[39] D Stosic D Stosic T Stosic and H E Stanley ldquoMultifractalanalysis of managed and independent float exchange ratesrdquoPhysica A Statistical Mechanics and its Applications vol 428pp 13ndash18 2015

[40] K Matia Y Ashkenazy and H E Stanley ldquoMultifractalproperties of price fluctuations of stocks and commoditiesrdquoEurophysics Letters (EPL) vol 61 no 3 pp 422ndash428 2003

18 Complexity

2018516 2018814 201811122018215Date

2018516 2018814 201811122018215Date

2018516 2018814 201811122018215Date

0

50

100

150

200PM

25

0

50

100

150

200

PM2

5

0

50

100

150

200

PM2

5

(a)

2019516 2019814 201911122019215Date

2019516 2019814 201911122019215Date

2019516 2019814 201911122019215Date

0

50

100

150

200

PM2

5

0

50

100

150

200

PM2

5

0

50

100

150

200

PM2

5

(b)

Figure 6 Time series of PM25 (a) before and (b) after implementation of the law From top to bottom are Seoul Busan and Daegu

Complexity 7

055

06

065

07

075

08

085

09

095h(

2)

100 150 200 250 30050Time series length

03

035

04

045

05

h(2)

05

055

06

065

07

075

08

h(2)

100 150 200 250 30050Time series length

100 150 200 250 30050Time series length

(a)

100 150 200 250 30050Time series length

05

055

06

065

07

075

08

085

h(2)

025

03

035

04

045

05

h(2)

03

035

04

045

h(2)

100 150 200 250 30050Time series length

100 150 200 250 30050Time series length

(b)

Figure 7 Confidence intervals of h(2) versus the length of PM25 time series (a) before and (b) after implementation of the law withconfidence level by 950 From top to bottom are Seoul Busan and Daegu

8 Complexity

q = 10

q = 10

q = 10

q = ndash10

q = ndash10

q = ndash10

ndash1

0

1

2

3

4

ln(F

q(s)

)

ndash1

0

1

2

3

4

ln(F

q(s)

)

ndash2

ndash1

0

1

2

3

4

ln(F

q(s)

)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

(a)

q = 10

q = 10

q = 10

q = ndash10

q = ndash10

q = ndash10

0

1

2

3

4

ln(F

q(s)

)

ndash1

0

1

2

3

4

ln(F

q(s)

)

ndash1

0

1

2

3

4

ln(F

q(s)

)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

(b)

Figure 8 Log-log plots of the fluctuation function Fq(s) for PM25 (a) before and (b) after implementation of the law From top to bottomare Seoul Busan and Daegu

Complexity 9

ndash5 0 5 10ndash10q

05

1

15

2

25

3

h(q)

Before lawAfter law

(a)

ndash5 0 5 10ndash10q

05

1

15

2

25

h(q)

Before lawAfter law

(b)

ndash5 0 5 10ndash10q

05

1

15

2

25

3

35

h(q)

Before lawAfter law

(c)

Figure 9 Hurst exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

Before lawAfter law

(a)

ndash5 0 5 10ndash10q

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

Before lawAfter law

(b)

Figure 10 Continued

10 Complexity

ndash40

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

Before lawAfter law

(c)

Figure 10 Renyi exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu

ndash15

ndash1

ndash05

0

05

1

f(α)

1 15 2 25 305α

Before lawAfter law

(a)

1 15 2 25 305α

ndash15

ndash1

ndash05

0

05

1f(α

)

Before lawAfter law

(b)

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

f(α)

1 2 3 40α

Before lawAfter law

(c)

Figure 11 Holder exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu e triangle and circle symbols represent beforeand after the law respectively

Complexity 11

Table 3 the conclusion we described above about the Hurstexponent and the Renyi exponent is discussed Moreover asshown in Figure 5 the multifractal spectra are stronglyasymmetrical and as an important indicator of fractal or-ganization the asymmetry coefficient which is used toquantify the asymmetry of the multifractal spectrum hasbeen estimated in [37]

Based on Table 3 the multifractal features can be clearlyconfirmed for all time series and it can be seen that exceptfor spring the Hurst exponents and multifractal spectrumwidths of PM25 are obviously larger than those of NO2 andthe multifractal degrees of PM25 are stronger than those ofNO2

42 Implications of the ldquoHaze Special Lawrdquo In this sectionwe focus on the enactment of the ldquohaze special lawrdquo andstudy whether the law is effective in the governance of at-mospheric pollution We select daily data from the sameperiod described in Section 3 As we studied above themultifractal degree of PM25 is stronger than that of NO2us in this section we focus on the comparison of PM25concentrations We show the PM25 time series before andafter the law was implemented in Seoul Busan and Daegu inFigure 6

As the fluctuation statistics are of asymptotic feature asufficiently long-length sequence guarantees the accurateestimation of the Hurst exponent and for a small-lengthsequence the asymptotic value in the slope of ln(Fq(s))

versus ln(s) will be lacked [38] As the total number ofobservations is selected to be 305 to eliminate the doubt ofwhether the selected time series can give representativeresults for a multifractal analysis in the investigation wedealt with this concern by computing confidence intervals ofHurst exponent h(2) to check whether there is a differencecaused by the law for all the three cities e number ofsamples increased from 50 to 300 and every additionalinterval is 10 e confidence level is selected as 950 Foreach set of sample lengths we generate the shuffled timeseries 1000 times and then the confidence interval is plottedin Figure 7

As shown in Figure 7 and as expected the domain sizedecreases with an increase in the length of the time seriesHowever when the length of the time series is larger than 200the decrease in the domain size of the confidence intervals israther slow and of a distinct asymptotic behavior Besides we

see that the confidence intervals do not overlap for Seoulwhile for Busan ([034 039] before and [032 037] after) andDaegu ([055 064] before and [058 069] after) the confi-dence intervals overlap a lot erefore we can conclude thatthere is a difference caused by the law for Seoul however thesame conclusion is inconclusive for Busan and Daegu

Now we first depict the log-log plots of the fluctuationfunction Fq(s) versus q and s for the PM25 time series Asillustrated in Figure 8 for all series the fluctuation valueincreases linearly with s indicating that a power-law be-havior and long-range correlations exist in each series paire decreasing gradient shows that all time series havemultifractal characteristics

e generalized Hurst exponent h(q) Renyi exponentτ(q) and Holder exponent α of the time series before andafter the enactment of the ldquohaze special lawrdquo for all the threecities are shown in Figures 9ndash11 respectively

In Figure 9 it is shown that the generalized Hurst ex-ponents h(q) of PM25 of all three cities decrease with variedq thereby showing the multifractal behaviors of these twoseries We present h(2) for these series in Table 4 All Hurstexponents are confirmed to be larger than 05 which in-dicates that all the time series exhibit positive persistenceBesides we observe that h(2) before the implementation ofthe law is larger than after the law implying that the fluc-tuations before the law have more significant persistenceSubsequently we calculate the range of h(q) for each timeseries and list them in Table 4 We find that for all threecities ΔH(q) after the law is larger which indicates that themultifractal characteristics are stronger and that the mul-tifractalities of the time series before the law are more stable

e Renyi exponents τ(q) plotted in Figure 10 arenonlinear along q which provides further evidence of theexistence of multifractality We notice that the curvature ofthe Renyi exponents before the law are higher which impliesthat the time series before the implementation of the lawhave stronger multifractal characteristics All of these areconsistent with the results calculated from the generalizedHurst exponents

At last the multifractal spectrums of these time series areexamined We depict the curves between α and f(α) inFigure 11 We see that the multifractal spectra are not shownas points which indicates all the time series have multi-fractality en Δα are calculated and presented in Table 4From Table 4 it can be seen that ΔH(q) and Δα of the timeseries after the enactment of law are lower than those before

Table 4 Multifractality for PM25 concentration

City Multifractality Before law After law

Seoulh(2) 12270 12085ΔH(q) 17329 11410Δα 22414 13953

Busanh(2) 12905 11041ΔH(q) 13539 09565Δα 17463 12708

Daeguh(2) 11975 10878ΔH(q) 22910 19516Δα 29610 25043

12 Complexity

05

1

15

2

25

3

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

35

h(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

0

05

1

15

2

h(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

h(q)

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 12 Hurst exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 13

ndash30

ndash20

ndash10

0

10

τ(q)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash40

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 13 Renyi exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

14 Complexity

ndash3

ndash2

ndash1

0

1

f(α)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

1 2 3 40α

05 1 15 2 25 30α

05 1 15 2 25 30α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

05 1 15 2 25 30α

05 1 15 20α

05 1 15 2 250α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 14 Holder exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 15

the implementation of the law for all three cities implyingthat the multifractality of the time series becomes morestable with the measures taken by the government and thatthe efficiency of the ldquohaze special lawrdquo in three cities isdemonstrable this conclusion can be referred to [16 39] inwhich it was pointed out that the decrease in multifractalitymeans the efficiency is improved when some measures aretaken Moreover we also notice that the efficiency of theldquohaze special lawrdquo in Seoul is most obvious when comparedwith Busan and in last place is Daegu Interestingly theorder is the same as the scale and prosperity of the citiesWe can therefore conclude that the law has a higher effecton a larger city

43 Sources of Multifractal Features e two universallyacknowledged sources of multifractality are long-rangecorrelations and a fat-tailed distribution [7 40] Now weinvestigate the major source of multifractality of the PM25concentration series To check the contribution of long-range correlations and fat-tailed distribution quantitativelywe shuffle and phase-randomize the time series

Subsequently we calculate the Hurst exponent h(q) theRenyi exponent τ(q) and the Holder exponent Δα of theoriginal shuffled and phase-randomized time series of thethree cities As shown in Figures 12ndash14 all the time series arestrongly multifractal From Table 5 all h(2) of the originalseries are larger than those of the shuffled and phase-ran-domized time series which indicates that the persistence hasbeen moved after shuffling and phase-randomizationMoreover the original series of all three cities has thegreatest spectrum width Δα while the multifractality de-creased after shuffling and phase-randomizing the seriese findings can also be confirmed in Figures 12ndash14

To examine the main source of multifractality the firstand third rows of Figure 14 show that the multifractalspectra of phase-randomized series are the narrowest ones

which means that the main source of multifractality of theseries before and after the law in Seoul and Daegu are the fat-tailed distributions e second row in Figure 14 shows thatthe multifractal spectrum of the shuffled series is the nar-rowest which implies that the long-range correlation resultsin multifractality of the series before and after imple-mentation of the law in Busan

5 Conclusions

In this study we examined the multifractal characteristics ofPM25 and NO2 time series of all seasons We determinedthat multifractality existed in each season and the multi-fractal property of PM25 was stronger than that of NO2 ineach season We also validated the effectiveness of the ldquohazespecial lawrdquo which was implemented to improve the gov-ernance of air pollution in South Korea We checkedwhether there is a difference caused by the law for these citiesand calculated confidence intervals of Hurst exponent h(2)e results showed that the confidence intervals do notoverlap for Seoul while the confidence intervals overlap a lotfor Busan and Daegu indicating that there is a differencecaused by the law for Seoul however the same conclusion isinconclusive for Busan and Daegu By comparing the Hurstexponent Renyi exponent and Holder exponent the timeseries of the PM25 concentration before the implementationof the law was found to have a higher multifractal degreewhich decreased after the enactment of the law is phe-nomenon reflected the fact that the law has played a role inimproving the efficiency of air pollution control in SouthKorea We also concluded that the effect of the law will bemore significant in a larger city To explore the major causesof multifractality we shuffled and phase-randomized theoriginal series of PM25 By analyzing the width of themultifractal spectrum Δα the results showed that the fat-tailed distribution contributed to the multifractality of bothtimes series before and after the implementation of the ldquohazespecial lawrdquo in Seoul and Daegu whereas long-range cor-relations resulted in the multifractality of the series beforeand after the implementation of the law in Busan Weconcluded that the implementation of the law was verysuccessful and effective for the improvement of PM25 levelserefore we believe that the ldquohaze special lawrdquo is a possiblecause for the change in API in South Korea

Data Availability

e data will be available upon request with the corre-sponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e first author (Jian Wang) was supported by the ChinaScholarship Council (201808260026) J S Kim expressesgratitude for the support from the BK21 PLUS programe

Table 5 Multifractality of PM25 time series

h(2) Δα LRC FTD

Seoul

BeforeOriginal 12270 22414

No YesShuffled 07232 17946Phase-randomized 06965 09574

AfterOriginal 12085 13953

No YesShuffled 03177 10978Phase-randomized 03049 08569

Busan

BeforeOriginal 12905 17463

Yes NoShuffled 03982 09999Phase-randomized 03993 13512

AfterOriginal 11041 12708

Yes NoShuffled 03678 07175Phase-randomized 03794 08673

Daegu

BeforeOriginal 11975 29610

No YesShuffled 05901 22093Phase-randomized 05868 20587

AfterOriginal 10878 25043

No YesShuffled 06756 25600Phase-randomized 06407 22981

LRC and FTD represent the long-range correlations and fat-tailed distri-bution respectively

16 Complexity

corresponding author Wei Shao thanks Professor DaraeJeong for helpful discussion

References

[1] G Shaddick and J Wakefield ldquoModelling daily multivariatepollutant data at multiple sitesrdquo Journal of the Royal StatisticalSociety Series C (Applied Statistics) vol 51 no 3 pp 351ndash3722002

[2] M OrsquoDwyer Air Quality in Ireland 2010 Key Indicators ofAmbient Air Quality EPA Wexford Ireland 2011

[3] A Challoner and L Gill ldquoIndooroutdoor air pollution re-lationships in ten commercial buildings PM25 and NO2rdquoBuilding and Environment vol 80 pp 159ndash173 2014

[4] M Hazenkamp-von Arx T Gotschi U Ackermann-Liebrichet al ldquoPM25 and NO2 assessment in 21 European studycentres of ECRHS II annual means and seasonal differencesrdquoAtmospheric Environment vol 38 no 13 pp 1943ndash19532004

[5] H K Lai L Bayer-Oglesby R Colvile et al ldquoDeterminants ofindoor air concentrations of PM25 black smoke and NO2 insix European cities (EXPOLIS study)rdquo Atmospheric Envi-ronment vol 40 no 7 pp 1299ndash1313 2006

[6] K de Hoogh J Gulliver A V Donkelaar et al ldquoDevelopmentof West-European PM25 and NO2 land use regression modelsincorporating satellite-derived and chemical transport mod-elling datardquo Environmental Research vol 151 pp 1ndash10 2016

[7] S Drozdz J Kwapien P Oswiecimka and R Rak ldquoQuan-titative features of multifractal subtleties in time seriesrdquo EPL(Europhysics Letters) vol 88 no 6 p 60003 2010

[8] R J Buonocore T Aste and T Di Matteo ldquoMeasuringmultiscaling in financial time-seriesrdquo Chaos Solitons ampFractals vol 88 pp 38ndash47 2016

[9] P Oswiecimka S Drozdz J Kwapien and A Z GorskildquoFractals log-periodicity and financial crashesrdquo Acta PhysicaPolonica A vol 117 no 4 pp 637ndash639 2010

[10] B B Mandelbrot and N Goldenfeld ldquoFractals and scaling infinance discontinuity concentration riskrdquo Physics Todayvol 51 no 8 pp 59-60 1998

[11] S Drozdz R Kowalski P Oswiecimka R Rak andR Gebarowski ldquoDynamical variety of shapes in financialmultifractalityrdquo Complexity vol 2018 Article ID 701572113 pages 2018

[12] S Lahmiri ldquoMultifractal analysis of Moroccan family businessstock returnsrdquo Physica A Statistical Mechanics and Its Ap-plications vol 486 pp 183ndash191 2017

[13] X Zhao P Shang and Q Jin ldquoMultifractal detrended cross-correlation analysis of Chinese stock markets based on timedelayrdquo Fractals vol 19 no 03 pp 329ndash338 2011

[14] W Shao and J Wang ldquoDoes the ldquoice-breakingrdquo of South andNorth Korea affect the South Korean financial marketrdquoChaos Solitons amp Fractals vol 132 Article ID 109564 2020

[15] D-H Wang X-W Yu and Y-Y Suo ldquoStatistical propertiesof the yuan exchange rate indexrdquo Physica A Statistical Me-chanics and Its Applications vol 391 no 12 pp 3503ndash35122012

[16] Y Ning Y Wang and C-W Su ldquoHow did Chinarsquos foreignexchange reform affect the efficiency of foreign exchangemarketrdquo Physica A Statistical Mechanics and Its Applicationsvol 483 pp 219ndash226 2017

[17] S Lahmiri and S Bekiros ldquoChaos randomness and multi-fractality in bitcoin marketrdquo Chaos Solitons amp Fractalsvol 106 pp 28ndash34 2018

[18] P Manimaran and A C Narayana ldquoMultifractal detrendedcross-correlation analysis on air pollutants of University ofHyderabad Campus Indiardquo Physica A Statistical Mechanicsand Its Applications vol 502 pp 228ndash235 2018

[19] L Zhao W Li C Yang J Han Z Su and Y Zou ldquoMul-tifractality and network analysis of phase transitionrdquo PLoSOne vol 12 Article ID e0170467 2017

[20] N Kalamaras K Philippopoulos D Deligiorgi C G Tzanisand G Karvounis ldquoMultifractal scaling properties of daily airtemperature time seriesrdquo Chaos Solitons amp Fractals vol 98pp 38ndash43 2017

[21] J Li C Chen Q Yao et al ldquoe effect of circadian rhythm onthe correlation and multifractality of heart rate signals duringexerciserdquo Physica A Statistical Mechanics and Its Applica-tions vol 509 pp 1207ndash1213 2018

[22] P Castiglioni D Lazzeroni P Coruzzi and A FainildquoMultifractal-multiscale analysis of cardiovascular signals aDFA-based characterization of blood pressure and heart-ratecomplexity by genderrdquo Complexity vol 2018 Article ID4801924 14 pages 2018

[23] W Shi R-B Zou F Wang and L Su ldquoA new image seg-mentation method based on multifractal detrended movingaverage analysisrdquo Physica A Statistical Mechanics and ItsApplications vol 432 pp 197ndash205 2015

[24] F Wang Q Fan and H E Stanley ldquoMultiscale multifractaldetrended-fluctuation analysis of two-dimensional surfacesrdquoPhysical Review E vol 93 Article ID 042213 2016

[25] X Li M Zheng J Pu et al ldquoIdentification of abnormallyexpressed lncrnas induced by PM25 in human bronchialepithelial cellsrdquo Bioscience Reports vol 38 no 5 Article IDBSR20171577 2018

[26] K Shi C Liu and Y Huang ldquoMultifractal processes and self-organized criticality of PM25 during a typical haze period inChengdu Chinardquo Aerosol and Air Quality Research vol 15no 3 pp 926ndash934 2015

[27] C Zhang Z Ni and L Ni ldquoMultifractal detrended cross-correlation analysis between PM25 and meteorological fac-torsrdquo Physica A Statistical Mechanics and Its Applicationsvol 438 pp 114ndash123 2015

[28] C Zhang X Wang S Chen L Zou X Zhang and C TangldquoA study on daily PM25 concentrations in Hong Kong usingthe EMD-based MFDFA methodrdquo Physica A StatisticalMechanics and Its Applications vol 530 p 121182 2019

[29] H-D He ldquoMultifractal analysis of interactive patterns be-tween meteorological factors and pollutants in urban andrural areasrdquo Atmospheric Environment vol 149 pp 47ndash542017

[30] J W Kantelhardt S A Zschiegner E Koscielny-BundeS Havlin A Bunde and H E Stanley ldquoMultifractaldetrended fluctuation analysis of nonstationary time seriesrdquoPhysica A Statistical Mechanics and Its Applications vol 316no 1ndash4 pp 87ndash114 2002

[31] L Zunino B M Tabak A Figliola D G PerezM Garavaglia and O A Rosso ldquoA multifractal approach forstock market inefficiencyrdquo Physica A Statistical Mechanicsand Its Applications vol 387 no 26 pp 6558ndash6566 2008

[32] L Zunino A Figliola B M Tabak D G PerezM Garavaglia and O A Rosso ldquoMultifractal structure inLatin-American market indicesrdquo Chaos Solitons amp Fractalsvol 41 no 5 pp 2331ndash2340 2009

[33] B Lashermes P Abry and P Chainais ldquoNew insights into theestimation of scaling exponentsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 2no 4 pp 497ndash523 2004

Complexity 17

[34] E A F Ihlen ldquoIntroduction to multifractal detrended fluc-tuation analysis in Matlabrdquo Frontiers in Physiology vol 3p 141 2012

[35] L Jiang X Zhao N Li F Li and Z Guo ldquoDifferent mul-tifractal scaling of the 0 cm average ground surface tem-perature of four representative weather stations over ChinardquoAdvances inMeteorology vol 2013 Article ID 341934 8 pages2013

[36] P C Ivanov L A N Amaral A L Goldberger et alldquoMultifractality in human heartbeat dynamicsrdquo Naturevol 399 no 6735 pp 461ndash465 1999

[37] S Drozdz and P Oswiecimka ldquoDetecting and interpretingdistortions in hierarchical organization of complex time se-riesrdquo Physical Review E vol 91 Article ID 030902 2015

[38] C Ibarra-Valdez J Alvarez and J Alvarez-Ramirez ldquoRan-domness confidence bands of fractal scaling exponents forfinancial price returnsrdquo Chaos Solitons amp Fractals vol 83pp 119ndash124 2016

[39] D Stosic D Stosic T Stosic and H E Stanley ldquoMultifractalanalysis of managed and independent float exchange ratesrdquoPhysica A Statistical Mechanics and its Applications vol 428pp 13ndash18 2015

[40] K Matia Y Ashkenazy and H E Stanley ldquoMultifractalproperties of price fluctuations of stocks and commoditiesrdquoEurophysics Letters (EPL) vol 61 no 3 pp 422ndash428 2003

18 Complexity

055

06

065

07

075

08

085

09

095h(

2)

100 150 200 250 30050Time series length

03

035

04

045

05

h(2)

05

055

06

065

07

075

08

h(2)

100 150 200 250 30050Time series length

100 150 200 250 30050Time series length

(a)

100 150 200 250 30050Time series length

05

055

06

065

07

075

08

085

h(2)

025

03

035

04

045

05

h(2)

03

035

04

045

h(2)

100 150 200 250 30050Time series length

100 150 200 250 30050Time series length

(b)

Figure 7 Confidence intervals of h(2) versus the length of PM25 time series (a) before and (b) after implementation of the law withconfidence level by 950 From top to bottom are Seoul Busan and Daegu

8 Complexity

q = 10

q = 10

q = 10

q = ndash10

q = ndash10

q = ndash10

ndash1

0

1

2

3

4

ln(F

q(s)

)

ndash1

0

1

2

3

4

ln(F

q(s)

)

ndash2

ndash1

0

1

2

3

4

ln(F

q(s)

)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

(a)

q = 10

q = 10

q = 10

q = ndash10

q = ndash10

q = ndash10

0

1

2

3

4

ln(F

q(s)

)

ndash1

0

1

2

3

4

ln(F

q(s)

)

ndash1

0

1

2

3

4

ln(F

q(s)

