Correlation and scaling behaviors of P M 2.5 concentration in China
Yongwen Zhang, Dean Chen, Jingfang Fan, Shlomo Havlin, Xiaosong Chen
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Correlation and scaling behaviors of
P M . concentration in China Yongwen. Zhang , , Dean. Chen , Jingfang. Fan , Shlomo. Havlin and Xiaosong. Chen , Data Science Research Center, Kunming University of Science and Technology, Kunming 650500, Yunnan, China CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, P. O.Box 2735, Beijing 100190, China Department of Physics, University of Helsinki, P.O. Box 48, 00014 Helsinki, Finland Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China;
PACS – Complex systems
PACS – Environmental studies
PACS – Nonlinear dynamics and chaos
Abstract – Air pollution has become a major issue and caused widespread environmental andhealth problems. Aerosols or particulate matters are an important component of the atmosphereand can transport under complex meteorological conditions. Based on the data of
P M . ob-servations, we develop a network approach to study and quantify their spreading and diffusionpatterns. We calculate cross-correlation functions of time lag between sites within different season.The probability distribution of correlation changes with season. It is found that the probabilitydistributions in four seasons can be scaled into one scaling function with averages and standarddeviations of correlation. This seasonal scaling behavior indicates there is the same mechanismbehind correlations of P M . concentration in different seasons. Further, from weighted and direc-tional degrees of complex network, different properties of P M . concentration are studied. Theweighted degrees reveal the strongest correlations of P M . concentration in winter and in theNorth China plain. These directional degrees show net influences of P M . along Gobi and innerMongolia, the North China plain, Central China, and Yangtze River Delta. Introduction. –
Aerosols or particulate matters,which control process from low visibility events to pre-cipitation, are important components of the atmosphere.They play a critical role in global climate pattern andpublic health. Chen et al. [1] have reported the impact onlife expectancy of sustained exposure to air pollution fromChina ’s Huai River policy. Due to anthropogenic emis-sions, the concentration of particulate matters is growingsharply. In the past few years, China has witnessed rapidgrowth both in industry and in citys population. As a re-sult, air pollution, especially the pollution caused by high
P M . concentration, has become a serious issue [2].Most previous studies on P M . concentrated on obser-vation in one site. Winter and summer P M . chemicalcompositions in 14 cities of China have been analysed byCao et al. [3]. The publishing of hourly data since 2013provided possibility to study spatial distribution and sea-sonal variation of P M . in China [4]. Using the data (a) E-mail: [email protected] of monitoring network in the North China Plain and theYangtze River Delta, Hu et al. [5] found strong temporalcorrelation between cities within 250 km. For 81 cities inChina, Gao et al. [6] studied air pollution of city clustersfrom June 2004 to June 2007. The relation between airquality over Beijing and its surroundings and circulationpatterns was studied by Zhang et al. [7]. The spatiotem-poral variations of
P M . and P M concentrations of 31Chinese cities from March 2013 to March 2014 were re-lated to SO , NO , CO and O [8]. At a suburban sitebetween Beijing and Tianjin, the correlation of pollutantsto meteorological conditions was discussed [9].The studies [5,6] have shown that P M . concentrationsin different cities are not localized and related each other.It is of great interest to investigate how far the P M . concentrations in different cities of China are correlated.Using the hourly data of monitoring sites over China, thespatial correlations of P M . concentrations in 2015 havebeen studied using the principal component analysis [10].In the last decade, network has emerged as an importantp-1 a r X i v : . [ phy s i c s . a o - ph ] M a r ongwen. Zhang et al. tool in studies of complex systems and has been appliedto a wide variety of disciplines [11–13]. Recently, complexnetwork theory has been used to study climate systems[14–21]. For a climate system, geographical locations orgrid points are regarded as nodes of network and the linksbetween them are defined from a cross correlation function[18, 21] or event synchronization [22].In this letter, we study the P M . concentrations inChina from the aspect of complex networks. The nodesof P M . concentration network can be defined from themonitoring stations. Using P M . concentration data, wecan calculate the correlation between nodes and definetheir links. The global properties of P M . concentra-tions in China will be studied from the aspect of network.Our work is organized as follows. In the next section, wedescribe the data and introduce the methodology. Theresults are presented and discussed in the third section.Finally, a short summary is given. Data and Methodology. –
Data.