)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

(b)

Figure 8 Log-log plots of the fluctuation function Fq(s) for PM25 (a) before and (b) after implementation of the law From top to bottomare Seoul Busan and Daegu

Complexity 9

ndash5 0 5 10ndash10q

05

1

15

2

25

3

h(q)

Before lawAfter law

(a)

ndash5 0 5 10ndash10q

05

1

15

2

25

h(q)

Before lawAfter law

(b)

ndash5 0 5 10ndash10q

05

1

15

2

25

3

35

h(q)

Before lawAfter law

(c)

Figure 9 Hurst exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

Before lawAfter law

(a)

ndash5 0 5 10ndash10q

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

Before lawAfter law

(b)

Figure 10 Continued

10 Complexity

ndash40

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

Before lawAfter law

(c)

Figure 10 Renyi exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu

ndash15

ndash1

ndash05

0

05

1

f(α)

1 15 2 25 305α

Before lawAfter law

(a)

1 15 2 25 305α

ndash15

ndash1

ndash05

0

05

1f(α

)

Before lawAfter law

(b)

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

f(α)

1 2 3 40α

Before lawAfter law

(c)

Figure 11 Holder exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu e triangle and circle symbols represent beforeand after the law respectively

Complexity 11

Table 3 the conclusion we described above about the Hurstexponent and the Renyi exponent is discussed Moreover asshown in Figure 5 the multifractal spectra are stronglyasymmetrical and as an important indicator of fractal or-ganization the asymmetry coefficient which is used toquantify the asymmetry of the multifractal spectrum hasbeen estimated in [37]

Based on Table 3 the multifractal features can be clearlyconfirmed for all time series and it can be seen that exceptfor spring the Hurst exponents and multifractal spectrumwidths of PM25 are obviously larger than those of NO2 andthe multifractal degrees of PM25 are stronger than those ofNO2

42 Implications of the ldquoHaze Special Lawrdquo In this sectionwe focus on the enactment of the ldquohaze special lawrdquo andstudy whether the law is effective in the governance of at-mospheric pollution We select daily data from the sameperiod described in Section 3 As we studied above themultifractal degree of PM25 is stronger than that of NO2us in this section we focus on the comparison of PM25concentrations We show the PM25 time series before andafter the law was implemented in Seoul Busan and Daegu inFigure 6

As the fluctuation statistics are of asymptotic feature asufficiently long-length sequence guarantees the accurateestimation of the Hurst exponent and for a small-lengthsequence the asymptotic value in the slope of ln(Fq(s))

versus ln(s) will be lacked [38] As the total number ofobservations is selected to be 305 to eliminate the doubt ofwhether the selected time series can give representativeresults for a multifractal analysis in the investigation wedealt with this concern by computing confidence intervals ofHurst exponent h(2) to check whether there is a differencecaused by the law for all the three cities e number ofsamples increased from 50 to 300 and every additionalinterval is 10 e confidence level is selected as 950 Foreach set of sample lengths we generate the shuffled timeseries 1000 times and then the confidence interval is plottedin Figure 7

As shown in Figure 7 and as expected the domain sizedecreases with an increase in the length of the time seriesHowever when the length of the time series is larger than 200the decrease in the domain size of the confidence intervals israther slow and of a distinct asymptotic behavior Besides we

see that the confidence intervals do not overlap for Seoulwhile for Busan ([034 039] before and [032 037] after) andDaegu ([055 064] before and [058 069] after) the confi-dence intervals overlap a lot erefore we can conclude thatthere is a difference caused by the law for Seoul however thesame conclusion is inconclusive for Busan and Daegu

Now we first depict the log-log plots of the fluctuationfunction Fq(s) versus q and s for the PM25 time series Asillustrated in Figure 8 for all series the fluctuation valueincreases linearly with s indicating that a power-law be-havior and long-range correlations exist in each series paire decreasing gradient shows that all time series havemultifractal characteristics

e generalized Hurst exponent h(q) Renyi exponentτ(q) and Holder exponent α of the time series before andafter the enactment of the ldquohaze special lawrdquo for all the threecities are shown in Figures 9ndash11 respectively

In Figure 9 it is shown that the generalized Hurst ex-ponents h(q) of PM25 of all three cities decrease with variedq thereby showing the multifractal behaviors of these twoseries We present h(2) for these series in Table 4 All Hurstexponents are confirmed to be larger than 05 which in-dicates that all the time series exhibit positive persistenceBesides we observe that h(2) before the implementation ofthe law is larger than after the law implying that the fluc-tuations before the law have more significant persistenceSubsequently we calculate the range of h(q) for each timeseries and list them in Table 4 We find that for all threecities ΔH(q) after the law is larger which indicates that themultifractal characteristics are stronger and that the mul-tifractalities of the time series before the law are more stable

e Renyi exponents τ(q) plotted in Figure 10 arenonlinear along q which provides further evidence of theexistence of multifractality We notice that the curvature ofthe Renyi exponents before the law are higher which impliesthat the time series before the implementation of the lawhave stronger multifractal characteristics All of these areconsistent with the results calculated from the generalizedHurst exponents

At last the multifractal spectrums of these time series areexamined We depict the curves between α and f(α) inFigure 11 We see that the multifractal spectra are not shownas points which indicates all the time series have multi-fractality en Δα are calculated and presented in Table 4From Table 4 it can be seen that ΔH(q) and Δα of the timeseries after the enactment of law are lower than those before

Table 4 Multifractality for PM25 concentration

City Multifractality Before law After law

Seoulh(2) 12270 12085ΔH(q) 17329 11410Δα 22414 13953

Busanh(2) 12905 11041ΔH(q) 13539 09565Δα 17463 12708

Daeguh(2) 11975 10878ΔH(q) 22910 19516Δα 29610 25043

12 Complexity

05

1

15

2

25

3

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

35

h(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

0

05

1

15

2

h(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

h(q)

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 12 Hurst exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 13

ndash30

ndash20

ndash10

0

10

τ(q)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash40

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 13 Renyi exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

14 Complexity

ndash3

ndash2

ndash1

0

1

f(α)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

1 2 3 40α

05 1 15 2 25 30α

05 1 15 2 25 30α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

05 1 15 2 25 30α

05 1 15 20α

05 1 15 2 250α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 14 Holder exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 15

the implementation of the law for all three cities implyingthat the multifractality of the time series becomes morestable with the measures taken by the government and thatthe efficiency of the ldquohaze special lawrdquo in three cities isdemonstrable this conclusion can be referred to [16 39] inwhich it was pointed out that the decrease in multifractalitymeans the efficiency is improved when some measures aretaken Moreover we also notice that the efficiency of theldquohaze special lawrdquo in Seoul is most obvious when comparedwith Busan and in last place is Daegu Interestingly theorder is the same as the scale and prosperity of the citiesWe can therefore conclude that the law has a higher effecton a larger city

43 Sources of Multifractal Features e two universallyacknowledged sources of multifractality are long-rangecorrelations and a fat-tailed distribution [7 40] Now weinvestigate the major source of multifractality of the PM25concentration series To check the contribution of long-range correlations and fat-tailed distribution quantitativelywe shuffle and phase-randomize the time series

Subsequently we calculate the Hurst exponent h(q) theRenyi exponent τ(q) and the Holder exponent Δα of theoriginal shuffled and phase-randomized time series of thethree cities As shown in Figures 12ndash14 all the time series arestrongly multifractal From Table 5 all h(2) of the originalseries are larger than those of the shuffled and phase-ran-domized time series which indicates that the persistence hasbeen moved after shuffling and phase-randomizationMoreover the original series of all three cities has thegreatest spectrum width Δα while the multifractality de-creased after shuffling and phase-randomizing the seriese findings can also be confirmed in Figures 12ndash14

To examine the main source of multifractality the firstand third rows of Figure 14 show that the multifractalspectra of phase-randomized series are the narrowest ones

which means that the main source of multifractality of theseries before and after the law in Seoul and Daegu are the fat-tailed distributions e second row in Figure 14 shows thatthe multifractal spectrum of the shuffled series is the nar-rowest which implies that the long-range correlation resultsin multifractality of the series before and after imple-mentation of the law in Busan

5 Conclusions

In this study we examined the multifractal characteristics ofPM25 and NO2 time series of all seasons We determinedthat multifractality existed in each season and the multi-fractal property of PM25 was stronger than that of NO2 ineach season We also validated the effectiveness of the ldquohazespecial lawrdquo which was implemented to improve the gov-ernance of air pollution in South Korea We checkedwhether there is a difference caused by the law for these citiesand calculated confidence intervals of Hurst exponent h(2)e results showed that the confidence intervals do notoverlap for Seoul while the confidence intervals overlap a lotfor Busan and Daegu indicating that there is a differencecaused by the law for Seoul however the same conclusion isinconclusive for Busan and Daegu By comparing the Hurstexponent Renyi exponent and Holder exponent the timeseries of the PM25 concentration before the implementationof the law was found to have a higher multifractal degreewhich decreased after the enactment of the law is phe-nomenon reflected the fact that the law has played a role inimproving the efficiency of air pollution control in SouthKorea We also concluded that the effect of the law will bemore significant in a larger city To explore the major causesof multifractality we shuffled and phase-randomized theoriginal series of PM25 By analyzing the width of themultifractal spectrum Δα the results showed that the fat-tailed distribution contributed to the multifractality of bothtimes series before and after the implementation of the ldquohazespecial lawrdquo in Seoul and Daegu whereas long-range cor-relations resulted in the multifractality of the series beforeand after the implementation of the law in Busan Weconcluded that the implementation of the law was verysuccessful and effective for the improvement of PM25 levelserefore we believe that the ldquohaze special lawrdquo is a possiblecause for the change in API in South Korea

Data Availability

e data will be available upon request with the corre-sponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e first author (Jian Wang) was supported by the ChinaScholarship Council (201808260026) J S Kim expressesgratitude for the support from the BK21 PLUS programe

Table 5 Multifractality of PM25 time series

h(2) Δα LRC FTD

Seoul

BeforeOriginal 12270 22414

No YesShuffled 07232 17946Phase-randomized 06965 09574

AfterOriginal 12085 13953

No YesShuffled 03177 10978Phase-randomized 03049 08569

Busan

BeforeOriginal 12905 17463

Yes NoShuffled 03982 09999Phase-randomized 03993 13512

AfterOriginal 11041 12708

Yes NoShuffled 03678 07175Phase-randomized 03794 08673

Daegu

BeforeOriginal 11975 29610

No YesShuffled 05901 22093Phase-randomized 05868 20587

AfterOriginal 10878 25043

No YesShuffled 06756 25600Phase-randomized 06407 22981

LRC and FTD represent the long-range correlations and fat-tailed distri-bution respectively

16 Complexity

corresponding author Wei Shao thanks Professor DaraeJeong for helpful discussion

References

[1] G Shaddick and J Wakefield ldquoModelling daily multivariatepollutant data at multiple sitesrdquo Journal of the Royal StatisticalSociety Series C (Applied Statistics) vol 51 no 3 pp 351ndash3722002

[2] M OrsquoDwyer Air Quality in Ireland 2010 Key Indicators ofAmbient Air Quality EPA Wexford Ireland 2011

[3] A Challoner and L Gill ldquoIndooroutdoor air pollution re-lationships in ten commercial buildings PM25 and NO2rdquoBuilding and Environment vol 80 pp 159ndash173 2014

[4] M Hazenkamp-von Arx T Gotschi U Ackermann-Liebrichet al ldquoPM25 and NO2 assessment in 21 European studycentres of ECRHS II annual means and seasonal differencesrdquoAtmospheric Environment vol 38 no 13 pp 1943ndash19532004

[5] H K Lai L Bayer-Oglesby R Colvile et al ldquoDeterminants ofindoor air concentrations of PM25 black smoke and NO2 insix European cities (EXPOLIS study)rdquo Atmospheric Envi-ronment vol 40 no 7 pp 1299ndash1313 2006

[6] K de Hoogh J Gulliver A V Donkelaar et al ldquoDevelopmentof West-European PM25 and NO2 land use regression modelsincorporating satellite-derived and chemical transport mod-elling datardquo Environmental Research vol 151 pp 1ndash10 2016

[7] S Drozdz J Kwapien P Oswiecimka and R Rak ldquoQuan-titative features of multifractal subtleties in time seriesrdquo EPL(Europhysics Letters) vol 88 no 6 p 60003 2010

[8] R J Buonocore T Aste and T Di Matteo ldquoMeasuringmultiscaling in financial time-seriesrdquo Chaos Solitons ampFractals vol 88 pp 38ndash47 2016

[9] P Oswiecimka S Drozdz J Kwapien and A Z GorskildquoFractals log-periodicity and financial crashesrdquo Acta PhysicaPolonica A vol 117 no 4 pp 637ndash639 2010

[10] B B Mandelbrot and N Goldenfeld ldquoFractals and scaling infinance discontinuity concentration riskrdquo Physics Todayvol 51 no 8 pp 59-60 1998

[11] S Drozdz R Kowalski P Oswiecimka R Rak andR Gebarowski ldquoDynamical variety of shapes in financialmultifractalityrdquo Complexity vol 2018 Article ID 701572113 pages 2018

[12] S Lahmiri ldquoMultifractal analysis of Moroccan family businessstock returnsrdquo Physica A Statistical Mechanics and Its Ap-plications vol 486 pp 183ndash191 2017

[13] X Zhao P Shang and Q Jin ldquoMultifractal detrended cross-correlation analysis of Chinese stock markets based on timedelayrdquo Fractals vol 19 no 03 pp 329ndash338 2011

[14] W Shao and J Wang ldquoDoes the ldquoice-breakingrdquo of South andNorth Korea affect the South Korean financial marketrdquoChaos Solitons amp Fractals vol 132 Article ID 109564 2020

[15] D-H Wang X-W Yu and Y-Y Suo ldquoStatistical propertiesof the yuan exchange rate indexrdquo Physica A Statistical Me-chanics and Its Applications vol 391 no 12 pp 3503ndash35122012

[16] Y Ning Y Wang and C-W Su ldquoHow did Chinarsquos foreignexchange reform affect the efficiency of foreign exchangemarketrdquo Physica A Statistical Mechanics and Its Applicationsvol 483 pp 219ndash226 2017

[17] S Lahmiri and S Bekiros ldquoChaos randomness and multi-fractality in bitcoin marketrdquo Chaos Solitons amp Fractalsvol 106 pp 28ndash34 2018

[18] P Manimaran and A C Narayana ldquoMultifractal detrendedcross-correlation analysis on air pollutants of University ofHyderabad Campus Indiardquo Physica A Statistical Mechanicsand Its Applications vol 502 pp 228ndash235 2018

[19] L Zhao W Li C Yang J Han Z Su and Y Zou ldquoMul-tifractality and network analysis of phase transitionrdquo PLoSOne vol 12 Article ID e0170467 2017

[20] N Kalamaras K Philippopoulos D Deligiorgi C G Tzanisand G Karvounis ldquoMultifractal scaling properties of daily airtemperature time seriesrdquo Chaos Solitons amp Fractals vol 98pp 38ndash43 2017

[21] J Li C Chen Q Yao et al ldquoe effect of circadian rhythm onthe correlation and multifractality of heart rate signals duringexerciserdquo Physica A Statistical Mechanics and Its Applica-tions vol 509 pp 1207ndash1213 2018

[22] P Castiglioni D Lazzeroni P Coruzzi and A FainildquoMultifractal-multiscale analysis of cardiovascular signals aDFA-based characterization of blood pressure and heart-ratecomplexity by genderrdquo Complexity vol 2018 Article ID4801924 14 pages 2018

[23] W Shi R-B Zou F Wang and L Su ldquoA new image seg-mentation method based on multifractal detrended movingaverage analysisrdquo Physica A Statistical Mechanics and ItsApplications vol 432 pp 197ndash205 2015

[24] F Wang Q Fan and H E Stanley ldquoMultiscale multifractaldetrended-fluctuation analysis of two-dimensional surfacesrdquoPhysical Review E vol 93 Article ID 042213 2016

[25] X Li M Zheng J Pu et al ldquoIdentification of abnormallyexpressed lncrnas induced by PM25 in human bronchialepithelial cellsrdquo Bioscience Reports vol 38 no 5 Article IDBSR20171577 2018

[26] K Shi C Liu and Y Huang ldquoMultifractal processes and self-organized criticality of PM25 during a typical haze period inChengdu Chinardquo Aerosol and Air Quality Research vol 15no 3 pp 926ndash934 2015

[27] C Zhang Z Ni and L Ni ldquoMultifractal detrended cross-correlation analysis between PM25 and meteorological fac-torsrdquo Physica A Statistical Mechanics and Its Applicationsvol 438 pp 114ndash123 2015

[28] C Zhang X Wang S Chen L Zou X Zhang and C TangldquoA study on daily PM25 concentrations in Hong Kong usingthe EMD-based MFDFA methodrdquo Physica A StatisticalMechanics and Its Applications vol 530 p 121182 2019

[29] H-D He ldquoMultifractal analysis of interactive patterns be-tween meteorological factors and pollutants in urban andrural areasrdquo Atmospheric Environment vol 149 pp 47ndash542017

[30] J W Kantelhardt S A Zschiegner E Koscielny-BundeS Havlin A Bunde and H E Stanley ldquoMultifractaldetrended fluctuation analysis of nonstationary time seriesrdquoPhysica A Statistical Mechanics and Its Applications vol 316no 1ndash4 pp 87ndash114 2002

[31] L Zunino B M Tabak A Figliola D G PerezM Garavaglia and O A Rosso ldquoA multifractal approach forstock market inefficiencyrdquo Physica A Statistical Mechanicsand Its Applications vol 387 no 26 pp 6558ndash6566 2008

[32] L Zunino A Figliola B M Tabak D G PerezM Garavaglia and O A Rosso ldquoMultifractal structure inLatin-American market indicesrdquo Chaos Solitons amp Fractalsvol 41 no 5 pp 2331ndash2340 2009

[33] B Lashermes P Abry and P Chainais ldquoNew insights into theestimation of scaling exponentsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 2no 4 pp 497ndash523 2004

Complexity 17

[34] E A F Ihlen ldquoIntroduction to multifractal detrended fluc-tuation analysis in Matlabrdquo Frontiers in Physiology vol 3p 141 2012

[35] L Jiang X Zhao N Li F Li and Z Guo ldquoDifferent mul-tifractal scaling of the 0 cm average ground surface tem-perature of four representative weather stations over ChinardquoAdvances inMeteorology vol 2013 Article ID 341934 8 pages2013

[36] P C Ivanov L A N Amaral A L Goldberger et alldquoMultifractality in human heartbeat dynamicsrdquo Naturevol 399 no 6735 pp 461ndash465 1999

[37] S Drozdz and P Oswiecimka ldquoDetecting and interpretingdistortions in hierarchical organization of complex time se-riesrdquo Physical Review E vol 91 Article ID 030902 2015

[38] C Ibarra-Valdez J Alvarez and J Alvarez-Ramirez ldquoRan-domness confidence bands of fractal scaling exponents forfinancial price returnsrdquo Chaos Solitons amp Fractals vol 83pp 119ndash124 2016

[39] D Stosic D Stosic T Stosic and H E Stanley ldquoMultifractalanalysis of managed and independent float exchange ratesrdquoPhysica A Statistical Mechanics and its Applications vol 428pp 13ndash18 2015

[40] K Matia Y Ashkenazy and H E Stanley ldquoMultifractalproperties of price fluctuations of stocks and commoditiesrdquoEurophysics Letters (EPL) vol 61 no 3 pp 422ndash428 2003

18 Complexity

q = 10

q = 10

q = 10

q = ndash10

q = ndash10

q = ndash10

ndash1

0

1

2

3

4

ln(F

q(s)

)

ndash1

0

1

2

3

4

ln(F

q(s)

)

ndash2

ndash1

0

1

2

3

4

ln(F

q(s)

)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

(a)

q = 10

q = 10

q = 10

q = ndash10

q = ndash10

q = ndash10

0

1

2

3

4

ln(F

q(s)

)

ndash1

0

1

2

3

4

ln(F

q(s)

)

ndash1

0

1

2

3

4

ln(F

q(s)

)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

2 22 24 2618ln(s)

(b)

Figure 8 Log-log plots of the fluctuation function Fq(s) for PM25 (a) before and (b) after implementation of the law From top to bottomare Seoul Busan and Daegu

Complexity 9

ndash5 0 5 10ndash10q

05

1

15

2

25

3

h(q)

Before lawAfter law

(a)

ndash5 0 5 10ndash10q

05

1

15

2

25

h(q)

Before lawAfter law

(b)

ndash5 0 5 10ndash10q

05

1

15

2

25

3

35

h(q)

Before lawAfter law

(c)

Figure 9 Hurst exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

Before lawAfter law

(a)

ndash5 0 5 10ndash10q

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

Before lawAfter law

(b)

Figure 10 Continued

10 Complexity

ndash40

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

Before lawAfter law

(c)

Figure 10 Renyi exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu

ndash15

ndash1

ndash05

0

05

1

f(α)

1 15 2 25 305α

Before lawAfter law

(a)

1 15 2 25 305α

ndash15

ndash1

ndash05

0

05

1f(α

)

Before lawAfter law

(b)

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

f(α)

1 2 3 40α

Before lawAfter law

(c)

Figure 11 Holder exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu e triangle and circle symbols represent beforeand after the law respectively

Complexity 11

Table 3 the conclusion we described above about the Hurstexponent and the Renyi exponent is discussed Moreover asshown in Figure 5 the multifractal spectra are stronglyasymmetrical and as an important indicator of fractal or-ganization the asymmetry coefficient which is used toquantify the asymmetry of the multifractal spectrum hasbeen estimated in [37]

Based on Table 3 the multifractal features can be clearlyconfirmed for all time series and it can be seen that exceptfor spring the Hurst exponents and multifractal spectrumwidths of PM25 are obviously larger than those of NO2 andthe multifractal degrees of PM25 are stronger than those ofNO2

42 Implications of the ldquoHaze Special Lawrdquo In this sectionwe focus on the enactment of the ldquohaze special lawrdquo andstudy whether the law is effective in the governance of at-mospheric pollution We select daily data from the sameperiod described in Section 3 As we studied above themultifractal degree of PM25 is stronger than that of NO2us in this section we focus on the comparison of PM25concentrations We show the PM25 time series before andafter the law was implemented in Seoul Busan and Daegu inFigure 6

As the fluctuation statistics are of asymptotic feature asufficiently long-length sequence guarantees the accurateestimation of the Hurst exponent and for a small-lengthsequence the asymptotic value in the slope of ln(Fq(s))

versus ln(s) will be lacked [38] As the total number ofobservations is selected to be 305 to eliminate the doubt ofwhether the selected time series can give representativeresults for a multifractal analysis in the investigation wedealt with this concern by computing confidence intervals ofHurst exponent h(2) to check whether there is a differencecaused by the law for all the three cities e number ofsamples increased from 50 to 300 and every additionalinterval is 10 e confidence level is selected as 950 Foreach set of sample lengths we generate the shuffled timeseries 1000 times and then the confidence interval is plottedin Figure 7