The Ministry of Environmental Protectionof China has been publishing air quality index since2013 and provide data for us to study atmosphericpollution. We use the hourly
P M . concentrationdata of 754 monitoring sites over China from Dec.2014to Nov.2015 (http://113.108.142.147:20035/emcpublish/).In pre-processing, we transform 754 monitoring stationsinto 163 sites with the area 1 ◦ × ◦ . The concentrationof a site is defined by the average of monitoring stationsinside this site. Since the strong seasonal dependence of P M . concentration, we divide the data into four groupscorresponding to winter(Dec, Jan, Feb), spring(Mar, Apr,May), summer(Jun, Jul, Aug) and autumn(Sep, Oct,Nov). Methodology.
During a time period T , the P M . con-centration of site i has a series X i ( t ). With respect to itsaverage (cid:104) X i (cid:105) = T (cid:80) Tt =1 X i ( t ), there is a fluctuation se-ries δX i ( t ) = X i ( t ) − (cid:104) X i (cid:105) . To study the correlation of P M . concentration between sites i and j , we calculatethe cross-correlation function [18],ˆ C ij ( τ ) = (cid:104) δX i ( t ) · δX j ( t + τ ) (cid:105) (cid:114)(cid:68) [ δX i ( t )] (cid:69) · (cid:114)(cid:68) [ δX j ( t + τ )] (cid:69) , (1)where − τ max ≤ τ ≤ τ max is the time lag. On the basisof time-reversal symmetry, there is a relation ˆ C ij ( − τ ) =ˆ C ji ( τ ). The cross-correlation in the interval [ − τ max , τ max ]can be calculated by ˆ C ij ( τ ≥
0) and ˆ C ji ( τ ≥ C ij ( τ ) and denotethe corresponding time lag as τ ∗ ij . The correlation betweensites i and j is defined as C ij ≡ ˆ C ij ( τ ∗ ). If τ ∗ ij (cid:54) = 0,the correlation between sites i and j is directional. Thedirection of correlation is from i to j when τ ∗ ij > j to i when τ ∗ ij < N nodes, there are ( N − N/ ρ ( C ).For the definition of a network, a threshold ∆ of correla-tion is introduced to exclude noise. The adjacency matrixof the network is defined with the threshold as A ij = (cid:40) − δ ij | C ij | > ∆0 | C ij | ≤ ∆ , (2)where the Kronecker’s delta δ ij = 0 for i (cid:54) = j and δ ij = 1for i = j so that self-loop is excluded.The importance of site i in the network is characterizedusually by its degree k Ci = (cid:80) Nj =1 A ij [11]. More informa-tion can be taken into account with a weighted degree¯ k Ci = N (cid:88) j =1 A ij | C ij | . (3)The direction from sites i to j is described by a unitvector (cid:126)e ij = d ( δφ, δθ ) with d = (cid:112) δφ + δθ , where δφ and δθ are the longitude and latitude differences of i and j respectively. We can further introduce a directional degreeas (cid:126)k Ci = N (cid:88) j =1 ,τ ∗ ij > A ij | C ij | (cid:126)e ij + N (cid:88) j =1 ,τ ∗ ij < A ij | C ij | ( − (cid:126)e ij )(4)to quantify the P M . concentration directional influencesof site i .Alternatively, we can determine network links accordingto G ij = C ij − mean ( ˆ C ij ( τ )) std ( ˆ C ij ( τ )) , (5)where “mean” and “std” represent the mean and standarddeviation of the cross-correlation function [19, 23, 24].The adjacency matrix of network is now defined as B ij = (cid:40) − δ ij | G ij | > Θ0 | G ij | ≤ Θ . (6)with the threshold Θ of G . With C ij replaced by G ij inEq. (3) and Eq. (4), we can obtain the weighted degree¯ k Gi and the directional degree (cid:126)k Gi of G . Results. –
We calculate firstly the mean