As shown in Figure 7 and as expected the domain sizedecreases with an increase in the length of the time seriesHowever when the length of the time series is larger than 200the decrease in the domain size of the confidence intervals israther slow and of a distinct asymptotic behavior Besides we

see that the confidence intervals do not overlap for Seoulwhile for Busan ([034 039] before and [032 037] after) andDaegu ([055 064] before and [058 069] after) the confi-dence intervals overlap a lot erefore we can conclude thatthere is a difference caused by the law for Seoul however thesame conclusion is inconclusive for Busan and Daegu

Now we first depict the log-log plots of the fluctuationfunction Fq(s) versus q and s for the PM25 time series Asillustrated in Figure 8 for all series the fluctuation valueincreases linearly with s indicating that a power-law be-havior and long-range correlations exist in each series paire decreasing gradient shows that all time series havemultifractal characteristics

e generalized Hurst exponent h(q) Renyi exponentτ(q) and Holder exponent α of the time series before andafter the enactment of the ldquohaze special lawrdquo for all the threecities are shown in Figures 9ndash11 respectively

In Figure 9 it is shown that the generalized Hurst ex-ponents h(q) of PM25 of all three cities decrease with variedq thereby showing the multifractal behaviors of these twoseries We present h(2) for these series in Table 4 All Hurstexponents are confirmed to be larger than 05 which in-dicates that all the time series exhibit positive persistenceBesides we observe that h(2) before the implementation ofthe law is larger than after the law implying that the fluc-tuations before the law have more significant persistenceSubsequently we calculate the range of h(q) for each timeseries and list them in Table 4 We find that for all threecities ΔH(q) after the law is larger which indicates that themultifractal characteristics are stronger and that the mul-tifractalities of the time series before the law are more stable

e Renyi exponents τ(q) plotted in Figure 10 arenonlinear along q which provides further evidence of theexistence of multifractality We notice that the curvature ofthe Renyi exponents before the law are higher which impliesthat the time series before the implementation of the lawhave stronger multifractal characteristics All of these areconsistent with the results calculated from the generalizedHurst exponents

At last the multifractal spectrums of these time series areexamined We depict the curves between α and f(α) inFigure 11 We see that the multifractal spectra are not shownas points which indicates all the time series have multi-fractality en Δα are calculated and presented in Table 4From Table 4 it can be seen that ΔH(q) and Δα of the timeseries after the enactment of law are lower than those before

Table 4 Multifractality for PM25 concentration

City Multifractality Before law After law

Seoulh(2) 12270 12085ΔH(q) 17329 11410Δα 22414 13953

Busanh(2) 12905 11041ΔH(q) 13539 09565Δα 17463 12708

Daeguh(2) 11975 10878ΔH(q) 22910 19516Δα 29610 25043

12 Complexity

05

1

15

2

25

3

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

35

h(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

0

05

1

15

2

h(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

h(q)

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 12 Hurst exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 13

ndash30

ndash20

ndash10

0

10

τ(q)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash40

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 13 Renyi exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

14 Complexity

ndash3

ndash2

ndash1

0

1

f(α)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

1 2 3 40α

05 1 15 2 25 30α

05 1 15 2 25 30α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

05 1 15 2 25 30α

05 1 15 20α

05 1 15 2 250α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 14 Holder exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 15

the implementation of the law for all three cities implyingthat the multifractality of the time series becomes morestable with the measures taken by the government and thatthe efficiency of the ldquohaze special lawrdquo in three cities isdemonstrable this conclusion can be referred to [16 39] inwhich it was pointed out that the decrease in multifractalitymeans the efficiency is improved when some measures aretaken Moreover we also notice that the efficiency of theldquohaze special lawrdquo in Seoul is most obvious when comparedwith Busan and in last place is Daegu Interestingly theorder is the same as the scale and prosperity of the citiesWe can therefore conclude that the law has a higher effecton a larger city

43 Sources of Multifractal Features e two universallyacknowledged sources of multifractality are long-rangecorrelations and a fat-tailed distribution [7 40] Now weinvestigate the major source of multifractality of the PM25concentration series To check the contribution of long-range correlations and fat-tailed distribution quantitativelywe shuffle and phase-randomize the time series

Subsequently we calculate the Hurst exponent h(q) theRenyi exponent τ(q) and the Holder exponent Δα of theoriginal shuffled and phase-randomized time series of thethree cities As shown in Figures 12ndash14 all the time series arestrongly multifractal From Table 5 all h(2) of the originalseries are larger than those of the shuffled and phase-ran-domized time series which indicates that the persistence hasbeen moved after shuffling and phase-randomizationMoreover the original series of all three cities has thegreatest spectrum width Δα while the multifractality de-creased after shuffling and phase-randomizing the seriese findings can also be confirmed in Figures 12ndash14

To examine the main source of multifractality the firstand third rows of Figure 14 show that the multifractalspectra of phase-randomized series are the narrowest ones

which means that the main source of multifractality of theseries before and after the law in Seoul and Daegu are the fat-tailed distributions e second row in Figure 14 shows thatthe multifractal spectrum of the shuffled series is the nar-rowest which implies that the long-range correlation resultsin multifractality of the series before and after imple-mentation of the law in Busan

5 Conclusions

In this study we examined the multifractal characteristics ofPM25 and NO2 time series of all seasons We determinedthat multifractality existed in each season and the multi-fractal property of PM25 was stronger than that of NO2 ineach season We also validated the effectiveness of the ldquohazespecial lawrdquo which was implemented to improve the gov-ernance of air pollution in South Korea We checkedwhether there is a difference caused by the law for these citiesand calculated confidence intervals of Hurst exponent h(2)e results showed that the confidence intervals do notoverlap for Seoul while the confidence intervals overlap a lotfor Busan and Daegu indicating that there is a differencecaused by the law for Seoul however the same conclusion isinconclusive for Busan and Daegu By comparing the Hurstexponent Renyi exponent and Holder exponent the timeseries of the PM25 concentration before the implementationof the law was found to have a higher multifractal degreewhich decreased after the enactment of the law is phe-nomenon reflected the fact that the law has played a role inimproving the efficiency of air pollution control in SouthKorea We also concluded that the effect of the law will bemore significant in a larger city To explore the major causesof multifractality we shuffled and phase-randomized theoriginal series of PM25 By analyzing the width of themultifractal spectrum Δα the results showed that the fat-tailed distribution contributed to the multifractality of bothtimes series before and after the implementation of the ldquohazespecial lawrdquo in Seoul and Daegu whereas long-range cor-relations resulted in the multifractality of the series beforeand after the implementation of the law in Busan Weconcluded that the implementation of the law was verysuccessful and effective for the improvement of PM25 levelserefore we believe that the ldquohaze special lawrdquo is a possiblecause for the change in API in South Korea

Data Availability

e data will be available upon request with the corre-sponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e first author (Jian Wang) was supported by the ChinaScholarship Council (201808260026) J S Kim expressesgratitude for the support from the BK21 PLUS programe

Table 5 Multifractality of PM25 time series

h(2) Δα LRC FTD

Seoul

BeforeOriginal 12270 22414

No YesShuffled 07232 17946Phase-randomized 06965 09574

AfterOriginal 12085 13953

No YesShuffled 03177 10978Phase-randomized 03049 08569

Busan

BeforeOriginal 12905 17463

Yes NoShuffled 03982 09999Phase-randomized 03993 13512

AfterOriginal 11041 12708

Yes NoShuffled 03678 07175Phase-randomized 03794 08673

Daegu

BeforeOriginal 11975 29610

No YesShuffled 05901 22093Phase-randomized 05868 20587

AfterOriginal 10878 25043

No YesShuffled 06756 25600Phase-randomized 06407 22981

LRC and FTD represent the long-range correlations and fat-tailed distri-bution respectively

16 Complexity

corresponding author Wei Shao thanks Professor DaraeJeong for helpful discussion

References

[1] G Shaddick and J Wakefield ldquoModelling daily multivariatepollutant data at multiple sitesrdquo Journal of the Royal StatisticalSociety Series C (Applied Statistics) vol 51 no 3 pp 351ndash3722002

[2] M OrsquoDwyer Air Quality in Ireland 2010 Key Indicators ofAmbient Air Quality EPA Wexford Ireland 2011

[3] A Challoner and L Gill ldquoIndooroutdoor air pollution re-lationships in ten commercial buildings PM25 and NO2rdquoBuilding and Environment vol 80 pp 159ndash173 2014

[4] M Hazenkamp-von Arx T Gotschi U Ackermann-Liebrichet al ldquoPM25 and NO2 assessment in 21 European studycentres of ECRHS II annual means and seasonal differencesrdquoAtmospheric Environment vol 38 no 13 pp 1943ndash19532004

[5] H K Lai L Bayer-Oglesby R Colvile et al ldquoDeterminants ofindoor air concentrations of PM25 black smoke and NO2 insix European cities (EXPOLIS study)rdquo Atmospheric Envi-ronment vol 40 no 7 pp 1299ndash1313 2006

[6] K de Hoogh J Gulliver A V Donkelaar et al ldquoDevelopmentof West-European PM25 and NO2 land use regression modelsincorporating satellite-derived and chemical transport mod-elling datardquo Environmental Research vol 151 pp 1ndash10 2016

[7] S Drozdz J Kwapien P Oswiecimka and R Rak ldquoQuan-titative features of multifractal subtleties in time seriesrdquo EPL(Europhysics Letters) vol 88 no 6 p 60003 2010

[8] R J Buonocore T Aste and T Di Matteo ldquoMeasuringmultiscaling in financial time-seriesrdquo Chaos Solitons ampFractals vol 88 pp 38ndash47 2016

[9] P Oswiecimka S Drozdz J Kwapien and A Z GorskildquoFractals log-periodicity and financial crashesrdquo Acta PhysicaPolonica A vol 117 no 4 pp 637ndash639 2010

[10] B B Mandelbrot and N Goldenfeld ldquoFractals and scaling infinance discontinuity concentration riskrdquo Physics Todayvol 51 no 8 pp 59-60 1998

[11] S Drozdz R Kowalski P Oswiecimka R Rak andR Gebarowski ldquoDynamical variety of shapes in financialmultifractalityrdquo Complexity vol 2018 Article ID 701572113 pages 2018

[12] S Lahmiri ldquoMultifractal analysis of Moroccan family businessstock returnsrdquo Physica A Statistical Mechanics and Its Ap-plications vol 486 pp 183ndash191 2017

[13] X Zhao P Shang and Q Jin ldquoMultifractal detrended cross-correlation analysis of Chinese stock markets based on timedelayrdquo Fractals vol 19 no 03 pp 329ndash338 2011

[14] W Shao and J Wang ldquoDoes the ldquoice-breakingrdquo of South andNorth Korea affect the South Korean financial marketrdquoChaos Solitons amp Fractals vol 132 Article ID 109564 2020

[15] D-H Wang X-W Yu and Y-Y Suo ldquoStatistical propertiesof the yuan exchange rate indexrdquo Physica A Statistical Me-chanics and Its Applications vol 391 no 12 pp 3503ndash35122012

[16] Y Ning Y Wang and C-W Su ldquoHow did Chinarsquos foreignexchange reform affect the efficiency of foreign exchangemarketrdquo Physica A Statistical Mechanics and Its Applicationsvol 483 pp 219ndash226 2017

[17] S Lahmiri and S Bekiros ldquoChaos randomness and multi-fractality in bitcoin marketrdquo Chaos Solitons amp Fractalsvol 106 pp 28ndash34 2018

[18] P Manimaran and A C Narayana ldquoMultifractal detrendedcross-correlation analysis on air pollutants of University ofHyderabad Campus Indiardquo Physica A Statistical Mechanicsand Its Applications vol 502 pp 228ndash235 2018

[19] L Zhao W Li C Yang J Han Z Su and Y Zou ldquoMul-tifractality and network analysis of phase transitionrdquo PLoSOne vol 12 Article ID e0170467 2017

[20] N Kalamaras K Philippopoulos D Deligiorgi C G Tzanisand G Karvounis ldquoMultifractal scaling properties of daily airtemperature time seriesrdquo Chaos Solitons amp Fractals vol 98pp 38ndash43 2017

[21] J Li C Chen Q Yao et al ldquoe effect of circadian rhythm onthe correlation and multifractality of heart rate signals duringexerciserdquo Physica A Statistical Mechanics and Its Applica-tions vol 509 pp 1207ndash1213 2018

[22] P Castiglioni D Lazzeroni P Coruzzi and A FainildquoMultifractal-multiscale analysis of cardiovascular signals aDFA-based characterization of blood pressure and heart-ratecomplexity by genderrdquo Complexity vol 2018 Article ID4801924 14 pages 2018

[23] W Shi R-B Zou F Wang and L Su ldquoA new image seg-mentation method based on multifractal detrended movingaverage analysisrdquo Physica A Statistical Mechanics and ItsApplications vol 432 pp 197ndash205 2015

[24] F Wang Q Fan and H E Stanley ldquoMultiscale multifractaldetrended-fluctuation analysis of two-dimensional surfacesrdquoPhysical Review E vol 93 Article ID 042213 2016

[25] X Li M Zheng J Pu et al ldquoIdentification of abnormallyexpressed lncrnas induced by PM25 in human bronchialepithelial cellsrdquo Bioscience Reports vol 38 no 5 Article IDBSR20171577 2018

[26] K Shi C Liu and Y Huang ldquoMultifractal processes and self-organized criticality of PM25 during a typical haze period inChengdu Chinardquo Aerosol and Air Quality Research vol 15no 3 pp 926ndash934 2015

[27] C Zhang Z Ni and L Ni ldquoMultifractal detrended cross-correlation analysis between PM25 and meteorological fac-torsrdquo Physica A Statistical Mechanics and Its Applicationsvol 438 pp 114ndash123 2015

[28] C Zhang X Wang S Chen L Zou X Zhang and C TangldquoA study on daily PM25 concentrations in Hong Kong usingthe EMD-based MFDFA methodrdquo Physica A StatisticalMechanics and Its Applications vol 530 p 121182 2019

[29] H-D He ldquoMultifractal analysis of interactive patterns be-tween meteorological factors and pollutants in urban andrural areasrdquo Atmospheric Environment vol 149 pp 47ndash542017

[30] J W Kantelhardt S A Zschiegner E Koscielny-BundeS Havlin A Bunde and H E Stanley ldquoMultifractaldetrended fluctuation analysis of nonstationary time seriesrdquoPhysica A Statistical Mechanics and Its Applications vol 316no 1ndash4 pp 87ndash114 2002

[31] L Zunino B M Tabak A Figliola D G PerezM Garavaglia and O A Rosso ldquoA multifractal approach forstock market inefficiencyrdquo Physica A Statistical Mechanicsand Its Applications vol 387 no 26 pp 6558ndash6566 2008

[32] L Zunino A Figliola B M Tabak D G PerezM Garavaglia and O A Rosso ldquoMultifractal structure inLatin-American market indicesrdquo Chaos Solitons amp Fractalsvol 41 no 5 pp 2331ndash2340 2009

[33] B Lashermes P Abry and P Chainais ldquoNew insights into theestimation of scaling exponentsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 2no 4 pp 497ndash523 2004

Complexity 17

[34] E A F Ihlen ldquoIntroduction to multifractal detrended fluc-tuation analysis in Matlabrdquo Frontiers in Physiology vol 3p 141 2012

[35] L Jiang X Zhao N Li F Li and Z Guo ldquoDifferent mul-tifractal scaling of the 0 cm average ground surface tem-perature of four representative weather stations over ChinardquoAdvances inMeteorology vol 2013 Article ID 341934 8 pages2013

[36] P C Ivanov L A N Amaral A L Goldberger et alldquoMultifractality in human heartbeat dynamicsrdquo Naturevol 399 no 6735 pp 461ndash465 1999

[37] S Drozdz and P Oswiecimka ldquoDetecting and interpretingdistortions in hierarchical organization of complex time se-riesrdquo Physical Review E vol 91 Article ID 030902 2015

[38] C Ibarra-Valdez J Alvarez and J Alvarez-Ramirez ldquoRan-domness confidence bands of fractal scaling exponents forfinancial price returnsrdquo Chaos Solitons amp Fractals vol 83pp 119ndash124 2016

[39] D Stosic D Stosic T Stosic and H E Stanley ldquoMultifractalanalysis of managed and independent float exchange ratesrdquoPhysica A Statistical Mechanics and its Applications vol 428pp 13ndash18 2015

[40] K Matia Y Ashkenazy and H E Stanley ldquoMultifractalproperties of price fluctuations of stocks and commoditiesrdquoEurophysics Letters (EPL) vol 61 no 3 pp 422ndash428 2003

18 Complexity

ndash5 0 5 10ndash10q

05

1

15

2

25

3

h(q)

Before lawAfter law

(a)

ndash5 0 5 10ndash10q

05

1

15

2

25

h(q)

Before lawAfter law

(b)

ndash5 0 5 10ndash10q

05

1

15

2

25

3

35

h(q)

Before lawAfter law

(c)

Figure 9 Hurst exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

Before lawAfter law

(a)

ndash5 0 5 10ndash10q

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

Before lawAfter law

(b)

Figure 10 Continued

10 Complexity

ndash40

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

Before lawAfter law

(c)

Figure 10 Renyi exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu

ndash15

ndash1

ndash05

0

05

1

f(α)

1 15 2 25 305α

Before lawAfter law

(a)

1 15 2 25 305α

ndash15

ndash1

ndash05

0

05

1f(α

)

Before lawAfter law

(b)

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

f(α)

1 2 3 40α

Before lawAfter law

(c)

Figure 11 Holder exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu e triangle and circle symbols represent beforeand after the law respectively

Complexity 11

Table 3 the conclusion we described above about the Hurstexponent and the Renyi exponent is discussed Moreover asshown in Figure 5 the multifractal spectra are stronglyasymmetrical and as an important indicator of fractal or-ganization the asymmetry coefficient which is used toquantify the asymmetry of the multifractal spectrum hasbeen estimated in [37]

Based on Table 3 the multifractal features can be clearlyconfirmed for all time series and it can be seen that exceptfor spring the Hurst exponents and multifractal spectrumwidths of PM25 are obviously larger than those of NO2 andthe multifractal degrees of PM25 are stronger than those ofNO2

42 Implications of the ldquoHaze Special Lawrdquo In this sectionwe focus on the enactment of the ldquohaze special lawrdquo andstudy whether the law is effective in the governance of at-mospheric pollution We select daily data from the sameperiod described in Section 3 As we studied above themultifractal degree of PM25 is stronger than that of NO2us in this section we focus on the comparison of PM25concentrations We show the PM25 time series before andafter the law was implemented in Seoul Busan and Daegu inFigure 6

As the fluctuation statistics are of asymptotic feature asufficiently long-length sequence guarantees the accurateestimation of the Hurst exponent and for a small-lengthsequence the asymptotic value in the slope of ln(Fq(s))

versus ln(s) will be lacked [38] As the total number ofobservations is selected to be 305 to eliminate the doubt ofwhether the selected time series can give representativeresults for a multifractal analysis in the investigation wedealt with this concern by computing confidence intervals ofHurst exponent h(2) to check whether there is a differencecaused by the law for all the three cities e number ofsamples increased from 50 to 300 and every additionalinterval is 10 e confidence level is selected as 950 Foreach set of sample lengths we generate the shuffled timeseries 1000 times and then the confidence interval is plottedin Figure 7

As shown in Figure 7 and as expected the domain sizedecreases with an increase in the length of the time seriesHowever when the length of the time series is larger than 200the decrease in the domain size of the confidence intervals israther slow and of a distinct asymptotic behavior Besides we

see that the confidence intervals do not overlap for Seoulwhile for Busan ([034 039] before and [032 037] after) andDaegu ([055 064] before and [058 069] after) the confi-dence intervals overlap a lot erefore we can conclude thatthere is a difference caused by the law for Seoul however thesame conclusion is inconclusive for Busan and Daegu

Now we first depict the log-log plots of the fluctuationfunction Fq(s) versus q and s for the PM25 time series Asillustrated in Figure 8 for all series the fluctuation valueincreases linearly with s indicating that a power-law be-havior and long-range correlations exist in each series paire decreasing gradient shows that all time series havemultifractal characteristics

e generalized Hurst exponent h(q) Renyi exponentτ(q) and Holder exponent α of the time series before andafter the enactment of the ldquohaze special lawrdquo for all the threecities are shown in Figures 9ndash11 respectively

In Figure 9 it is shown that the generalized Hurst ex-ponents h(q) of PM25 of all three cities decrease with variedq thereby showing the multifractal behaviors of these twoseries We present h(2) for these series in Table 4 All Hurstexponents are confirmed to be larger than 05 which in-dicates that all the time series exhibit positive persistenceBesides we observe that h(2) before the implementation ofthe law is larger than after the law implying that the fluc-tuations before the law have more significant persistenceSubsequently we calculate the range of h(q) for each timeseries and list them in Table 4 We find that for all threecities ΔH(q) after the law is larger which indicates that themultifractal characteristics are stronger and that the mul-tifractalities of the time series before the law are more stable

e Renyi exponents τ(q) plotted in Figure 10 arenonlinear along q which provides further evidence of theexistence of multifractality We notice that the curvature ofthe Renyi exponents before the law are higher which impliesthat the time series before the implementation of the lawhave stronger multifractal characteristics All of these areconsistent with the results calculated from the generalizedHurst exponents

At last the multifractal spectrums of these time series areexamined We depict the curves between α and f(α) inFigure 11 We see that the multifractal spectra are not shownas points which indicates all the time series have multi-fractality en Δα are calculated and presented in Table 4From Table 4 it can be seen that ΔH(q) and Δα of the timeseries after the enactment of law are lower than those before

Table 4 Multifractality for PM25 concentration

City Multifractality Before law After law

Seoulh(2) 12270 12085ΔH(q) 17329 11410Δα 22414 13953

Busanh(2) 12905 11041ΔH(q) 13539 09565Δα 17463 12708

Daeguh(2) 11975 10878ΔH(q) 22910 19516Δα 29610 25043

12 Complexity

05

1

15

2

25

3

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

35

h(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

0

05

1

15

2

h(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

h(q)

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 12 Hurst exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 13

ndash30

ndash20

ndash10

0

10

τ(q)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash40

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 13 Renyi exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

14 Complexity

ndash3

ndash2

ndash1

0

1

f(α)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

1 2 3 40α

05 1 15 2 25 30α

05 1 15 2 25 30α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

05 1 15 2 25 30α

05 1 15 20α

05 1 15 2 250α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 14 Holder exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 15

the implementation of the law for all three cities implyingthat the multifractality of the time series becomes morestable with the measures taken by the government and thatthe efficiency of the ldquohaze special lawrdquo in three cities isdemonstrable this conclusion can be referred to [16 39] inwhich it was pointed out that the decrease in multifractalitymeans the efficiency is improved when some measures aretaken Moreover we also notice that the efficiency of theldquohaze special lawrdquo in Seoul is most obvious when comparedwith Busan and in last place is Daegu Interestingly theorder is the same as the scale and prosperity of the citiesWe can therefore conclude that the law has a higher effecton a larger city

43 Sources of Multifractal Features e two universallyacknowledged sources of multifractality are long-rangecorrelations and a fat-tailed distribution [7 40] Now weinvestigate the major source of multifractality of the PM25concentration series To check the contribution of long-range correlations and fat-tailed distribution quantitativelywe shuffle and phase-randomize the time series

Subsequently we calculate the Hurst exponent h(q) theRenyi exponent τ(q) and the Holder exponent Δα of theoriginal shuffled and phase-randomized time series of thethree cities As shown in Figures 12ndash14 all the time series arestrongly multifractal From Table 5 all h(2) of the originalseries are larger than those of the shuffled and phase-ran-domized time series which indicates that the persistence hasbeen moved after shuffling and phase-randomizationMoreover the original series of all three cities has thegreatest spectrum width Δα while the multifractality de-creased after shuffling and phase-randomizing the seriese findings can also be confirmed in Figures 12ndash14

To examine the main source of multifractality the firstand third rows of Figure 14 show that the multifractalspectra of phase-randomized series are the narrowest ones

which means that the main source of multifractality of theseries before and after the law in Seoul and Daegu are the fat-tailed distributions e second row in Figure 14 shows thatthe multifractal spectrum of the shuffled series is the nar-rowest which implies that the long-range correlation resultsin multifractality of the series before and after imple-mentation of the law in Busan

5 Conclusions

In this study we examined the multifractal characteristics ofPM25 and NO2 time series of all seasons We determinedthat multifractality existed in each season and the multi-fractal property of PM25 was stronger than that of NO2 ineach season We also validated the effectiveness of the ldquohazespecial lawrdquo which was implemented to improve the gov-ernance of air pollution in South Korea We checkedwhether there is a difference caused by the law for these citiesand calculated confidence intervals of Hurst exponent h(2)e results showed that the confidence intervals do notoverlap for Seoul while the confidence intervals overlap a lotfor Busan and Daegu indicating that there is a differencecaused by the law for Seoul however the same conclusion isinconclusive for Busan and Daegu By comparing the Hurstexponent Renyi exponent and Holder exponent the timeseries of the PM25 concentration before the implementationof the law was found to have a higher multifractal degreewhich decreased after the enactment of the law is phe-nomenon reflected the fact that the law has played a role inimproving the efficiency of air pollution control in SouthKorea We also concluded that the effect of the law will bemore significant in a larger city To explore the major causesof multifractality we shuffled and phase-randomized theoriginal series of PM25 By analyzing the width of themultifractal spectrum Δα the results showed that the fat-tailed distribution contributed to the multifractality of bothtimes series before and after the implementation of the ldquohazespecial lawrdquo in Seoul and Daegu whereas long-range cor-relations resulted in the multifractality of the series beforeand after the implementation of the law in Busan Weconcluded that the implementation of the law was verysuccessful and effective for the improvement of PM25 levelserefore we believe that the ldquohaze special lawrdquo is a possiblecause for the change in API in South Korea

Data Availability

e data will be available upon request with the corre-sponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e first author (Jian Wang) was supported by the ChinaScholarship Council (201808260026) J S Kim expressesgratitude for the support from the BK21 PLUS programe

Table 5 Multifractality of PM25 time series

h(2) Δα LRC FTD

Seoul

BeforeOriginal 12270 22414

No YesShuffled 07232 17946Phase-randomized 06965 09574

AfterOriginal 12085 13953

No YesShuffled 03177 10978Phase-randomized 03049 08569

Busan

BeforeOriginal 12905 17463

Yes NoShuffled 03982 09999Phase-randomized 03993 13512

AfterOriginal 11041 12708

Yes NoShuffled 03678 07175Phase-randomized 03794 08673

Daegu

BeforeOriginal 11975 29610

No YesShuffled 05901 22093Phase-randomized 05868 20587

AfterOriginal 10878 25043

No YesShuffled 06756 25600Phase-randomized 06407 22981

LRC and FTD represent the long-range correlations and fat-tailed distri-bution respectively

16 Complexity

corresponding author Wei Shao thanks Professor DaraeJeong for helpful discussion

References

[1] G Shaddick and J Wakefield ldquoModelling daily multivariatepollutant data at multiple sitesrdquo Journal of the Royal StatisticalSociety Series C (Applied Statistics) vol 51 no 3 pp 351ndash3722002

[2] M OrsquoDwyer Air Quality in Ireland 2010 Key Indicators ofAmbient Air Quality EPA Wexford Ireland 2011

[3] A Challoner and L Gill ldquoIndooroutdoor air pollution re-lationships in ten commercial buildings PM25 and NO2rdquoBuilding and Environment vol 80 pp 159ndash173 2014

[4] M Hazenkamp-von Arx T Gotschi U Ackermann-Liebrichet al ldquoPM25 and NO2 assessment in 21 European studycentres of ECRHS II annual means and seasonal differencesrdquoAtmospheric Environment vol 38 no 13 pp 1943ndash19532004

[5] H K Lai L Bayer-Oglesby R Colvile et al ldquoDeterminants ofindoor air concentrations of PM25 black smoke and NO2 insix European cities (EXPOLIS study)rdquo Atmospheric Envi-ronment vol 40 no 7 pp 1299ndash1313 2006

[6] K de Hoogh J Gulliver A V Donkelaar et al ldquoDevelopmentof West-European PM25 and NO2 land use regression modelsincorporating satellite-derived and chemical transport mod-elling datardquo Environmental Research vol 151 pp 1ndash10 2016

[7] S Drozdz J Kwapien P Oswiecimka and R Rak ldquoQuan-titative features of multifractal subtleties in time seriesrdquo EPL(Europhysics Letters) vol 88 no 6 p 60003 2010

[8] R J Buonocore T Aste and T Di Matteo ldquoMeasuringmultiscaling in financial time-seriesrdquo Chaos Solitons ampFractals vol 88 pp 38ndash47 2016

[9] P Oswiecimka S Drozdz J Kwapien and A Z GorskildquoFractals log-periodicity and financial crashesrdquo Acta PhysicaPolonica A vol 117 no 4 pp 637ndash639 2010

[10] B B Mandelbrot and N Goldenfeld ldquoFractals and scaling infinance discontinuity concentration riskrdquo Physics Todayvol 51 no 8 pp 59-60 1998

[11] S Drozdz R Kowalski P Oswiecimka R Rak andR Gebarowski ldquoDynamical variety of shapes in financialmultifractalityrdquo Complexity vol 2018 Article ID 701572113 pages 2018

[12] S Lahmiri ldquoMultifractal analysis of Moroccan family businessstock returnsrdquo Physica A Statistical Mechanics and Its Ap-plications vol 486 pp 183ndash191 2017

[13] X Zhao P Shang and Q Jin ldquoMultifractal detrended cross-correlation analysis of Chinese stock markets based on timedelayrdquo Fractals vol 19 no 03 pp 329ndash338 2011

[14] W Shao and J Wang ldquoDoes the ldquoice-breakingrdquo of South andNorth Korea affect the South Korean financial marketrdquoChaos Solitons amp Fractals vol 132 Article ID 109564 2020

[15] D-H Wang X-W Yu and Y-Y Suo ldquoStatistical propertiesof the yuan exchange rate indexrdquo Physica A Statistical Me-chanics and Its Applications vol 391 no 12 pp 3503ndash35122012

[16] Y Ning Y Wang and C-W Su ldquoHow did Chinarsquos foreignexchange reform affect the efficiency of foreign exchangemarketrdquo Physica A Statistical Mechanics and Its Applicationsvol 483 pp 219ndash226 2017

[17] S Lahmiri and S Bekiros ldquoChaos randomness and multi-fractality in bitcoin marketrdquo Chaos Solitons amp Fractalsvol 106 pp 28ndash34 2018

[18] P Manimaran and A C Narayana ldquoMultifractal detrendedcross-correlation analysis on air pollutants of University ofHyderabad Campus Indiardquo Physica A Statistical Mechanicsand Its Applications vol 502 pp 228ndash235 2018

[19] L Zhao W Li C Yang J Han Z Su and Y Zou ldquoMul-tifractality and network analysis of phase transitionrdquo PLoSOne vol 12 Article ID e0170467 2017

[20] N Kalamaras K Philippopoulos D Deligiorgi C G Tzanisand G Karvounis ldquoMultifractal scaling properties of daily airtemperature time seriesrdquo Chaos Solitons amp Fractals vol 98pp 38ndash43 2017

[21] J Li C Chen Q Yao et al ldquoe effect of circadian rhythm onthe correlation and multifractality of heart rate signals duringexerciserdquo Physica A Statistical Mechanics and Its Applica-tions vol 509 pp 1207ndash1213 2018

[22] P Castiglioni D Lazzeroni P Coruzzi and A FainildquoMultifractal-multiscale analysis of cardiovascular signals aDFA-based characterization of blood pressure and heart-ratecomplexity by genderrdquo Complexity vol 2018 Article ID4801924 14 pages 2018

[23] W Shi R-B Zou F Wang and L Su ldquoA new image seg-mentation method based on multifractal detrended movingaverage analysisrdquo Physica A Statistical Mechanics and ItsApplications vol 432 pp 197ndash205 2015

[24] F Wang Q Fan and H E Stanley ldquoMultiscale multifractaldetrended-fluctuation analysis of two-dimensional surfacesrdquoPhysical Review E vol 93 Article ID 042213 2016

[25] X Li M Zheng J Pu et al ldquoIdentification of abnormallyexpressed lncrnas induced by PM25 in human bronchialepithelial cellsrdquo Bioscience Reports vol 38 no 5 Article IDBSR20171577 2018

[26] K Shi C Liu and Y Huang ldquoMultifractal processes and self-organized criticality of PM25 during a typical haze period inChengdu Chinardquo Aerosol and Air Quality Research vol 15no 3 pp 926ndash934 2015

[27] C Zhang Z Ni and L Ni ldquoMultifractal detrended cross-correlation analysis between PM25 and meteorological fac-torsrdquo Physica A Statistical Mechanics and Its Applicationsvol 438 pp 114ndash123 2015

[28] C Zhang X Wang S Chen L Zou X Zhang and C TangldquoA study on daily PM25 concentrations in Hong Kong usingthe EMD-based MFDFA methodrdquo Physica A StatisticalMechanics and Its Applications vol 530 p 121182 2019

[29] H-D He ldquoMultifractal analysis of interactive patterns be-tween meteorological factors and pollutants in urban andrural areasrdquo Atmospheric Environment vol 149 pp 47ndash542017

[30] J W Kantelhardt S A Zschiegner E Koscielny-BundeS Havlin A Bunde and H E Stanley ldquoMultifractaldetrended fluctuation analysis of nonstationary time seriesrdquoPhysica A Statistical Mechanics and Its Applications vol 316no 1ndash4 pp 87ndash114 2002

[31] L Zunino B M Tabak A Figliola D G PerezM Garavaglia and O A Rosso ldquoA multifractal approach forstock market inefficiencyrdquo Physica A Statistical Mechanicsand Its Applications vol 387 no 26 pp 6558ndash6566 2008

[32] L Zunino A Figliola B M Tabak D G PerezM Garavaglia and O A Rosso ldquoMultifractal structure inLatin-American market indicesrdquo Chaos Solitons amp Fractalsvol 41 no 5 pp 2331ndash2340 2009

[33] B Lashermes P Abry and P Chainais ldquoNew insights into theestimation of scaling exponentsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 2no 4 pp 497ndash523 2004

Complexity 17

[34] E A F Ihlen ldquoIntroduction to multifractal detrended fluc-tuation analysis in Matlabrdquo Frontiers in Physiology vol 3p 141 2012

[35] L Jiang X Zhao N Li F Li and Z Guo ldquoDifferent mul-tifractal scaling of the 0 cm average ground surface tem-perature of four representative weather stations over ChinardquoAdvances inMeteorology vol 2013 Article ID 341934 8 pages2013

[36] P C Ivanov L A N Amaral A L Goldberger et alldquoMultifractality in human heartbeat dynamicsrdquo Naturevol 399 no 6735 pp 461ndash465 1999

[37] S Drozdz and P Oswiecimka ldquoDetecting and interpretingdistortions in hierarchical organization of complex time se-riesrdquo Physical Review E vol 91 Article ID 030902 2015

[38] C Ibarra-Valdez J Alvarez and J Alvarez-Ramirez ldquoRan-domness confidence bands of fractal scaling exponents forfinancial price returnsrdquo Chaos Solitons amp Fractals vol 83pp 119ndash124 2016

[39] D Stosic D Stosic T Stosic and H E Stanley ldquoMultifractalanalysis of managed and independent float exchange ratesrdquoPhysica A Statistical Mechanics and its Applications vol 428pp 13ndash18 2015

[40] K Matia Y Ashkenazy and H E Stanley ldquoMultifractalproperties of price fluctuations of stocks and commoditiesrdquoEurophysics Letters (EPL) vol 61 no 3 pp 422ndash428 2003

18 Complexity

ndash40

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

Before lawAfter law

(c)

Figure 10 Renyi exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu

ndash15

ndash1

ndash05

0

05

1

f(α)

1 15 2 25 305α

Before lawAfter law

(a)

1 15 2 25 305α

ndash15

ndash1

ndash05

0

05

1f(α

)

Before lawAfter law

(b)

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

f(α)

1 2 3 40α

Before lawAfter law

(c)

Figure 11 Holder exponent of PM25 concentration of (a) Seoul (b) Busan and (c) Daegu e triangle and circle symbols represent beforeand after the law respectively

Complexity 11

Table 3 the conclusion we described above about the Hurstexponent and the Renyi exponent is discussed Moreover asshown in Figure 5 the multifractal spectra are stronglyasymmetrical and as an important indicator of fractal or-ganization the asymmetry coefficient which is used toquantify the asymmetry of the multifractal spectrum hasbeen estimated in [37]

Based on Table 3 the multifractal features can be clearlyconfirmed for all time series and it can be seen that exceptfor spring the Hurst exponents and multifractal spectrumwidths of PM25 are obviously larger than those of NO2 andthe multifractal degrees of PM25 are stronger than those ofNO2

42 Implications of the ldquoHaze Special Lawrdquo In this sectionwe focus on the enactment of the ldquohaze special lawrdquo andstudy whether the law is effective in the governance of at-mospheric pollution We select daily data from the sameperiod described in Section 3 As we studied above themultifractal degree of PM25 is stronger than that of NO2us in this section we focus on the comparison of PM25concentrations We show the PM25 time series before andafter the law was implemented in Seoul Busan and Daegu inFigure 6

As the fluctuation statistics are of asymptotic feature asufficiently long-length sequence guarantees the accurateestimation of the Hurst exponent and for a small-lengthsequence the asymptotic value in the slope of ln(Fq(s))

versus ln(s) will be lacked [38] As the total number ofobservations is selected to be 305 to eliminate the doubt ofwhether the selected time series can give representativeresults for a multifractal analysis in the investigation wedealt with this concern by computing confidence intervals ofHurst exponent h(2) to check whether there is a differencecaused by the law for all the three cities e number ofsamples increased from 50 to 300 and every additionalinterval is 10 e confidence level is selected as 950 Foreach set of sample lengths we generate the shuffled timeseries 1000 times and then the confidence interval is plottedin Figure 7

As shown in Figure 7 and as expected the domain sizedecreases with an increase in the length of the time seriesHowever when the length of the time series is larger than 200the decrease in the domain size of the confidence intervals israther slow and of a distinct asymptotic behavior Besides we

see that the confidence intervals do not overlap for Seoulwhile for Busan ([034 039] before and [032 037] after) andDaegu ([055 064] before and [058 069] after) the confi-dence intervals overlap a lot erefore we can conclude thatthere is a difference caused by the law for Seoul however thesame conclusion is inconclusive for Busan and Daegu

Now we first depict the log-log plots of the fluctuationfunction Fq(s) versus q and s for the PM25 time series Asillustrated in Figure 8 for all series the fluctuation valueincreases linearly with s indicating that a power-law be-havior and long-range correlations exist in each series paire decreasing gradient shows that all time series havemultifractal characteristics

e generalized Hurst exponent h(q) Renyi exponentτ(q) and Holder exponent α of the time series before andafter the enactment of the ldquohaze special lawrdquo for all the threecities are shown in Figures 9ndash11 respectively

In Figure 9 it is shown that the generalized Hurst ex-ponents h(q) of PM25 of all three cities decrease with variedq thereby showing the multifractal behaviors of these twoseries We present h(2) for these series in Table 4 All Hurstexponents are confirmed to be larger than 05 which in-dicates that all the time series exhibit positive persistenceBesides we observe that h(2) before the implementation ofthe law is larger than after the law implying that the fluc-tuations before the law have more significant persistenceSubsequently we calculate the range of h(q) for each timeseries and list them in Table 4 We find that for all threecities ΔH(q) after the law is larger which indicates that themultifractal characteristics are stronger and that the mul-tifractalities of the time series before the law are more stable

e Renyi exponents τ(q) plotted in Figure 10 arenonlinear along q which provides further evidence of theexistence of multifractality We notice that the curvature ofthe Renyi exponents before the law are higher which impliesthat the time series before the implementation of the lawhave stronger multifractal characteristics All of these areconsistent with the results calculated from the generalizedHurst exponents

At last the multifractal spectrums of these time series areexamined We depict the curves between α and f(α) inFigure 11 We see that the multifractal spectra are not shownas points which indicates all the time series have multi-fractality en Δα are calculated and presented in Table 4From Table 4 it can be seen that ΔH(q) and Δα of the timeseries after the enactment of law are lower than those before

Table 4 Multifractality for PM25 concentration

City Multifractality Before law After law

Seoulh(2) 12270 12085ΔH(q) 17329 11410Δα 22414 13953

Busanh(2) 12905 11041ΔH(q) 13539 09565Δα 17463 12708

Daeguh(2) 11975 10878ΔH(q) 22910 19516Δα 29610 25043

12 Complexity

05

1

15

2

25

3

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

35

h(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

0

05

1

15

2

h(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

h(q)

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 12 Hurst exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 13

ndash30

ndash20

ndash10

0

10

τ(q)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash40

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 13 Renyi exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

14 Complexity

ndash3

ndash2

ndash1

0

1

f(α)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

1 2 3 40α

05 1 15 2 25 30α

05 1 15 2 25 30α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

05 1 15 2 25 30α

05 1 15 20α

05 1 15 2 250α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 14 Holder exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 15

the implementation of the law for all three cities implyingthat the multifractality of the time series becomes morestable with the measures taken by the government and thatthe efficiency of the ldquohaze special lawrdquo in three cities isdemonstrable this conclusion can be referred to [16 39] inwhich it was pointed out that the decrease in multifractalitymeans the efficiency is improved when some measures aretaken Moreover we also notice that the efficiency of theldquohaze special lawrdquo in Seoul is most obvious when comparedwith Busan and in last place is Daegu Interestingly theorder is the same as the scale and prosperity of the citiesWe can therefore conclude that the law has a higher effecton a larger city

43 Sources of Multifractal Features e two universallyacknowledged sources of multifractality are long-rangecorrelations and a fat-tailed distribution [7 40] Now weinvestigate the major source of multifractality of the PM25concentration series To check the contribution of long-range correlations and fat-tailed distribution quantitativelywe shuffle and phase-randomize the time series

Subsequently we calculate the Hurst exponent h(q) theRenyi exponent τ(q) and the Holder exponent Δα of theoriginal shuffled and phase-randomized time series of thethree cities As shown in Figures 12ndash14 all the time series arestrongly multifractal From Table 5 all h(2) of the originalseries are larger than those of the shuffled and phase-ran-domized time series which indicates that the persistence hasbeen moved after shuffling and phase-randomizationMoreover the original series of all three cities has thegreatest spectrum width Δα while the multifractality de-creased after shuffling and phase-randomizing the seriese findings can also be confirmed in Figures 12ndash14

To examine the main source of multifractality the firstand third rows of Figure 14 show that the multifractalspectra of phase-randomized series are the narrowest ones

which means that the main source of multifractality of theseries before and after the law in Seoul and Daegu are the fat-tailed distributions e second row in Figure 14 shows thatthe multifractal spectrum of the shuffled series is the nar-rowest which implies that the long-range correlation resultsin multifractality of the series before and after imple-mentation of the law in Busan

5 Conclusions

In this study we examined the multifractal characteristics ofPM25 and NO2 time series of all seasons We determinedthat multifractality existed in each season and the multi-fractal property of PM25 was stronger than that of NO2 ineach season We also validated the effectiveness of the ldquohazespecial lawrdquo which was implemented to improve the gov-ernance of air pollution in South Korea We checkedwhether there is a difference caused by the law for these citiesand calculated confidence intervals of Hurst exponent h(2)e results showed that the confidence intervals do notoverlap for Seoul while the confidence intervals overlap a lotfor Busan and Daegu indicating that there is a differencecaused by the law for Seoul however the same conclusion isinconclusive for Busan and Daegu By comparing the Hurstexponent Renyi exponent and Holder exponent the timeseries of the PM25 concentration before the implementationof the law was found to have a higher multifractal degreewhich decreased after the enactment of the law is phe-nomenon reflected the fact that the law has played a role inimproving the efficiency of air pollution control in SouthKorea We also concluded that the effect of the law will bemore significant in a larger city To explore the major causesof multifractality we shuffled and phase-randomized theoriginal series of PM25 By analyzing the width of themultifractal spectrum Δα the results showed that the fat-tailed distribution contributed to the multifractality of bothtimes series before and after the implementation of the ldquohazespecial lawrdquo in Seoul and Daegu whereas long-range cor-relations resulted in the multifractality of the series beforeand after the implementation of the law in Busan Weconcluded that the implementation of the law was verysuccessful and effective for the improvement of PM25 levelserefore we believe that the ldquohaze special lawrdquo is a possiblecause for the change in API in South Korea

Data Availability

e data will be available upon request with the corre-sponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e first author (Jian Wang) was supported by the ChinaScholarship Council (201808260026) J S Kim expressesgratitude for the support from the BK21 PLUS programe

Table 5 Multifractality of PM25 time series

h(2) Δα LRC FTD

Seoul

BeforeOriginal 12270 22414

No YesShuffled 07232 17946Phase-randomized 06965 09574

AfterOriginal 12085 13953

No YesShuffled 03177 10978Phase-randomized 03049 08569

Busan

BeforeOriginal 12905 17463

Yes NoShuffled 03982 09999Phase-randomized 03993 13512

AfterOriginal 11041 12708

Yes NoShuffled 03678 07175Phase-randomized 03794 08673

Daegu

BeforeOriginal 11975 29610

No YesShuffled 05901 22093Phase-randomized 05868 20587

AfterOriginal 10878 25043

No YesShuffled 06756 25600Phase-randomized 06407 22981

LRC and FTD represent the long-range correlations and fat-tailed distri-bution respectively

16 Complexity

corresponding author Wei Shao thanks Professor DaraeJeong for helpful discussion

References

[1] G Shaddick and J Wakefield ldquoModelling daily multivariatepollutant data at multiple sitesrdquo Journal of the Royal StatisticalSociety Series C (Applied Statistics) vol 51 no 3 pp 351ndash3722002

[2] M OrsquoDwyer Air Quality in Ireland 2010 Key Indicators ofAmbient Air Quality EPA Wexford Ireland 2011

[3] A Challoner and L Gill ldquoIndooroutdoor air pollution re-lationships in ten commercial buildings PM25 and NO2rdquoBuilding and Environment vol 80 pp 159ndash173 2014

[4] M Hazenkamp-von Arx T Gotschi U Ackermann-Liebrichet al ldquoPM25 and NO2 assessment in 21 European studycentres of ECRHS II annual means and seasonal differencesrdquoAtmospheric Environment vol 38 no 13 pp 1943ndash19532004

[5] H K Lai L Bayer-Oglesby R Colvile et al ldquoDeterminants ofindoor air concentrations of PM25 black smoke and NO2 insix European cities (EXPOLIS study)rdquo Atmospheric Envi-ronment vol 40 no 7 pp 1299ndash1313 2006

[6] K de Hoogh J Gulliver A V Donkelaar et al ldquoDevelopmentof West-European PM25 and NO2 land use regression modelsincorporating satellite-derived and chemical transport mod-elling datardquo Environmental Research vol 151 pp 1ndash10 2016

[7] S Drozdz J Kwapien P Oswiecimka and R Rak ldquoQuan-titative features of multifractal subtleties in time seriesrdquo EPL(Europhysics Letters) vol 88 no 6 p 60003 2010

[8] R J Buonocore T Aste and T Di Matteo ldquoMeasuringmultiscaling in financial time-seriesrdquo Chaos Solitons ampFractals vol 88 pp 38ndash47 2016

[9] P Oswiecimka S Drozdz J Kwapien and A Z GorskildquoFractals log-periodicity and financial crashesrdquo Acta PhysicaPolonica A vol 117 no 4 pp 637ndash639 2010

[10] B B Mandelbrot and N Goldenfeld ldquoFractals and scaling infinance discontinuity concentration riskrdquo Physics Todayvol 51 no 8 pp 59-60 1998

[11] S Drozdz R Kowalski P Oswiecimka R Rak andR Gebarowski ldquoDynamical variety of shapes in financialmultifractalityrdquo Complexity vol 2018 Article ID 701572113 pages 2018

[12] S Lahmiri ldquoMultifractal analysis of Moroccan family businessstock returnsrdquo Physica A Statistical Mechanics and Its Ap-plications vol 486 pp 183ndash191 2017

[13] X Zhao P Shang and Q Jin ldquoMultifractal detrended cross-correlation analysis of Chinese stock markets based on timedelayrdquo Fractals vol 19 no 03 pp 329ndash338 2011

[14] W Shao and J Wang ldquoDoes the ldquoice-breakingrdquo of South andNorth Korea affect the South Korean financial marketrdquoChaos Solitons amp Fractals vol 132 Article ID 109564 2020

[15] D-H Wang X-W Yu and Y-Y Suo ldquoStatistical propertiesof the yuan exchange rate indexrdquo Physica A Statistical Me-chanics and Its Applications vol 391 no 12 pp 3503ndash35122012

[16] Y Ning Y Wang and C-W Su ldquoHow did Chinarsquos foreignexchange reform affect the efficiency of foreign exchangemarketrdquo Physica A Statistical Mechanics and Its Applicationsvol 483 pp 219ndash226 2017

[17] S Lahmiri and S Bekiros ldquoChaos randomness and multi-fractality in bitcoin marketrdquo Chaos Solitons amp Fractalsvol 106 pp 28ndash34 2018

[18] P Manimaran and A C Narayana ldquoMultifractal detrendedcross-correlation analysis on air pollutants of University ofHyderabad Campus Indiardquo Physica A Statistical Mechanicsand Its Applications vol 502 pp 228ndash235 2018

[19] L Zhao W Li C Yang J Han Z Su and Y Zou ldquoMul-tifractality and network analysis of phase transitionrdquo PLoSOne vol 12 Article ID e0170467 2017

[20] N Kalamaras K Philippopoulos D Deligiorgi C G Tzanisand G Karvounis ldquoMultifractal scaling properties of daily airtemperature time seriesrdquo Chaos Solitons amp Fractals vol 98pp 38ndash43 2017

[21] J Li C Chen Q Yao et al ldquoe effect of circadian rhythm onthe correlation and multifractality of heart rate signals duringexerciserdquo Physica A Statistical Mechanics and Its Applica-tions vol 509 pp 1207ndash1213 2018

[22] P Castiglioni D Lazzeroni P Coruzzi and A FainildquoMultifractal-multiscale analysis of cardiovascular signals aDFA-based characterization of blood pressure and heart-ratecomplexity by genderrdquo Complexity vol 2018 Article ID4801924 14 pages 2018

[23] W Shi R-B Zou F Wang and L Su ldquoA new image seg-mentation method based on multifractal detrended movingaverage analysisrdquo Physica A Statistical Mechanics and ItsApplications vol 432 pp 197ndash205 2015

[24] F Wang Q Fan and H E Stanley ldquoMultiscale multifractaldetrended-fluctuation analysis of two-dimensional surfacesrdquoPhysical Review E vol 93 Article ID 042213 2016

[25] X Li M Zheng J Pu et al ldquoIdentification of abnormallyexpressed lncrnas induced by PM25 in human bronchialepithelial cellsrdquo Bioscience Reports vol 38 no 5 Article IDBSR20171577 2018

[26] K Shi C Liu and Y Huang ldquoMultifractal processes and self-organized criticality of PM25 during a typical haze period inChengdu Chinardquo Aerosol and Air Quality Research vol 15no 3 pp 926ndash934 2015

[27] C Zhang Z Ni and L Ni ldquoMultifractal detrended cross-correlation analysis between PM25 and meteorological fac-torsrdquo Physica A Statistical Mechanics and Its Applicationsvol 438 pp 114ndash123 2015

[28] C Zhang X Wang S Chen L Zou X Zhang and C TangldquoA study on daily PM25 concentrations in Hong Kong usingthe EMD-based MFDFA methodrdquo Physica A StatisticalMechanics and Its Applications vol 530 p 121182 2019

[29] H-D He ldquoMultifractal analysis of interactive patterns be-tween meteorological factors and pollutants in urban andrural areasrdquo Atmospheric Environment vol 149 pp 47ndash542017

[30] J W Kantelhardt S A Zschiegner E Koscielny-BundeS Havlin A Bunde and H E Stanley ldquoMultifractaldetrended fluctuation analysis of nonstationary time seriesrdquoPhysica A Statistical Mechanics and Its Applications vol 316no 1ndash4 pp 87ndash114 2002

[31] L Zunino B M Tabak A Figliola D G PerezM Garavaglia and O A Rosso ldquoA multifractal approach forstock market inefficiencyrdquo Physica A Statistical Mechanicsand Its Applications vol 387 no 26 pp 6558ndash6566 2008

[32] L Zunino A Figliola B M Tabak D G PerezM Garavaglia and O A Rosso ldquoMultifractal structure inLatin-American market indicesrdquo Chaos Solitons amp Fractalsvol 41 no 5 pp 2331ndash2340 2009

[33] B Lashermes P Abry and P Chainais ldquoNew insights into theestimation of scaling exponentsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 2no 4 pp 497ndash523 2004

Complexity 17

[34] E A F Ihlen ldquoIntroduction to multifractal detrended fluc-tuation analysis in Matlabrdquo Frontiers in Physiology vol 3p 141 2012

[35] L Jiang X Zhao N Li F Li and Z Guo ldquoDifferent mul-tifractal scaling of the 0 cm average ground surface tem-perature of four representative weather stations over ChinardquoAdvances inMeteorology vol 2013 Article ID 341934 8 pages2013

[36] P C Ivanov L A N Amaral A L Goldberger et alldquoMultifractality in human heartbeat dynamicsrdquo Naturevol 399 no 6735 pp 461ndash465 1999

[37] S Drozdz and P Oswiecimka ldquoDetecting and interpretingdistortions in hierarchical organization of complex time se-riesrdquo Physical Review E vol 91 Article ID 030902 2015

[38] C Ibarra-Valdez J Alvarez and J Alvarez-Ramirez ldquoRan-domness confidence bands of fractal scaling exponents forfinancial price returnsrdquo Chaos Solitons amp Fractals vol 83pp 119ndash124 2016

[39] D Stosic D Stosic T Stosic and H E Stanley ldquoMultifractalanalysis of managed and independent float exchange ratesrdquoPhysica A Statistical Mechanics and its Applications vol 428pp 13ndash18 2015

[40] K Matia Y Ashkenazy and H E Stanley ldquoMultifractalproperties of price fluctuations of stocks and commoditiesrdquoEurophysics Letters (EPL) vol 61 no 3 pp 422ndash428 2003

18 Complexity

Table 3 the conclusion we described above about the Hurstexponent and the Renyi exponent is discussed Moreover asshown in Figure 5 the multifractal spectra are stronglyasymmetrical and as an important indicator of fractal or-ganization the asymmetry coefficient which is used toquantify the asymmetry of the multifractal spectrum hasbeen estimated in [37]

Based on Table 3 the multifractal features can be clearlyconfirmed for all time series and it can be seen that exceptfor spring the Hurst exponents and multifractal spectrumwidths of PM25 are obviously larger than those of NO2 andthe multifractal degrees of PM25 are stronger than those ofNO2

42 Implications of the ldquoHaze Special Lawrdquo In this sectionwe focus on the enactment of the ldquohaze special lawrdquo andstudy whether the law is effective in the governance of at-mospheric pollution We select daily data from the sameperiod described in Section 3 As we studied above themultifractal degree of PM25 is stronger than that of NO2us in this section we focus on the comparison of PM25concentrations We show the PM25 time series before andafter the law was implemented in Seoul Busan and Daegu inFigure 6

As the fluctuation statistics are of asymptotic feature asufficiently long-length sequence guarantees the accurateestimation of the Hurst exponent and for a small-lengthsequence the asymptotic value in the slope of ln(Fq(s))

versus ln(s) will be lacked [38] As the total number ofobservations is selected to be 305 to eliminate the doubt ofwhether the selected time series can give representativeresults for a multifractal analysis in the investigation wedealt with this concern by computing confidence intervals ofHurst exponent h(2) to check whether there is a differencecaused by the law for all the three cities e number ofsamples increased from 50 to 300 and every additionalinterval is 10 e confidence level is selected as 950 Foreach set of sample lengths we generate the shuffled timeseries 1000 times and then the confidence interval is plottedin Figure 7

As shown in Figure 7 and as expected the domain sizedecreases with an increase in the length of the time seriesHowever when the length of the time series is larger than 200the decrease in the domain size of the confidence intervals israther slow and of a distinct asymptotic behavior Besides we

see that the confidence intervals do not overlap for Seoulwhile for Busan ([034 039] before and [032 037] after) andDaegu ([055 064] before and [058 069] after) the confi-dence intervals overlap a lot erefore we can conclude thatthere is a difference caused by the law for Seoul however thesame conclusion is inconclusive for Busan and Daegu

Now we first depict the log-log plots of the fluctuationfunction Fq(s) versus q and s for the PM25 time series Asillustrated in Figure 8 for all series the fluctuation valueincreases linearly with s indicating that a power-law be-havior and long-range correlations exist in each series paire decreasing gradient shows that all time series havemultifractal characteristics

e generalized Hurst exponent h(q) Renyi exponentτ(q) and Holder exponent α of the time series before andafter the enactment of the ldquohaze special lawrdquo for all the threecities are shown in Figures 9ndash11 respectively

In Figure 9 it is shown that the generalized Hurst ex-ponents h(q) of PM25 of all three cities decrease with variedq thereby showing the multifractal behaviors of these twoseries We present h(2) for these series in Table 4 All Hurstexponents are confirmed to be larger than 05 which in-dicates that all the time series exhibit positive persistenceBesides we observe that h(2) before the implementation ofthe law is larger than after the law implying that the fluc-tuations before the law have more significant persistenceSubsequently we calculate the range of h(q) for each timeseries and list them in Table 4 We find that for all threecities ΔH(q) after the law is larger which indicates that themultifractal characteristics are stronger and that the mul-tifractalities of the time series before the law are more stable

e Renyi exponents τ(q) plotted in Figure 10 arenonlinear along q which provides further evidence of theexistence of multifractality We notice that the curvature ofthe Renyi exponents before the law are higher which impliesthat the time series before the implementation of the lawhave stronger multifractal characteristics All of these areconsistent with the results calculated from the generalizedHurst exponents

At last the multifractal spectrums of these time series areexamined We depict the curves between α and f(α) inFigure 11 We see that the multifractal spectra are not shownas points which indicates all the time series have multi-fractality en Δα are calculated and presented in Table 4From Table 4 it can be seen that ΔH(q) and Δα of the timeseries after the enactment of law are lower than those before

Table 4 Multifractality for PM25 concentration

City Multifractality Before law After law

Seoulh(2) 12270 12085ΔH(q) 17329 11410Δα 22414 13953

Busanh(2) 12905 11041ΔH(q) 13539 09565Δα 17463 12708

Daeguh(2) 11975 10878ΔH(q) 22910 19516Δα 29610 25043

12 Complexity

05

1

15

2

25

3

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

35

h(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

0

05

1

15

2

h(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

h(q)

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 12 Hurst exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 13

ndash30

ndash20

ndash10

0

10

τ(q)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash40

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 13 Renyi exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

14 Complexity

ndash3

ndash2

ndash1

0

1

f(α)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

1 2 3 40α

05 1 15 2 25 30α

05 1 15 2 25 30α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

05 1 15 2 25 30α

05 1 15 20α

05 1 15 2 250α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 14 Holder exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 15

the implementation of the law for all three cities implyingthat the multifractality of the time series becomes morestable with the measures taken by the government and thatthe efficiency of the ldquohaze special lawrdquo in three cities isdemonstrable this conclusion can be referred to [16 39] inwhich it was pointed out that the decrease in multifractalitymeans the efficiency is improved when some measures aretaken Moreover we also notice that the efficiency of theldquohaze special lawrdquo in Seoul is most obvious when comparedwith Busan and in last place is Daegu Interestingly theorder is the same as the scale and prosperity of the citiesWe can therefore conclude that the law has a higher effecton a larger city

43 Sources of Multifractal Features e two universallyacknowledged sources of multifractality are long-rangecorrelations and a fat-tailed distribution [7 40] Now weinvestigate the major source of multifractality of the PM25concentration series To check the contribution of long-range correlations and fat-tailed distribution quantitativelywe shuffle and phase-randomize the time series

Subsequently we calculate the Hurst exponent h(q) theRenyi exponent τ(q) and the Holder exponent Δα of theoriginal shuffled and phase-randomized time series of thethree cities As shown in Figures 12ndash14 all the time series arestrongly multifractal From Table 5 all h(2) of the originalseries are larger than those of the shuffled and phase-ran-domized time series which indicates that the persistence hasbeen moved after shuffling and phase-randomizationMoreover the original series of all three cities has thegreatest spectrum width Δα while the multifractality de-creased after shuffling and phase-randomizing the seriese findings can also be confirmed in Figures 12ndash14

To examine the main source of multifractality the firstand third rows of Figure 14 show that the multifractalspectra of phase-randomized series are the narrowest ones

which means that the main source of multifractality of theseries before and after the law in Seoul and Daegu are the fat-tailed distributions e second row in Figure 14 shows thatthe multifractal spectrum of the shuffled series is the nar-rowest which implies that the long-range correlation resultsin multifractality of the series before and after imple-mentation of the law in Busan

5 Conclusions

In this study we examined the multifractal characteristics ofPM25 and NO2 time series of all seasons We determinedthat multifractality existed in each season and the multi-fractal property of PM25 was stronger than that of NO2 ineach season We also validated the effectiveness of the ldquohazespecial lawrdquo which was implemented to improve the gov-ernance of air pollution in South Korea We checkedwhether there is a difference caused by the law for these citiesand calculated confidence intervals of Hurst exponent h(2)e results showed that the confidence intervals do notoverlap for Seoul while the confidence intervals overlap a lotfor Busan and Daegu indicating that there is a differencecaused by the law for Seoul however the same conclusion isinconclusive for Busan and Daegu By comparing the Hurstexponent Renyi exponent and Holder exponent the timeseries of the PM25 concentration before the implementationof the law was found to have a higher multifractal degreewhich decreased after the enactment of the law is phe-nomenon reflected the fact that the law has played a role inimproving the efficiency of air pollution control in SouthKorea We also concluded that the effect of the law will bemore significant in a larger city To explore the major causesof multifractality we shuffled and phase-randomized theoriginal series of PM25 By analyzing the width of themultifractal spectrum Δα the results showed that the fat-tailed distribution contributed to the multifractality of bothtimes series before and after the implementation of the ldquohazespecial lawrdquo in Seoul and Daegu whereas long-range cor-relations resulted in the multifractality of the series beforeand after the implementation of the law in Busan Weconcluded that the implementation of the law was verysuccessful and effective for the improvement of PM25 levelserefore we believe that the ldquohaze special lawrdquo is a possiblecause for the change in API in South Korea

Data Availability

e data will be available upon request with the corre-sponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e first author (Jian Wang) was supported by the ChinaScholarship Council (201808260026) J S Kim expressesgratitude for the support from the BK21 PLUS programe

Table 5 Multifractality of PM25 time series

h(2) Δα LRC FTD

Seoul

BeforeOriginal 12270 22414

No YesShuffled 07232 17946Phase-randomized 06965 09574

AfterOriginal 12085 13953

No YesShuffled 03177 10978Phase-randomized 03049 08569

Busan

BeforeOriginal 12905 17463

Yes NoShuffled 03982 09999Phase-randomized 03993 13512

AfterOriginal 11041 12708

Yes NoShuffled 03678 07175Phase-randomized 03794 08673

Daegu

BeforeOriginal 11975 29610

No YesShuffled 05901 22093Phase-randomized 05868 20587

AfterOriginal 10878 25043

No YesShuffled 06756 25600Phase-randomized 06407 22981

LRC and FTD represent the long-range correlations and fat-tailed distri-bution respectively

16 Complexity

corresponding author Wei Shao thanks Professor DaraeJeong for helpful discussion

References

[1] G Shaddick and J Wakefield ldquoModelling daily multivariatepollutant data at multiple sitesrdquo Journal of the Royal StatisticalSociety Series C (Applied Statistics) vol 51 no 3 pp 351ndash3722002

[2] M OrsquoDwyer Air Quality in Ireland 2010 Key Indicators ofAmbient Air Quality EPA Wexford Ireland 2011

[3] A Challoner and L Gill ldquoIndooroutdoor air pollution re-lationships in ten commercial buildings PM25 and NO2rdquoBuilding and Environment vol 80 pp 159ndash173 2014

[4] M Hazenkamp-von Arx T Gotschi U Ackermann-Liebrichet al ldquoPM25 and NO2 assessment in 21 European studycentres of ECRHS II annual means and seasonal differencesrdquoAtmospheric Environment vol 38 no 13 pp 1943ndash19532004

[5] H K Lai L Bayer-Oglesby R Colvile et al ldquoDeterminants ofindoor air concentrations of PM25 black smoke and NO2 insix European cities (EXPOLIS study)rdquo Atmospheric Envi-ronment vol 40 no 7 pp 1299ndash1313 2006

[6] K de Hoogh J Gulliver A V Donkelaar et al ldquoDevelopmentof West-European PM25 and NO2 land use regression modelsincorporating satellite-derived and chemical transport mod-elling datardquo Environmental Research vol 151 pp 1ndash10 2016

[7] S Drozdz J Kwapien P Oswiecimka and R Rak ldquoQuan-titative features of multifractal subtleties in time seriesrdquo EPL(Europhysics Letters) vol 88 no 6 p 60003 2010

[8] R J Buonocore T Aste and T Di Matteo ldquoMeasuringmultiscaling in financial time-seriesrdquo Chaos Solitons ampFractals vol 88 pp 38ndash47 2016

[9] P Oswiecimka S Drozdz J Kwapien and A Z GorskildquoFractals log-periodicity and financial crashesrdquo Acta PhysicaPolonica A vol 117 no 4 pp 637ndash639 2010

[10] B B Mandelbrot and N Goldenfeld ldquoFractals and scaling infinance discontinuity concentration riskrdquo Physics Todayvol 51 no 8 pp 59-60 1998

[11] S Drozdz R Kowalski P Oswiecimka R Rak andR Gebarowski ldquoDynamical variety of shapes in financialmultifractalityrdquo Complexity vol 2018 Article ID 701572113 pages 2018

[12] S Lahmiri ldquoMultifractal analysis of Moroccan family businessstock returnsrdquo Physica A Statistical Mechanics and Its Ap-plications vol 486 pp 183ndash191 2017

[13] X Zhao P Shang and Q Jin ldquoMultifractal detrended cross-correlation analysis of Chinese stock markets based on timedelayrdquo Fractals vol 19 no 03 pp 329ndash338 2011

[14] W Shao and J Wang ldquoDoes the ldquoice-breakingrdquo of South andNorth Korea affect the South Korean financial marketrdquoChaos Solitons amp Fractals vol 132 Article ID 109564 2020

[15] D-H Wang X-W Yu and Y-Y Suo ldquoStatistical propertiesof the yuan exchange rate indexrdquo Physica A Statistical Me-chanics and Its Applications vol 391 no 12 pp 3503ndash35122012

[16] Y Ning Y Wang and C-W Su ldquoHow did Chinarsquos foreignexchange reform affect the efficiency of foreign exchangemarketrdquo Physica A Statistical Mechanics and Its Applicationsvol 483 pp 219ndash226 2017

[17] S Lahmiri and S Bekiros ldquoChaos randomness and multi-fractality in bitcoin marketrdquo Chaos Solitons amp Fractalsvol 106 pp 28ndash34 2018

[18] P Manimaran and A C Narayana ldquoMultifractal detrendedcross-correlation analysis on air pollutants of University ofHyderabad Campus Indiardquo Physica A Statistical Mechanicsand Its Applications vol 502 pp 228ndash235 2018

[19] L Zhao W Li C Yang J Han Z Su and Y Zou ldquoMul-tifractality and network analysis of phase transitionrdquo PLoSOne vol 12 Article ID e0170467 2017

[20] N Kalamaras K Philippopoulos D Deligiorgi C G Tzanisand G Karvounis ldquoMultifractal scaling properties of daily airtemperature time seriesrdquo Chaos Solitons amp Fractals vol 98pp 38ndash43 2017

[21] J Li C Chen Q Yao et al ldquoe effect of circadian rhythm onthe correlation and multifractality of heart rate signals duringexerciserdquo Physica A Statistical Mechanics and Its Applica-tions vol 509 pp 1207ndash1213 2018

[22] P Castiglioni D Lazzeroni P Coruzzi and A FainildquoMultifractal-multiscale analysis of cardiovascular signals aDFA-based characterization of blood pressure and heart-ratecomplexity by genderrdquo Complexity vol 2018 Article ID4801924 14 pages 2018

[23] W Shi R-B Zou F Wang and L Su ldquoA new image seg-mentation method based on multifractal detrended movingaverage analysisrdquo Physica A Statistical Mechanics and ItsApplications vol 432 pp 197ndash205 2015

[24] F Wang Q Fan and H E Stanley ldquoMultiscale multifractaldetrended-fluctuation analysis of two-dimensional surfacesrdquoPhysical Review E vol 93 Article ID 042213 2016

[25] X Li M Zheng J Pu et al ldquoIdentification of abnormallyexpressed lncrnas induced by PM25 in human bronchialepithelial cellsrdquo Bioscience Reports vol 38 no 5 Article IDBSR20171577 2018

[26] K Shi C Liu and Y Huang ldquoMultifractal processes and self-organized criticality of PM25 during a typical haze period inChengdu Chinardquo Aerosol and Air Quality Research vol 15no 3 pp 926ndash934 2015

[27] C Zhang Z Ni and L Ni ldquoMultifractal detrended cross-correlation analysis between PM25 and meteorological fac-torsrdquo Physica A Statistical Mechanics and Its Applicationsvol 438 pp 114ndash123 2015

[28] C Zhang X Wang S Chen L Zou X Zhang and C TangldquoA study on daily PM25 concentrations in Hong Kong usingthe EMD-based MFDFA methodrdquo Physica A StatisticalMechanics and Its Applications vol 530 p 121182 2019

[29] H-D He ldquoMultifractal analysis of interactive patterns be-tween meteorological factors and pollutants in urban andrural areasrdquo Atmospheric Environment vol 149 pp 47ndash542017

[30] J W Kantelhardt S A Zschiegner E Koscielny-BundeS Havlin A Bunde and H E Stanley ldquoMultifractaldetrended fluctuation analysis of nonstationary time seriesrdquoPhysica A Statistical Mechanics and Its Applications vol 316no 1ndash4 pp 87ndash114 2002

[31] L Zunino B M Tabak A Figliola D G PerezM Garavaglia and O A Rosso ldquoA multifractal approach forstock market inefficiencyrdquo Physica A Statistical Mechanicsand Its Applications vol 387 no 26 pp 6558ndash6566 2008

[32] L Zunino A Figliola B M Tabak D G PerezM Garavaglia and O A Rosso ldquoMultifractal structure inLatin-American market indicesrdquo Chaos Solitons amp Fractalsvol 41 no 5 pp 2331ndash2340 2009

[33] B Lashermes P Abry and P Chainais ldquoNew insights into theestimation of scaling exponentsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 2no 4 pp 497ndash523 2004

Complexity 17

[34] E A F Ihlen ldquoIntroduction to multifractal detrended fluc-tuation analysis in Matlabrdquo Frontiers in Physiology vol 3p 141 2012

[35] L Jiang X Zhao N Li F Li and Z Guo ldquoDifferent mul-tifractal scaling of the 0 cm average ground surface tem-perature of four representative weather stations over ChinardquoAdvances inMeteorology vol 2013 Article ID 341934 8 pages2013

[36] P C Ivanov L A N Amaral A L Goldberger et alldquoMultifractality in human heartbeat dynamicsrdquo Naturevol 399 no 6735 pp 461ndash465 1999

[37] S Drozdz and P Oswiecimka ldquoDetecting and interpretingdistortions in hierarchical organization of complex time se-riesrdquo Physical Review E vol 91 Article ID 030902 2015

[38] C Ibarra-Valdez J Alvarez and J Alvarez-Ramirez ldquoRan-domness confidence bands of fractal scaling exponents forfinancial price returnsrdquo Chaos Solitons amp Fractals vol 83pp 119ndash124 2016

[39] D Stosic D Stosic T Stosic and H E Stanley ldquoMultifractalanalysis of managed and independent float exchange ratesrdquoPhysica A Statistical Mechanics and its Applications vol 428pp 13ndash18 2015

[40] K Matia Y Ashkenazy and H E Stanley ldquoMultifractalproperties of price fluctuations of stocks and commoditiesrdquoEurophysics Letters (EPL) vol 61 no 3 pp 422ndash428 2003

18 Complexity

05

1

15

2

25

3

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

35

h(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash5 0 5 10ndash10q

0

05

1

15

2

25

h(q)

0

05

1

15

2

h(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

0

05

1

15

2

25

3

h(q)

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 12 Hurst exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 13

ndash30

ndash20

ndash10

0

10

τ(q)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash40

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 13 Renyi exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

14 Complexity

ndash3

ndash2

ndash1

0

1

f(α)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

1 2 3 40α

05 1 15 2 25 30α

05 1 15 2 25 30α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

05 1 15 2 25 30α

05 1 15 20α

05 1 15 2 250α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 14 Holder exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 15

the implementation of the law for all three cities implyingthat the multifractality of the time series becomes morestable with the measures taken by the government and thatthe efficiency of the ldquohaze special lawrdquo in three cities isdemonstrable this conclusion can be referred to [16 39] inwhich it was pointed out that the decrease in multifractalitymeans the efficiency is improved when some measures aretaken Moreover we also notice that the efficiency of theldquohaze special lawrdquo in Seoul is most obvious when comparedwith Busan and in last place is Daegu Interestingly theorder is the same as the scale and prosperity of the citiesWe can therefore conclude that the law has a higher effecton a larger city

43 Sources of Multifractal Features e two universallyacknowledged sources of multifractality are long-rangecorrelations and a fat-tailed distribution [7 40] Now weinvestigate the major source of multifractality of the PM25concentration series To check the contribution of long-range correlations and fat-tailed distribution quantitativelywe shuffle and phase-randomize the time series

Subsequently we calculate the Hurst exponent h(q) theRenyi exponent τ(q) and the Holder exponent Δα of theoriginal shuffled and phase-randomized time series of thethree cities As shown in Figures 12ndash14 all the time series arestrongly multifractal From Table 5 all h(2) of the originalseries are larger than those of the shuffled and phase-ran-domized time series which indicates that the persistence hasbeen moved after shuffling and phase-randomizationMoreover the original series of all three cities has thegreatest spectrum width Δα while the multifractality de-creased after shuffling and phase-randomizing the seriese findings can also be confirmed in Figures 12ndash14

To examine the main source of multifractality the firstand third rows of Figure 14 show that the multifractalspectra of phase-randomized series are the narrowest ones

which means that the main source of multifractality of theseries before and after the law in Seoul and Daegu are the fat-tailed distributions e second row in Figure 14 shows thatthe multifractal spectrum of the shuffled series is the nar-rowest which implies that the long-range correlation resultsin multifractality of the series before and after imple-mentation of the law in Busan

5 Conclusions

In this study we examined the multifractal characteristics ofPM25 and NO2 time series of all seasons We determinedthat multifractality existed in each season and the multi-fractal property of PM25 was stronger than that of NO2 ineach season We also validated the effectiveness of the ldquohazespecial lawrdquo which was implemented to improve the gov-ernance of air pollution in South Korea We checkedwhether there is a difference caused by the law for these citiesand calculated confidence intervals of Hurst exponent h(2)e results showed that the confidence intervals do notoverlap for Seoul while the confidence intervals overlap a lotfor Busan and Daegu indicating that there is a differencecaused by the law for Seoul however the same conclusion isinconclusive for Busan and Daegu By comparing the Hurstexponent Renyi exponent and Holder exponent the timeseries of the PM25 concentration before the implementationof the law was found to have a higher multifractal degreewhich decreased after the enactment of the law is phe-nomenon reflected the fact that the law has played a role inimproving the efficiency of air pollution control in SouthKorea We also concluded that the effect of the law will bemore significant in a larger city To explore the major causesof multifractality we shuffled and phase-randomized theoriginal series of PM25 By analyzing the width of themultifractal spectrum Δα the results showed that the fat-tailed distribution contributed to the multifractality of bothtimes series before and after the implementation of the ldquohazespecial lawrdquo in Seoul and Daegu whereas long-range cor-relations resulted in the multifractality of the series beforeand after the implementation of the law in Busan Weconcluded that the implementation of the law was verysuccessful and effective for the improvement of PM25 levelserefore we believe that the ldquohaze special lawrdquo is a possiblecause for the change in API in South Korea

Data Availability

e data will be available upon request with the corre-sponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e first author (Jian Wang) was supported by the ChinaScholarship Council (201808260026) J S Kim expressesgratitude for the support from the BK21 PLUS programe

Table 5 Multifractality of PM25 time series

h(2) Δα LRC FTD

Seoul

BeforeOriginal 12270 22414

No YesShuffled 07232 17946Phase-randomized 06965 09574

AfterOriginal 12085 13953

No YesShuffled 03177 10978Phase-randomized 03049 08569

Busan

BeforeOriginal 12905 17463

Yes NoShuffled 03982 09999Phase-randomized 03993 13512

AfterOriginal 11041 12708

Yes NoShuffled 03678 07175Phase-randomized 03794 08673

Daegu

BeforeOriginal 11975 29610

No YesShuffled 05901 22093Phase-randomized 05868 20587

AfterOriginal 10878 25043

No YesShuffled 06756 25600Phase-randomized 06407 22981

LRC and FTD represent the long-range correlations and fat-tailed distri-bution respectively

16 Complexity

corresponding author Wei Shao thanks Professor DaraeJeong for helpful discussion

References

[1] G Shaddick and J Wakefield ldquoModelling daily multivariatepollutant data at multiple sitesrdquo Journal of the Royal StatisticalSociety Series C (Applied Statistics) vol 51 no 3 pp 351ndash3722002

[2] M OrsquoDwyer Air Quality in Ireland 2010 Key Indicators ofAmbient Air Quality EPA Wexford Ireland 2011

[3] A Challoner and L Gill ldquoIndooroutdoor air pollution re-lationships in ten commercial buildings PM25 and NO2rdquoBuilding and Environment vol 80 pp 159ndash173 2014

[4] M Hazenkamp-von Arx T Gotschi U Ackermann-Liebrichet al ldquoPM25 and NO2 assessment in 21 European studycentres of ECRHS II annual means and seasonal differencesrdquoAtmospheric Environment vol 38 no 13 pp 1943ndash19532004

[5] H K Lai L Bayer-Oglesby R Colvile et al ldquoDeterminants ofindoor air concentrations of PM25 black smoke and NO2 insix European cities (EXPOLIS study)rdquo Atmospheric Envi-ronment vol 40 no 7 pp 1299ndash1313 2006

[6] K de Hoogh J Gulliver A V Donkelaar et al ldquoDevelopmentof West-European PM25 and NO2 land use regression modelsincorporating satellite-derived and chemical transport mod-elling datardquo Environmental Research vol 151 pp 1ndash10 2016

[7] S Drozdz J Kwapien P Oswiecimka and R Rak ldquoQuan-titative features of multifractal subtleties in time seriesrdquo EPL(Europhysics Letters) vol 88 no 6 p 60003 2010

[8] R J Buonocore T Aste and T Di Matteo ldquoMeasuringmultiscaling in financial time-seriesrdquo Chaos Solitons ampFractals vol 88 pp 38ndash47 2016

[9] P Oswiecimka S Drozdz J Kwapien and A Z GorskildquoFractals log-periodicity and financial crashesrdquo Acta PhysicaPolonica A vol 117 no 4 pp 637ndash639 2010

[10] B B Mandelbrot and N Goldenfeld ldquoFractals and scaling infinance discontinuity concentration riskrdquo Physics Todayvol 51 no 8 pp 59-60 1998

[11] S Drozdz R Kowalski P Oswiecimka R Rak andR Gebarowski ldquoDynamical variety of shapes in financialmultifractalityrdquo Complexity vol 2018 Article ID 701572113 pages 2018

[12] S Lahmiri ldquoMultifractal analysis of Moroccan family businessstock returnsrdquo Physica A Statistical Mechanics and Its Ap-plications vol 486 pp 183ndash191 2017

[13] X Zhao P Shang and Q Jin ldquoMultifractal detrended cross-correlation analysis of Chinese stock markets based on timedelayrdquo Fractals vol 19 no 03 pp 329ndash338 2011

[14] W Shao and J Wang ldquoDoes the ldquoice-breakingrdquo of South andNorth Korea affect the South Korean financial marketrdquoChaos Solitons amp Fractals vol 132 Article ID 109564 2020

[15] D-H Wang X-W Yu and Y-Y Suo ldquoStatistical propertiesof the yuan exchange rate indexrdquo Physica A Statistical Me-chanics and Its Applications vol 391 no 12 pp 3503ndash35122012

[16] Y Ning Y Wang and C-W Su ldquoHow did Chinarsquos foreignexchange reform affect the efficiency of foreign exchangemarketrdquo Physica A Statistical Mechanics and Its Applicationsvol 483 pp 219ndash226 2017

[17] S Lahmiri and S Bekiros ldquoChaos randomness and multi-fractality in bitcoin marketrdquo Chaos Solitons amp Fractalsvol 106 pp 28ndash34 2018

[18] P Manimaran and A C Narayana ldquoMultifractal detrendedcross-correlation analysis on air pollutants of University ofHyderabad Campus Indiardquo Physica A Statistical Mechanicsand Its Applications vol 502 pp 228ndash235 2018

[19] L Zhao W Li C Yang J Han Z Su and Y Zou ldquoMul-tifractality and network analysis of phase transitionrdquo PLoSOne vol 12 Article ID e0170467 2017

[20] N Kalamaras K Philippopoulos D Deligiorgi C G Tzanisand G Karvounis ldquoMultifractal scaling properties of daily airtemperature time seriesrdquo Chaos Solitons amp Fractals vol 98pp 38ndash43 2017

[21] J Li C Chen Q Yao et al ldquoe effect of circadian rhythm onthe correlation and multifractality of heart rate signals duringexerciserdquo Physica A Statistical Mechanics and Its Applica-tions vol 509 pp 1207ndash1213 2018

[22] P Castiglioni D Lazzeroni P Coruzzi and A FainildquoMultifractal-multiscale analysis of cardiovascular signals aDFA-based characterization of blood pressure and heart-ratecomplexity by genderrdquo Complexity vol 2018 Article ID4801924 14 pages 2018

[23] W Shi R-B Zou F Wang and L Su ldquoA new image seg-mentation method based on multifractal detrended movingaverage analysisrdquo Physica A Statistical Mechanics and ItsApplications vol 432 pp 197ndash205 2015

[24] F Wang Q Fan and H E Stanley ldquoMultiscale multifractaldetrended-fluctuation analysis of two-dimensional surfacesrdquoPhysical Review E vol 93 Article ID 042213 2016

[25] X Li M Zheng J Pu et al ldquoIdentification of abnormallyexpressed lncrnas induced by PM25 in human bronchialepithelial cellsrdquo Bioscience Reports vol 38 no 5 Article IDBSR20171577 2018

[26] K Shi C Liu and Y Huang ldquoMultifractal processes and self-organized criticality of PM25 during a typical haze period inChengdu Chinardquo Aerosol and Air Quality Research vol 15no 3 pp 926ndash934 2015

[27] C Zhang Z Ni and L Ni ldquoMultifractal detrended cross-correlation analysis between PM25 and meteorological fac-torsrdquo Physica A Statistical Mechanics and Its Applicationsvol 438 pp 114ndash123 2015

[28] C Zhang X Wang S Chen L Zou X Zhang and C TangldquoA study on daily PM25 concentrations in Hong Kong usingthe EMD-based MFDFA methodrdquo Physica A StatisticalMechanics and Its Applications vol 530 p 121182 2019

[29] H-D He ldquoMultifractal analysis of interactive patterns be-tween meteorological factors and pollutants in urban andrural areasrdquo Atmospheric Environment vol 149 pp 47ndash542017

[30] J W Kantelhardt S A Zschiegner E Koscielny-BundeS Havlin A Bunde and H E Stanley ldquoMultifractaldetrended fluctuation analysis of nonstationary time seriesrdquoPhysica A Statistical Mechanics and Its Applications vol 316no 1ndash4 pp 87ndash114 2002

[31] L Zunino B M Tabak A Figliola D G PerezM Garavaglia and O A Rosso ldquoA multifractal approach forstock market inefficiencyrdquo Physica A Statistical Mechanicsand Its Applications vol 387 no 26 pp 6558ndash6566 2008

[32] L Zunino A Figliola B M Tabak D G PerezM Garavaglia and O A Rosso ldquoMultifractal structure inLatin-American market indicesrdquo Chaos Solitons amp Fractalsvol 41 no 5 pp 2331ndash2340 2009

[33] B Lashermes P Abry and P Chainais ldquoNew insights into theestimation of scaling exponentsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 2no 4 pp 497ndash523 2004

Complexity 17

[34] E A F Ihlen ldquoIntroduction to multifractal detrended fluc-tuation analysis in Matlabrdquo Frontiers in Physiology vol 3p 141 2012

[35] L Jiang X Zhao N Li F Li and Z Guo ldquoDifferent mul-tifractal scaling of the 0 cm average ground surface tem-perature of four representative weather stations over ChinardquoAdvances inMeteorology vol 2013 Article ID 341934 8 pages2013

[36] P C Ivanov L A N Amaral A L Goldberger et alldquoMultifractality in human heartbeat dynamicsrdquo Naturevol 399 no 6735 pp 461ndash465 1999

[37] S Drozdz and P Oswiecimka ldquoDetecting and interpretingdistortions in hierarchical organization of complex time se-riesrdquo Physical Review E vol 91 Article ID 030902 2015

[38] C Ibarra-Valdez J Alvarez and J Alvarez-Ramirez ldquoRan-domness confidence bands of fractal scaling exponents forfinancial price returnsrdquo Chaos Solitons amp Fractals vol 83pp 119ndash124 2016

[39] D Stosic D Stosic T Stosic and H E Stanley ldquoMultifractalanalysis of managed and independent float exchange ratesrdquoPhysica A Statistical Mechanics and its Applications vol 428pp 13ndash18 2015

[40] K Matia Y Ashkenazy and H E Stanley ldquoMultifractalproperties of price fluctuations of stocks and commoditiesrdquoEurophysics Letters (EPL) vol 61 no 3 pp 422ndash428 2003

18 Complexity

ndash30

ndash20

ndash10

0

10

τ(q)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash40

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash5 0 5 10ndash10q

ndash5 0 5 10ndash10q

ndash20

ndash15

ndash10

ndash5

0

5

10

τ(q)

ndash30

ndash20

ndash10

0

10

τ(q)

ndash5 0 5 10ndash10q

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 13 Renyi exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

14 Complexity

ndash3

ndash2

ndash1

0

1

f(α)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

1 2 3 40α

05 1 15 2 25 30α

05 1 15 2 25 30α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

05 1 15 2 25 30α

05 1 15 20α

05 1 15 2 250α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 14 Holder exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 15

the implementation of the law for all three cities implyingthat the multifractality of the time series becomes morestable with the measures taken by the government and thatthe efficiency of the ldquohaze special lawrdquo in three cities isdemonstrable this conclusion can be referred to [16 39] inwhich it was pointed out that the decrease in multifractalitymeans the efficiency is improved when some measures aretaken Moreover we also notice that the efficiency of theldquohaze special lawrdquo in Seoul is most obvious when comparedwith Busan and in last place is Daegu Interestingly theorder is the same as the scale and prosperity of the citiesWe can therefore conclude that the law has a higher effecton a larger city

43 Sources of Multifractal Features e two universallyacknowledged sources of multifractality are long-rangecorrelations and a fat-tailed distribution [7 40] Now weinvestigate the major source of multifractality of the PM25concentration series To check the contribution of long-range correlations and fat-tailed distribution quantitativelywe shuffle and phase-randomize the time series

Subsequently we calculate the Hurst exponent h(q) theRenyi exponent τ(q) and the Holder exponent Δα of theoriginal shuffled and phase-randomized time series of thethree cities As shown in Figures 12ndash14 all the time series arestrongly multifractal From Table 5 all h(2) of the originalseries are larger than those of the shuffled and phase-ran-domized time series which indicates that the persistence hasbeen moved after shuffling and phase-randomizationMoreover the original series of all three cities has thegreatest spectrum width Δα while the multifractality de-creased after shuffling and phase-randomizing the seriese findings can also be confirmed in Figures 12ndash14

To examine the main source of multifractality the firstand third rows of Figure 14 show that the multifractalspectra of phase-randomized series are the narrowest ones

which means that the main source of multifractality of theseries before and after the law in Seoul and Daegu are the fat-tailed distributions e second row in Figure 14 shows thatthe multifractal spectrum of the shuffled series is the nar-rowest which implies that the long-range correlation resultsin multifractality of the series before and after imple-mentation of the law in Busan

5 Conclusions

In this study we examined the multifractal characteristics ofPM25 and NO2 time series of all seasons We determinedthat multifractality existed in each season and the multi-fractal property of PM25 was stronger than that of NO2 ineach season We also validated the effectiveness of the ldquohazespecial lawrdquo which was implemented to improve the gov-ernance of air pollution in South Korea We checkedwhether there is a difference caused by the law for these citiesand calculated confidence intervals of Hurst exponent h(2)e results showed that the confidence intervals do notoverlap for Seoul while the confidence intervals overlap a lotfor Busan and Daegu indicating that there is a differencecaused by the law for Seoul however the same conclusion isinconclusive for Busan and Daegu By comparing the Hurstexponent Renyi exponent and Holder exponent the timeseries of the PM25 concentration before the implementationof the law was found to have a higher multifractal degreewhich decreased after the enactment of the law is phe-nomenon reflected the fact that the law has played a role inimproving the efficiency of air pollution control in SouthKorea We also concluded that the effect of the law will bemore significant in a larger city To explore the major causesof multifractality we shuffled and phase-randomized theoriginal series of PM25 By analyzing the width of themultifractal spectrum Δα the results showed that the fat-tailed distribution contributed to the multifractality of bothtimes series before and after the implementation of the ldquohazespecial lawrdquo in Seoul and Daegu whereas long-range cor-relations resulted in the multifractality of the series beforeand after the implementation of the law in Busan Weconcluded that the implementation of the law was verysuccessful and effective for the improvement of PM25 levelserefore we believe that the ldquohaze special lawrdquo is a possiblecause for the change in API in South Korea

Data Availability

e data will be available upon request with the corre-sponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e first author (Jian Wang) was supported by the ChinaScholarship Council (201808260026) J S Kim expressesgratitude for the support from the BK21 PLUS programe

Table 5 Multifractality of PM25 time series

h(2) Δα LRC FTD

Seoul

BeforeOriginal 12270 22414

No YesShuffled 07232 17946Phase-randomized 06965 09574

AfterOriginal 12085 13953

No YesShuffled 03177 10978Phase-randomized 03049 08569

Busan

BeforeOriginal 12905 17463

Yes NoShuffled 03982 09999Phase-randomized 03993 13512

AfterOriginal 11041 12708

Yes NoShuffled 03678 07175Phase-randomized 03794 08673

Daegu

BeforeOriginal 11975 29610

No YesShuffled 05901 22093Phase-randomized 05868 20587

AfterOriginal 10878 25043

No YesShuffled 06756 25600Phase-randomized 06407 22981

LRC and FTD represent the long-range correlations and fat-tailed distri-bution respectively

16 Complexity

corresponding author Wei Shao thanks Professor DaraeJeong for helpful discussion

References

[1] G Shaddick and J Wakefield ldquoModelling daily multivariatepollutant data at multiple sitesrdquo Journal of the Royal StatisticalSociety Series C (Applied Statistics) vol 51 no 3 pp 351ndash3722002

[2] M OrsquoDwyer Air Quality in Ireland 2010 Key Indicators ofAmbient Air Quality EPA Wexford Ireland 2011

[3] A Challoner and L Gill ldquoIndooroutdoor air pollution re-lationships in ten commercial buildings PM25 and NO2rdquoBuilding and Environment vol 80 pp 159ndash173 2014

[4] M Hazenkamp-von Arx T Gotschi U Ackermann-Liebrichet al ldquoPM25 and NO2 assessment in 21 European studycentres of ECRHS II annual means and seasonal differencesrdquoAtmospheric Environment vol 38 no 13 pp 1943ndash19532004

[5] H K Lai L Bayer-Oglesby R Colvile et al ldquoDeterminants ofindoor air concentrations of PM25 black smoke and NO2 insix European cities (EXPOLIS study)rdquo Atmospheric Envi-ronment vol 40 no 7 pp 1299ndash1313 2006

[6] K de Hoogh J Gulliver A V Donkelaar et al ldquoDevelopmentof West-European PM25 and NO2 land use regression modelsincorporating satellite-derived and chemical transport mod-elling datardquo Environmental Research vol 151 pp 1ndash10 2016

[7] S Drozdz J Kwapien P Oswiecimka and R Rak ldquoQuan-titative features of multifractal subtleties in time seriesrdquo EPL(Europhysics Letters) vol 88 no 6 p 60003 2010

[8] R J Buonocore T Aste and T Di Matteo ldquoMeasuringmultiscaling in financial time-seriesrdquo Chaos Solitons ampFractals vol 88 pp 38ndash47 2016

[9] P Oswiecimka S Drozdz J Kwapien and A Z GorskildquoFractals log-periodicity and financial crashesrdquo Acta PhysicaPolonica A vol 117 no 4 pp 637ndash639 2010

[10] B B Mandelbrot and N Goldenfeld ldquoFractals and scaling infinance discontinuity concentration riskrdquo Physics Todayvol 51 no 8 pp 59-60 1998

[11] S Drozdz R Kowalski P Oswiecimka R Rak andR Gebarowski ldquoDynamical variety of shapes in financialmultifractalityrdquo Complexity vol 2018 Article ID 701572113 pages 2018

[12] S Lahmiri ldquoMultifractal analysis of Moroccan family businessstock returnsrdquo Physica A Statistical Mechanics and Its Ap-plications vol 486 pp 183ndash191 2017

[13] X Zhao P Shang and Q Jin ldquoMultifractal detrended cross-correlation analysis of Chinese stock markets based on timedelayrdquo Fractals vol 19 no 03 pp 329ndash338 2011

[14] W Shao and J Wang ldquoDoes the ldquoice-breakingrdquo of South andNorth Korea affect the South Korean financial marketrdquoChaos Solitons amp Fractals vol 132 Article ID 109564 2020

[15] D-H Wang X-W Yu and Y-Y Suo ldquoStatistical propertiesof the yuan exchange rate indexrdquo Physica A Statistical Me-chanics and Its Applications vol 391 no 12 pp 3503ndash35122012

[16] Y Ning Y Wang and C-W Su ldquoHow did Chinarsquos foreignexchange reform affect the efficiency of foreign exchangemarketrdquo Physica A Statistical Mechanics and Its Applicationsvol 483 pp 219ndash226 2017

[17] S Lahmiri and S Bekiros ldquoChaos randomness and multi-fractality in bitcoin marketrdquo Chaos Solitons amp Fractalsvol 106 pp 28ndash34 2018

[18] P Manimaran and A C Narayana ldquoMultifractal detrendedcross-correlation analysis on air pollutants of University ofHyderabad Campus Indiardquo Physica A Statistical Mechanicsand Its Applications vol 502 pp 228ndash235 2018

[19] L Zhao W Li C Yang J Han Z Su and Y Zou ldquoMul-tifractality and network analysis of phase transitionrdquo PLoSOne vol 12 Article ID e0170467 2017

[20] N Kalamaras K Philippopoulos D Deligiorgi C G Tzanisand G Karvounis ldquoMultifractal scaling properties of daily airtemperature time seriesrdquo Chaos Solitons amp Fractals vol 98pp 38ndash43 2017

[21] J Li C Chen Q Yao et al ldquoe effect of circadian rhythm onthe correlation and multifractality of heart rate signals duringexerciserdquo Physica A Statistical Mechanics and Its Applica-tions vol 509 pp 1207ndash1213 2018

[22] P Castiglioni D Lazzeroni P Coruzzi and A FainildquoMultifractal-multiscale analysis of cardiovascular signals aDFA-based characterization of blood pressure and heart-ratecomplexity by genderrdquo Complexity vol 2018 Article ID4801924 14 pages 2018

[23] W Shi R-B Zou F Wang and L Su ldquoA new image seg-mentation method based on multifractal detrended movingaverage analysisrdquo Physica A Statistical Mechanics and ItsApplications vol 432 pp 197ndash205 2015

[24] F Wang Q Fan and H E Stanley ldquoMultiscale multifractaldetrended-fluctuation analysis of two-dimensional surfacesrdquoPhysical Review E vol 93 Article ID 042213 2016

[25] X Li M Zheng J Pu et al ldquoIdentification of abnormallyexpressed lncrnas induced by PM25 in human bronchialepithelial cellsrdquo Bioscience Reports vol 38 no 5 Article IDBSR20171577 2018

[26] K Shi C Liu and Y Huang ldquoMultifractal processes and self-organized criticality of PM25 during a typical haze period inChengdu Chinardquo Aerosol and Air Quality Research vol 15no 3 pp 926ndash934 2015

[27] C Zhang Z Ni and L Ni ldquoMultifractal detrended cross-correlation analysis between PM25 and meteorological fac-torsrdquo Physica A Statistical Mechanics and Its Applicationsvol 438 pp 114ndash123 2015

[28] C Zhang X Wang S Chen L Zou X Zhang and C TangldquoA study on daily PM25 concentrations in Hong Kong usingthe EMD-based MFDFA methodrdquo Physica A StatisticalMechanics and Its Applications vol 530 p 121182 2019

[29] H-D He ldquoMultifractal analysis of interactive patterns be-tween meteorological factors and pollutants in urban andrural areasrdquo Atmospheric Environment vol 149 pp 47ndash542017

[30] J W Kantelhardt S A Zschiegner E Koscielny-BundeS Havlin A Bunde and H E Stanley ldquoMultifractaldetrended fluctuation analysis of nonstationary time seriesrdquoPhysica A Statistical Mechanics and Its Applications vol 316no 1ndash4 pp 87ndash114 2002

[31] L Zunino B M Tabak A Figliola D G PerezM Garavaglia and O A Rosso ldquoA multifractal approach forstock market inefficiencyrdquo Physica A Statistical Mechanicsand Its Applications vol 387 no 26 pp 6558ndash6566 2008

[32] L Zunino A Figliola B M Tabak D G PerezM Garavaglia and O A Rosso ldquoMultifractal structure inLatin-American market indicesrdquo Chaos Solitons amp Fractalsvol 41 no 5 pp 2331ndash2340 2009

[33] B Lashermes P Abry and P Chainais ldquoNew insights into theestimation of scaling exponentsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 2no 4 pp 497ndash523 2004

Complexity 17

[34] E A F Ihlen ldquoIntroduction to multifractal detrended fluc-tuation analysis in Matlabrdquo Frontiers in Physiology vol 3p 141 2012

[35] L Jiang X Zhao N Li F Li and Z Guo ldquoDifferent mul-tifractal scaling of the 0 cm average ground surface tem-perature of four representative weather stations over ChinardquoAdvances inMeteorology vol 2013 Article ID 341934 8 pages2013

[36] P C Ivanov L A N Amaral A L Goldberger et alldquoMultifractality in human heartbeat dynamicsrdquo Naturevol 399 no 6735 pp 461ndash465 1999

[37] S Drozdz and P Oswiecimka ldquoDetecting and interpretingdistortions in hierarchical organization of complex time se-riesrdquo Physical Review E vol 91 Article ID 030902 2015

[38] C Ibarra-Valdez J Alvarez and J Alvarez-Ramirez ldquoRan-domness confidence bands of fractal scaling exponents forfinancial price returnsrdquo Chaos Solitons amp Fractals vol 83pp 119ndash124 2016

[39] D Stosic D Stosic T Stosic and H E Stanley ldquoMultifractalanalysis of managed and independent float exchange ratesrdquoPhysica A Statistical Mechanics and its Applications vol 428pp 13ndash18 2015

[40] K Matia Y Ashkenazy and H E Stanley ldquoMultifractalproperties of price fluctuations of stocks and commoditiesrdquoEurophysics Letters (EPL) vol 61 no 3 pp 422ndash428 2003

18 Complexity

ndash3

ndash2

ndash1

0

1

f(α)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

1 2 3 40α

05 1 15 2 25 30α

05 1 15 2 25 30α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(a)

ndash15

ndash1

ndash05

0

05

1

f(α)

ndash05

0

05

1

f(α)

ndash3

ndash2

ndash1

0

1

f(α)

05 1 15 2 25 30α

05 1 15 20α

05 1 15 2 250α

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

OriginalShuffledPhase-randomized

(b)

Figure 14 Holder exponent (a) before law and (b) after law From top to bottom are Seoul Busan and Daegu

Complexity 15

the implementation of the law for all three cities implyingthat the multifractality of the time series becomes morestable with the measures taken by the government and thatthe efficiency of the ldquohaze special lawrdquo in three cities isdemonstrable this conclusion can be referred to [16 39] inwhich it was pointed out that the decrease in multifractalitymeans the efficiency is improved when some measures aretaken Moreover we also notice that the efficiency of theldquohaze special lawrdquo in Seoul is most obvious when comparedwith Busan and in last place is Daegu Interestingly theorder is the same as the scale and prosperity of the citiesWe can therefore conclude that the law has a higher effecton a larger city

43 Sources of Multifractal Features e two universallyacknowledged sources of multifractality are long-rangecorrelations and a fat-tailed distribution [7 40] Now weinvestigate the major source of multifractality of the PM25concentration series To check the contribution of long-range correlations and fat-tailed distribution quantitativelywe shuffle and phase-randomize the time series

Subsequently we calculate the Hurst exponent h(q) theRenyi exponent τ(q) and the Holder exponent Δα of theoriginal shuffled and phase-randomized time series of thethree cities As shown in Figures 12ndash14 all the time series arestrongly multifractal From Table 5 all h(2) of the originalseries are larger than those of the shuffled and phase-ran-domized time series which indicates that the persistence hasbeen moved after shuffling and phase-randomizationMoreover the original series of all three cities has thegreatest spectrum width Δα while the multifractality de-creased after shuffling and phase-randomizing the seriese findings can also be confirmed in Figures 12ndash14

To examine the main source of multifractality the firstand third rows of Figure 14 show that the multifractalspectra of phase-randomized series are the narrowest ones

which means that the main source of multifractality of theseries before and after the law in Seoul and Daegu are the fat-tailed distributions e second row in Figure 14 shows thatthe multifractal spectrum of the shuffled series is the nar-rowest which implies that the long-range correlation resultsin multifractality of the series before and after imple-mentation of the law in Busan

5 Conclusions

In this study we examined the multifractal characteristics ofPM25 and NO2 time series of all seasons We determinedthat multifractality existed in each season and the multi-fractal property of PM25 was stronger than that of NO2 ineach season We also validated the effectiveness of the ldquohazespecial lawrdquo which was implemented to improve the gov-ernance of air pollution in South Korea We checkedwhether there is a difference caused by the law for these citiesand calculated confidence intervals of Hurst exponent h(2)e results showed that the confidence intervals do notoverlap for Seoul while the confidence intervals overlap a lotfor Busan and Daegu indicating that there is a differencecaused by the law for Seoul however the same conclusion isinconclusive for Busan and Daegu By comparing the Hurstexponent Renyi exponent and Holder exponent the timeseries of the PM25 concentration before the implementationof the law was found to have a higher multifractal degreewhich decreased after the enactment of the law is phe-nomenon reflected the fact that the law has played a role inimproving the efficiency of air pollution control in SouthKorea We also concluded that the effect of the law will bemore significant in a larger city To explore the major causesof multifractality we shuffled and phase-randomized theoriginal series of PM25 By analyzing the width of themultifractal spectrum Δα the results showed that the fat-tailed distribution contributed to the multifractality of bothtimes series before and after the implementation of the ldquohazespecial lawrdquo in Seoul and Daegu whereas long-range cor-relations resulted in the multifractality of the series beforeand after the implementation of the law in Busan Weconcluded that the implementation of the law was verysuccessful and effective for the improvement of PM25 levelserefore we believe that the ldquohaze special lawrdquo is a possiblecause for the change in API in South Korea

Data Availability

e data will be available upon request with the corre-sponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e first author (Jian Wang) was supported by the ChinaScholarship Council (201808260026) J S Kim expressesgratitude for the support from the BK21 PLUS programe

Table 5 Multifractality of PM25 time series

h(2) Δα LRC FTD

Seoul

BeforeOriginal 12270 22414

No YesShuffled 07232 17946Phase-randomized 06965 09574

AfterOriginal 12085 13953

No YesShuffled 03177 10978Phase-randomized 03049 08569

Busan

BeforeOriginal 12905 17463

Yes NoShuffled 03982 09999Phase-randomized 03993 13512

AfterOriginal 11041 12708

Yes NoShuffled 03678 07175Phase-randomized 03794 08673

Daegu

BeforeOriginal 11975 29610

No YesShuffled 05901 22093Phase-randomized 05868 20587

AfterOriginal 10878 25043

No YesShuffled 06756 25600Phase-randomized 06407 22981

LRC and FTD represent the long-range correlations and fat-tailed distri-bution respectively

16 Complexity

corresponding author Wei Shao thanks Professor DaraeJeong for helpful discussion

References

[1] G Shaddick and J Wakefield ldquoModelling daily multivariatepollutant data at multiple sitesrdquo Journal of the Royal StatisticalSociety Series C (Applied Statistics) vol 51 no 3 pp 351ndash3722002

[2] M OrsquoDwyer Air Quality in Ireland 2010 Key Indicators ofAmbient Air Quality EPA Wexford Ireland 2011

[3] A Challoner and L Gill ldquoIndooroutdoor air pollution re-lationships in ten commercial buildings PM25 and NO2rdquoBuilding and Environment vol 80 pp 159ndash173 2014

[4] M Hazenkamp-von Arx T Gotschi U Ackermann-Liebrichet al ldquoPM25 and NO2 assessment in 21 European studycentres of ECRHS II annual means and seasonal differencesrdquoAtmospheric Environment vol 38 no 13 pp 1943ndash19532004

[5] H K Lai L Bayer-Oglesby R Colvile et al ldquoDeterminants ofindoor air concentrations of PM25 black smoke and NO2 insix European cities (EXPOLIS study)rdquo Atmospheric Envi-ronment vol 40 no 7 pp 1299ndash1313 2006

[6] K de Hoogh J Gulliver A V Donkelaar et al ldquoDevelopmentof West-European PM25 and NO2 land use regression modelsincorporating satellite-derived and chemical transport mod-elling datardquo Environmental Research vol 151 pp 1ndash10 2016

[7] S Drozdz J Kwapien P Oswiecimka and R Rak ldquoQuan-titative features of multifractal subtleties in time seriesrdquo EPL(Europhysics Letters) vol 88 no 6 p 60003 2010

[8] R J Buonocore T Aste and T Di Matteo ldquoMeasuringmultiscaling in financial time-seriesrdquo Chaos Solitons ampFractals vol 88 pp 38ndash47 2016

[9] P Oswiecimka S Drozdz J Kwapien and A Z GorskildquoFractals log-periodicity and financial crashesrdquo Acta PhysicaPolonica A vol 117 no 4 pp 637ndash639 2010

[10] B B Mandelbrot and N Goldenfeld ldquoFractals and scaling infinance discontinuity concentration riskrdquo Physics Todayvol 51 no 8 pp 59-60 1998

[11] S Drozdz R Kowalski P Oswiecimka R Rak andR Gebarowski ldquoDynamical variety of shapes in financialmultifractalityrdquo Complexity vol 2018 Article ID 701572113 pages 2018

[12] S Lahmiri ldquoMultifractal analysis of Moroccan family businessstock returnsrdquo Physica A Statistical Mechanics and Its Ap-plications vol 486 pp 183ndash191 2017

[13] X Zhao P Shang and Q Jin ldquoMultifractal detrended cross-correlation analysis of Chinese stock markets based on timedelayrdquo Fractals vol 19 no 03 pp 329ndash338 2011

[14] W Shao and J Wang ldquoDoes the ldquoice-breakingrdquo of South andNorth Korea affect the South Korean financial marketrdquoChaos Solitons amp Fractals vol 132 Article ID 109564 2020

[15] D-H Wang X-W Yu and Y-Y Suo ldquoStatistical propertiesof the yuan exchange rate indexrdquo Physica A Statistical Me-chanics and Its Applications vol 391 no 12 pp 3503ndash35122012

[16] Y Ning Y Wang and C-W Su ldquoHow did Chinarsquos foreignexchange reform affect the efficiency of foreign exchangemarketrdquo Physica A Statistical Mechanics and Its Applicationsvol 483 pp 219ndash226 2017

[17] S Lahmiri and S Bekiros ldquoChaos randomness and multi-fractality in bitcoin marketrdquo Chaos Solitons amp Fractalsvol 106 pp 28ndash34 2018

[18] P Manimaran and A C Narayana ldquoMultifractal detrendedcross-correlation analysis on air pollutants of University ofHyderabad Campus Indiardquo Physica A Statistical Mechanicsand Its Applications vol 502 pp 228ndash235 2018

[19] L Zhao W Li C Yang J Han Z Su and Y Zou ldquoMul-tifractality and network analysis of phase transitionrdquo PLoSOne vol 12 Article ID e0170467 2017

[20] N Kalamaras K Philippopoulos D Deligiorgi C G Tzanisand G Karvounis ldquoMultifractal scaling properties of daily airtemperature time seriesrdquo Chaos Solitons amp Fractals vol 98pp 38ndash43 2017

[21] J Li C Chen Q Yao et al ldquoe effect of circadian rhythm onthe correlation and multifractality of heart rate signals duringexerciserdquo Physica A Statistical Mechanics and Its Applica-tions vol 509 pp 1207ndash1213 2018

[22] P Castiglioni D Lazzeroni P Coruzzi and A FainildquoMultifractal-multiscale analysis of cardiovascular signals aDFA-based characterization of blood pressure and heart-ratecomplexity by genderrdquo Complexity vol 2018 Article ID4801924 14 pages 2018

[23] W Shi R-B Zou F Wang and L Su ldquoA new image seg-mentation method based on multifractal detrended movingaverage analysisrdquo Physica A Statistical Mechanics and ItsApplications vol 432 pp 197ndash205 2015

[24] F Wang Q Fan and H E Stanley ldquoMultiscale multifractaldetrended-fluctuation analysis of two-dimensional surfacesrdquoPhysical Review E vol 93 Article ID 042213 2016

[25] X Li M Zheng J Pu et al ldquoIdentification of abnormallyexpressed lncrnas induced by PM25 in human bronchialepithelial cellsrdquo Bioscience Reports vol 38 no 5 Article IDBSR20171577 2018

[26] K Shi C Liu and Y Huang ldquoMultifractal processes and self-organized criticality of PM25 during a typical haze period inChengdu Chinardquo Aerosol and Air Quality Research vol 15no 3 pp 926ndash934 2015

[27] C Zhang Z Ni and L Ni ldquoMultifractal detrended cross-correlation analysis between PM25 and meteorological fac-torsrdquo Physica A Statistical Mechanics and Its Applicationsvol 438 pp 114ndash123 2015

[28] C Zhang X Wang S Chen L Zou X Zhang and C TangldquoA study on daily PM25 concentrations in Hong Kong usingthe EMD-based MFDFA methodrdquo Physica A StatisticalMechanics and Its Applications vol 530 p 121182 2019

[29] H-D He ldquoMultifractal analysis of interactive patterns be-tween meteorological factors and pollutants in urban andrural areasrdquo Atmospheric Environment vol 149 pp 47ndash542017

[30] J W Kantelhardt S A Zschiegner E Koscielny-BundeS Havlin A Bunde and H E Stanley ldquoMultifractaldetrended fluctuation analysis of nonstationary time seriesrdquoPhysica A Statistical Mechanics and Its Applications vol 316no 1ndash4 pp 87ndash114 2002

[31] L Zunino B M Tabak A Figliola D G PerezM Garavaglia and O A Rosso ldquoA multifractal approach forstock market inefficiencyrdquo Physica A Statistical Mechanicsand Its Applications vol 387 no 26 pp 6558ndash6566 2008

[32] L Zunino A Figliola B M Tabak D G PerezM Garavaglia and O A Rosso ldquoMultifractal structure inLatin-American market indicesrdquo Chaos Solitons amp Fractalsvol 41 no 5 pp 2331ndash2340 2009

[33] B Lashermes P Abry and P Chainais ldquoNew insights into theestimation of scaling exponentsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 2no 4 pp 497ndash523 2004

Complexity 17

[34] E A F Ihlen ldquoIntroduction to multifractal detrended fluc-tuation analysis in Matlabrdquo Frontiers in Physiology vol 3p 141 2012

[35] L Jiang X Zhao N Li F Li and Z Guo ldquoDifferent mul-tifractal scaling of the 0 cm average ground surface tem-perature of four representative weather stations over ChinardquoAdvances inMeteorology vol 2013 Article ID 341934 8 pages2013

[36] P C Ivanov L A N Amaral A L Goldberger et alldquoMultifractality in human heartbeat dynamicsrdquo Naturevol 399 no 6735 pp 461ndash465 1999

[37] S Drozdz and P Oswiecimka ldquoDetecting and interpretingdistortions in hierarchical organization of complex time se-riesrdquo Physical Review E vol 91 Article ID 030902 2015

[38] C Ibarra-Valdez J Alvarez and J Alvarez-Ramirez ldquoRan-domness confidence bands of fractal scaling exponents forfinancial price returnsrdquo Chaos Solitons amp Fractals vol 83pp 119ndash124 2016

[39] D Stosic D Stosic T Stosic and H E Stanley ldquoMultifractalanalysis of managed and independent float exchange ratesrdquoPhysica A Statistical Mechanics and its Applications vol 428pp 13ndash18 2015

[40] K Matia Y Ashkenazy and H E Stanley ldquoMultifractalproperties of price fluctuations of stocks and commoditiesrdquoEurophysics Letters (EPL) vol 61 no 3 pp 422ndash428 2003

18 Complexity

the implementation of the law for all three cities implyingthat the multifractality of the time series becomes morestable with the measures taken by the government and thatthe efficiency of the ldquohaze special lawrdquo in three cities isdemonstrable this conclusion can be referred to [16 39] inwhich it was pointed out that the decrease in multifractalitymeans the efficiency is improved when some measures aretaken Moreover we also notice that the efficiency of theldquohaze special lawrdquo in Seoul is most obvious when comparedwith Busan and in last place is Daegu Interestingly theorder is the same as the scale and prosperity of the citiesWe can therefore conclude that the law has a higher effecton a larger city

43 Sources of Multifractal Features e two universallyacknowledged sources of multifractality are long-rangecorrelations and a fat-tailed distribution [7 40] Now weinvestigate the major source of multifractality of the PM25concentration series To check the contribution of long-range correlations and fat-tailed distribution quantitativelywe shuffle and phase-randomize the time series

Subsequently we calculate the Hurst exponent h(q) theRenyi exponent τ(q) and the Holder exponent Δα of theoriginal shuffled and phase-randomized time series of thethree cities As shown in Figures 12ndash14 all the time series arestrongly multifractal From Table 5 all h(2) of the originalseries are larger than those of the shuffled and phase-ran-domized time series which indicates that the persistence hasbeen moved after shuffling and phase-randomizationMoreover the original series of all three cities has thegreatest spectrum width Δα while the multifractality de-creased after shuffling and phase-randomizing the seriese findings can also be confirmed in Figures 12ndash14

To examine the main source of multifractality the firstand third rows of Figure 14 show that the multifractalspectra of phase-randomized series are the narrowest ones

which means that the main source of multifractality of theseries before and after the law in Seoul and Daegu are the fat-tailed distributions e second row in Figure 14 shows thatthe multifractal spectrum of the shuffled series is the nar-rowest which implies that the long-range correlation resultsin multifractality of the series before and after imple-mentation of the law in Busan

5 Conclusions

In this study we examined the multifractal characteristics ofPM25 and NO2 time series of all seasons We determinedthat multifractality existed in each season and the multi-fractal property of PM25 was stronger than that of NO2 ineach season We also validated the effectiveness of the ldquohazespecial lawrdquo which was implemented to improve the gov-ernance of air pollution in South Korea We checkedwhether there is a difference caused by the law for these citiesand calculated confidence intervals of Hurst exponent h(2)e results showed that the confidence intervals do notoverlap for Seoul while the confidence intervals overlap a lotfor Busan and Daegu indicating that there is a differencecaused by the law for Seoul however the same conclusion isinconclusive for Busan and Daegu By comparing the Hurstexponent Renyi exponent and Holder exponent the timeseries of the PM25 concentration before the implementationof the law was found to have a higher multifractal degreewhich decreased after the enactment of the law is phe-nomenon reflected the fact that the law has played a role inimproving the efficiency of air pollution control in SouthKorea We also concluded that the effect of the law will bemore significant in a larger city To explore the major causesof multifractality we shuffled and phase-randomized theoriginal series of PM25 By analyzing the width of themultifractal spectrum Δα the results showed that the fat-tailed distribution contributed to the multifractality of bothtimes series before and after the implementation of the ldquohazespecial lawrdquo in Seoul and Daegu whereas long-range cor-relations resulted in the multifractality of the series beforeand after the implementation of the law in Busan Weconcluded that the implementation of the law was verysuccessful and effective for the improvement of PM25 levelserefore we believe that the ldquohaze special lawrdquo is a possiblecause for the change in API in South Korea

Data Availability

e data will be available upon request with the corre-sponding author

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

e first author (Jian Wang) was supported by the ChinaScholarship Council (201808260026) J S Kim expressesgratitude for the support from the BK21 PLUS programe

Table 5 Multifractality of PM25 time series

h(2) Δα LRC FTD

Seoul

BeforeOriginal 12270 22414

No YesShuffled 07232 17946Phase-randomized 06965 09574

AfterOriginal 12085 13953

No YesShuffled 03177 10978Phase-randomized 03049 08569

Busan

BeforeOriginal 12905 17463

Yes NoShuffled 03982 09999Phase-randomized 03993 13512

AfterOriginal 11041 12708

Yes NoShuffled 03678 07175Phase-randomized 03794 08673

Daegu

BeforeOriginal 11975 29610

No YesShuffled 05901 22093Phase-randomized 05868 20587

AfterOriginal 10878 25043

No YesShuffled 06756 25600Phase-randomized 06407 22981

LRC and FTD represent the long-range correlations and fat-tailed distri-bution respectively

16 Complexity

corresponding author Wei Shao thanks Professor DaraeJeong for helpful discussion

References

[1] G Shaddick and J Wakefield ldquoModelling daily multivariatepollutant data at multiple sitesrdquo Journal of the Royal StatisticalSociety Series C (Applied Statistics) vol 51 no 3 pp 351ndash3722002

[2] M OrsquoDwyer Air Quality in Ireland 2010 Key Indicators ofAmbient Air Quality EPA Wexford Ireland 2011

[3] A Challoner and L Gill ldquoIndooroutdoor air pollution re-lationships in ten commercial buildings PM25 and NO2rdquoBuilding and Environment vol 80 pp 159ndash173 2014

[4] M Hazenkamp-von Arx T Gotschi U Ackermann-Liebrichet al ldquoPM25 and NO2 assessment in 21 European studycentres of ECRHS II annual means and seasonal differencesrdquoAtmospheric Environment vol 38 no 13 pp 1943ndash19532004

[5] H K Lai L Bayer-Oglesby R Colvile et al ldquoDeterminants ofindoor air concentrations of PM25 black smoke and NO2 insix European cities (EXPOLIS study)rdquo Atmospheric Envi-ronment vol 40 no 7 pp 1299ndash1313 2006

[6] K de Hoogh J Gulliver A V Donkelaar et al ldquoDevelopmentof West-European PM25 and NO2 land use regression modelsincorporating satellite-derived and chemical transport mod-elling datardquo Environmental Research vol 151 pp 1ndash10 2016

[7] S Drozdz J Kwapien P Oswiecimka and R Rak ldquoQuan-titative features of multifractal subtleties in time seriesrdquo EPL(Europhysics Letters) vol 88 no 6 p 60003 2010

[8] R J Buonocore T Aste and T Di Matteo ldquoMeasuringmultiscaling in financial time-seriesrdquo Chaos Solitons ampFractals vol 88 pp 38ndash47 2016

[9] P Oswiecimka S Drozdz J Kwapien and A Z GorskildquoFractals log-periodicity and financial crashesrdquo Acta PhysicaPolonica A vol 117 no 4 pp 637ndash639 2010

[10] B B Mandelbrot and N Goldenfeld ldquoFractals and scaling infinance discontinuity concentration riskrdquo Physics Todayvol 51 no 8 pp 59-60 1998

[11] S Drozdz R Kowalski P Oswiecimka R Rak andR Gebarowski ldquoDynamical variety of shapes in financialmultifractalityrdquo Complexity vol 2018 Article ID 701572113 pages 2018

[12] S Lahmiri ldquoMultifractal analysis of Moroccan family businessstock returnsrdquo Physica A Statistical Mechanics and Its Ap-plications vol 486 pp 183ndash191 2017

[13] X Zhao P Shang and Q Jin ldquoMultifractal detrended cross-correlation analysis of Chinese stock markets based on timedelayrdquo Fractals vol 19 no 03 pp 329ndash338 2011

[14] W Shao and J Wang ldquoDoes the ldquoice-breakingrdquo of South andNorth Korea affect the South Korean financial marketrdquoChaos Solitons amp Fractals vol 132 Article ID 109564 2020

[15] D-H Wang X-W Yu and Y-Y Suo ldquoStatistical propertiesof the yuan exchange rate indexrdquo Physica A Statistical Me-chanics and Its Applications vol 391 no 12 pp 3503ndash35122012

[16] Y Ning Y Wang and C-W Su ldquoHow did Chinarsquos foreignexchange reform affect the efficiency of foreign exchangemarketrdquo Physica A Statistical Mechanics and Its Applicationsvol 483 pp 219ndash226 2017

[17] S Lahmiri and S Bekiros ldquoChaos randomness and multi-fractality in bitcoin marketrdquo Chaos Solitons amp Fractalsvol 106 pp 28ndash34 2018

[18] P Manimaran and A C Narayana ldquoMultifractal detrendedcross-correlation analysis on air pollutants of University ofHyderabad Campus Indiardquo Physica A Statistical Mechanicsand Its Applications vol 502 pp 228ndash235 2018

[19] L Zhao W Li C Yang J Han Z Su and Y Zou ldquoMul-tifractality and network analysis of phase transitionrdquo PLoSOne vol 12 Article ID e0170467 2017

[20] N Kalamaras K Philippopoulos D Deligiorgi C G Tzanisand G Karvounis ldquoMultifractal scaling properties of daily airtemperature time seriesrdquo Chaos Solitons amp Fractals vol 98pp 38ndash43 2017

[21] J Li C Chen Q Yao et al ldquoe effect of circadian rhythm onthe correlation and multifractality of heart rate signals duringexerciserdquo Physica A Statistical Mechanics and Its Applica-tions vol 509 pp 1207ndash1213 2018

[22] P Castiglioni D Lazzeroni P Coruzzi and A FainildquoMultifractal-multiscale analysis of cardiovascular signals aDFA-based characterization of blood pressure and heart-ratecomplexity by genderrdquo Complexity vol 2018 Article ID4801924 14 pages 2018

[23] W Shi R-B Zou F Wang and L Su ldquoA new image seg-mentation method based on multifractal detrended movingaverage analysisrdquo Physica A Statistical Mechanics and ItsApplications vol 432 pp 197ndash205 2015

[24] F Wang Q Fan and H E Stanley ldquoMultiscale multifractaldetrended-fluctuation analysis of two-dimensional surfacesrdquoPhysical Review E vol 93 Article ID 042213 2016

[25] X Li M Zheng J Pu et al ldquoIdentification of abnormallyexpressed lncrnas induced by PM25 in human bronchialepithelial cellsrdquo Bioscience Reports vol 38 no 5 Article IDBSR20171577 2018

[26] K Shi C Liu and Y Huang ldquoMultifractal processes and self-organized criticality of PM25 during a typical haze period inChengdu Chinardquo Aerosol and Air Quality Research vol 15no 3 pp 926ndash934 2015

[27] C Zhang Z Ni and L Ni ldquoMultifractal detrended cross-correlation analysis between PM25 and meteorological fac-torsrdquo Physica A Statistical Mechanics and Its Applicationsvol 438 pp 114ndash123 2015

[28] C Zhang X Wang S Chen L Zou X Zhang and C TangldquoA study on daily PM25 concentrations in Hong Kong usingthe EMD-based MFDFA methodrdquo Physica A StatisticalMechanics and Its Applications vol 530 p 121182 2019

[29] H-D He ldquoMultifractal analysis of interactive patterns be-tween meteorological factors and pollutants in urban andrural areasrdquo Atmospheric Environment vol 149 pp 47ndash542017

[30] J W Kantelhardt S A Zschiegner E Koscielny-BundeS Havlin A Bunde and H E Stanley ldquoMultifractaldetrended fluctuation analysis of nonstationary time seriesrdquoPhysica A Statistical Mechanics and Its Applications vol 316no 1ndash4 pp 87ndash114 2002

[31] L Zunino B M Tabak A Figliola D G PerezM Garavaglia and O A Rosso ldquoA multifractal approach forstock market inefficiencyrdquo Physica A Statistical Mechanicsand Its Applications vol 387 no 26 pp 6558ndash6566 2008

[32] L Zunino A Figliola B M Tabak D G PerezM Garavaglia and O A Rosso ldquoMultifractal structure inLatin-American market indicesrdquo Chaos Solitons amp Fractalsvol 41 no 5 pp 2331ndash2340 2009

[33] B Lashermes P Abry and P Chainais ldquoNew insights into theestimation of scaling exponentsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 2no 4 pp 497ndash523 2004

Complexity 17

[34] E A F Ihlen ldquoIntroduction to multifractal detrended fluc-tuation analysis in Matlabrdquo Frontiers in Physiology vol 3p 141 2012

[35] L Jiang X Zhao N Li F Li and Z Guo ldquoDifferent mul-tifractal scaling of the 0 cm average ground surface tem-perature of four representative weather stations over ChinardquoAdvances inMeteorology vol 2013 Article ID 341934 8 pages2013

[36] P C Ivanov L A N Amaral A L Goldberger et alldquoMultifractality in human heartbeat dynamicsrdquo Naturevol 399 no 6735 pp 461ndash465 1999

[37] S Drozdz and P Oswiecimka ldquoDetecting and interpretingdistortions in hierarchical organization of complex time se-riesrdquo Physical Review E vol 91 Article ID 030902 2015

[38] C Ibarra-Valdez J Alvarez and J Alvarez-Ramirez ldquoRan-domness confidence bands of fractal scaling exponents forfinancial price returnsrdquo Chaos Solitons amp Fractals vol 83pp 119ndash124 2016

[39] D Stosic D Stosic T Stosic and H E Stanley ldquoMultifractalanalysis of managed and independent float exchange ratesrdquoPhysica A Statistical Mechanics and its Applications vol 428pp 13ndash18 2015

[40] K Matia Y Ashkenazy and H E Stanley ldquoMultifractalproperties of price fluctuations of stocks and commoditiesrdquoEurophysics Letters (EPL) vol 61 no 3 pp 422ndash428 2003

18 Complexity

corresponding author Wei Shao thanks Professor DaraeJeong for helpful discussion

References

[1] G Shaddick and J Wakefield ldquoModelling daily multivariatepollutant data at multiple sitesrdquo Journal of the Royal StatisticalSociety Series C (Applied Statistics) vol 51 no 3 pp 351ndash3722002

[2] M OrsquoDwyer Air Quality in Ireland 2010 Key Indicators ofAmbient Air Quality EPA Wexford Ireland 2011

[3] A Challoner and L Gill ldquoIndooroutdoor air pollution re-lationships in ten commercial buildings PM25 and NO2rdquoBuilding and Environment vol 80 pp 159ndash173 2014

[4] M Hazenkamp-von Arx T Gotschi U Ackermann-Liebrichet al ldquoPM25 and NO2 assessment in 21 European studycentres of ECRHS II annual means and seasonal differencesrdquoAtmospheric Environment vol 38 no 13 pp 1943ndash19532004

[5] H K Lai L Bayer-Oglesby R Colvile et al ldquoDeterminants ofindoor air concentrations of PM25 black smoke and NO2 insix European cities (EXPOLIS study)rdquo Atmospheric Envi-ronment vol 40 no 7 pp 1299ndash1313 2006

[6] K de Hoogh J Gulliver A V Donkelaar et al ldquoDevelopmentof West-European PM25 and NO2 land use regression modelsincorporating satellite-derived and chemical transport mod-elling datardquo Environmental Research vol 151 pp 1ndash10 2016

[7] S Drozdz J Kwapien P Oswiecimka and R Rak ldquoQuan-titative features of multifractal subtleties in time seriesrdquo EPL(Europhysics Letters) vol 88 no 6 p 60003 2010

[8] R J Buonocore T Aste and T Di Matteo ldquoMeasuringmultiscaling in financial time-seriesrdquo Chaos Solitons ampFractals vol 88 pp 38ndash47 2016

[9] P Oswiecimka S Drozdz J Kwapien and A Z GorskildquoFractals log-periodicity and financial crashesrdquo Acta PhysicaPolonica A vol 117 no 4 pp 637ndash639 2010

[10] B B Mandelbrot and N Goldenfeld ldquoFractals and scaling infinance discontinuity concentration riskrdquo Physics Todayvol 51 no 8 pp 59-60 1998

[11] S Drozdz R Kowalski P Oswiecimka R Rak andR Gebarowski ldquoDynamical variety of shapes in financialmultifractalityrdquo Complexity vol 2018 Article ID 701572113 pages 2018

[12] S Lahmiri ldquoMultifractal analysis of Moroccan family businessstock returnsrdquo Physica A Statistical Mechanics and Its Ap-plications vol 486 pp 183ndash191 2017

[13] X Zhao P Shang and Q Jin ldquoMultifractal detrended cross-correlation analysis of Chinese stock markets based on timedelayrdquo Fractals vol 19 no 03 pp 329ndash338 2011

[14] W Shao and J Wang ldquoDoes the ldquoice-breakingrdquo of South andNorth Korea affect the South Korean financial marketrdquoChaos Solitons amp Fractals vol 132 Article ID 109564 2020

[15] D-H Wang X-W Yu and Y-Y Suo ldquoStatistical propertiesof the yuan exchange rate indexrdquo Physica A Statistical Me-chanics and Its Applications vol 391 no 12 pp 3503ndash35122012

[16] Y Ning Y Wang and C-W Su ldquoHow did Chinarsquos foreignexchange reform affect the efficiency of foreign exchangemarketrdquo Physica A Statistical Mechanics and Its Applicationsvol 483 pp 219ndash226 2017

[17] S Lahmiri and S Bekiros ldquoChaos randomness and multi-fractality in bitcoin marketrdquo Chaos Solitons amp Fractalsvol 106 pp 28ndash34 2018

[18] P Manimaran and A C Narayana ldquoMultifractal detrendedcross-correlation analysis on air pollutants of University ofHyderabad Campus Indiardquo Physica A Statistical Mechanicsand Its Applications vol 502 pp 228ndash235 2018

[19] L Zhao W Li C Yang J Han Z Su and Y Zou ldquoMul-tifractality and network analysis of phase transitionrdquo PLoSOne vol 12 Article ID e0170467 2017

[20] N Kalamaras K Philippopoulos D Deligiorgi C G Tzanisand G Karvounis ldquoMultifractal scaling properties of daily airtemperature time seriesrdquo Chaos Solitons amp Fractals vol 98pp 38ndash43 2017

[21] J Li C Chen Q Yao et al ldquoe effect of circadian rhythm onthe correlation and multifractality of heart rate signals duringexerciserdquo Physica A Statistical Mechanics and Its Applica-tions vol 509 pp 1207ndash1213 2018

[22] P Castiglioni D Lazzeroni P Coruzzi and A FainildquoMultifractal-multiscale analysis of cardiovascular signals aDFA-based characterization of blood pressure and heart-ratecomplexity by genderrdquo Complexity vol 2018 Article ID4801924 14 pages 2018

[23] W Shi R-B Zou F Wang and L Su ldquoA new image seg-mentation method based on multifractal detrended movingaverage analysisrdquo Physica A Statistical Mechanics and ItsApplications vol 432 pp 197ndash205 2015

[24] F Wang Q Fan and H E Stanley ldquoMultiscale multifractaldetrended-fluctuation analysis of two-dimensional surfacesrdquoPhysical Review E vol 93 Article ID 042213 2016

[25] X Li M Zheng J Pu et al ldquoIdentification of abnormallyexpressed lncrnas induced by PM25 in human bronchialepithelial cellsrdquo Bioscience Reports vol 38 no 5 Article IDBSR20171577 2018

[26] K Shi C Liu and Y Huang ldquoMultifractal processes and self-organized criticality of PM25 during a typical haze period inChengdu Chinardquo Aerosol and Air Quality Research vol 15no 3 pp 926ndash934 2015

[27] C Zhang Z Ni and L Ni ldquoMultifractal detrended cross-correlation analysis between PM25 and meteorological fac-torsrdquo Physica A Statistical Mechanics and Its Applicationsvol 438 pp 114ndash123 2015

[28] C Zhang X Wang S Chen L Zou X Zhang and C TangldquoA study on daily PM25 concentrations in Hong Kong usingthe EMD-based MFDFA methodrdquo Physica A StatisticalMechanics and Its Applications vol 530 p 121182 2019

[29] H-D He ldquoMultifractal analysis of interactive patterns be-tween meteorological factors and pollutants in urban andrural areasrdquo Atmospheric Environment vol 149 pp 47ndash542017

[30] J W Kantelhardt S A Zschiegner E Koscielny-BundeS Havlin A Bunde and H E Stanley ldquoMultifractaldetrended fluctuation analysis of nonstationary time seriesrdquoPhysica A Statistical Mechanics and Its Applications vol 316no 1ndash4 pp 87ndash114 2002

[31] L Zunino B M Tabak A Figliola D G PerezM Garavaglia and O A Rosso ldquoA multifractal approach forstock market inefficiencyrdquo Physica A Statistical Mechanicsand Its Applications vol 387 no 26 pp 6558ndash6566 2008

[32] L Zunino A Figliola B M Tabak D G PerezM Garavaglia and O A Rosso ldquoMultifractal structure inLatin-American market indicesrdquo Chaos Solitons amp Fractalsvol 41 no 5 pp 2331ndash2340 2009

[33] B Lashermes P Abry and P Chainais ldquoNew insights into theestimation of scaling exponentsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 2no 4 pp 497ndash523 2004

Complexity 17

[34] E A F Ihlen ldquoIntroduction to multifractal detrended fluc-tuation analysis in Matlabrdquo Frontiers in Physiology vol 3p 141 2012

[35] L Jiang X Zhao N Li F Li and Z Guo ldquoDifferent mul-tifractal scaling of the 0 cm average ground surface tem-perature of four representative weather stations over ChinardquoAdvances inMeteorology vol 2013 Article ID 341934 8 pages2013

[36] P C Ivanov L A N Amaral A L Goldberger et alldquoMultifractality in human heartbeat dynamicsrdquo Naturevol 399 no 6735 pp 461ndash465 1999

[37] S Drozdz and P Oswiecimka ldquoDetecting and interpretingdistortions in hierarchical organization of complex time se-riesrdquo Physical Review E vol 91 Article ID 030902 2015

[38] C Ibarra-Valdez J Alvarez and J Alvarez-Ramirez ldquoRan-domness confidence bands of fractal scaling exponents forfinancial price returnsrdquo Chaos Solitons amp Fractals vol 83pp 119ndash124 2016

[39] D Stosic D Stosic T Stosic and H E Stanley ldquoMultifractalanalysis of managed and independent float exchange ratesrdquoPhysica A Statistical Mechanics and its Applications vol 428pp 13ndash18 2015

[40] K Matia Y Ashkenazy and H E Stanley ldquoMultifractalproperties of price fluctuations of stocks and commoditiesrdquoEurophysics Letters (EPL) vol 61 no 3 pp 422ndash428 2003

18 Complexity

[34] E A F Ihlen ldquoIntroduction to multifractal detrended fluc-tuation analysis in Matlabrdquo Frontiers in Physiology vol 3p 141 2012

[35] L Jiang X Zhao N Li F Li and Z Guo ldquoDifferent mul-tifractal scaling of the 0 cm average ground surface tem-perature of four representative weather stations over ChinardquoAdvances inMeteorology vol 2013 Article ID 341934 8 pages2013

[36] P C Ivanov L A N Amaral A L Goldberger et alldquoMultifractality in human heartbeat dynamicsrdquo Naturevol 399 no 6735 pp 461ndash465 1999

[37] S Drozdz and P Oswiecimka ldquoDetecting and interpretingdistortions in hierarchical organization of complex time se-riesrdquo Physical Review E vol 91 Article ID 030902 2015

[38] C Ibarra-Valdez J Alvarez and J Alvarez-Ramirez ldquoRan-domness confidence bands of fractal scaling exponents forfinancial price returnsrdquo Chaos Solitons amp Fractals vol 83pp 119ndash124 2016

[39] D Stosic D Stosic T Stosic and H E Stanley ldquoMultifractalanalysis of managed and independent float exchange ratesrdquoPhysica A Statistical Mechanics and its Applications vol 428pp 13ndash18 2015

[40] K Matia Y Ashkenazy and H E Stanley ldquoMultifractalproperties of price fluctuations of stocks and commoditiesrdquoEurophysics Letters (EPL) vol 61 no 3 pp 422ndash428 2003

18 Complexity