P M . con-centration (cid:104) X i (cid:105) = T (cid:80) Tt =1 X i ( t ) of sites i = 1 , , ..., X = 1 N N (cid:88) i =1 (cid:104) X i (cid:105) (7)is 75 . µg/m in winter, 48 . µg/m in spring, 36 . µg/m in summer and 46 . µg/m in autumn. In winter, 45percent of sites has mean P M . concentration above75 µg/m and the percentage of the sites above 35 µg/m is95%. The maximum mean P M . concentration in winteris related to the enhanced anthropogenic emissions fromp-2orrelation and scaling behaviors of P M . concentration in Chinafossil fuel combustion, biomass burning and unfavorablemeteorological conditions for pollution dispersion [4]. Inspring, 8 percent of sites have mean concentrations above75 µg/m and 77 percent are above 35 µg/m . The lowestmean P M . concentration is reached in summer. Only 4percent of sites have mean concentrations above 75 µg/m and 47 percent are above 35 µg/m . In autumn, the per-centage of sites above 35 µg/m and 75 µg/m reaches 78%and 7%, respectively. DJF MAMJJA SON (cid:176) N30 (cid:176)
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P M . concentra-tion over China for the four seasons of 2015. The cross-correlation functions ˆ C ij ( τ ) between N siteswere calculated according to Eq. (1) and with τ max =10 days. From ˆ C ij ( τ ), we can obtain the correlation C ij between sites i and j . The PDF ρ ( C ) of correlation ispresented in Fig. 2 for four seasons. It can be seen that ρ ( C ) is separated into positive and negative parts. C r ( C ) DJFMAMJJASON
Fig. 2: Probability distribution function of correlation be-tween sites in the four seasons of 2015. Table 1: Proportion, average, and standard deviations of pos-itive and negative correlations.
DJF MAM JJA SON λ p
95% 78% 55% 84% (cid:104) C p (cid:105) σ p λ n
5% 22% 45% 16% (cid:104) C n (cid:105) -0.249 -0.255 -0.277 -0.254 σ n (a) C r p ( C ) DJFMAMJJASON (c) W p f p ( W p ) (b) C r n ( C ) (d) W n f n ( W n ) Fig. 3: (Color online) Probability distribution functions ρ p ( C )in (a) and ρ n ( C ) in (b). The variation of f ( W ) as a functionof the scaling quantity W for (c) positive and (d) negativecorrelations. The proportions of positive and negative correlationscan be calculated by λ p = (cid:90) ρ ( C ) dC , (8) λ n = (cid:90) − ρ ( C ) dC . (9)For positive correlations, we get λ p = 95% in winter, 78%in spring, 55% in summer and 84% in autumn. Corre-spondingly, the negative correlations have the proportion λ n = 1 − λ p = 5%, 22%, 45% and 16% in the four seasons.Further, we introduce probability distribution functions ρ p ( C ) = 1 λ p ρ ( C ) (10)for C > ρ n ( C ) = 1 λ n ρ ( C ) (11)for C <
0. They are presented in Fig.3 (a) and (b) anddepend on season. The averages (cid:104) C p (cid:105) , (cid:104) C n (cid:105) and standardp-3ongwen. Zhang et al. deviations σ p , σ n of positive and negative correlation canbe calculated with ρ p ( C ) and ρ n ( C ). Their results aresummarized in Table 1 for different seasons. λ p and (cid:104) C p (cid:105) have their maximum in winter and minimum in summer,which is in accord with the overall average of mean P M . concentration.In a system near its critical point, its physical propertiesfollow scaling behavior because of long-range correlation[25, 26]. The two-variable function of a physical propertycan be rewritten as a function of scaled variable, whichis universal. We take account of long-range correlationof P M . concentration and search for scaling behavior ofprobability distribution functions ρ p ( C ) and ρ n ( C ). Usingthe scaling variable W p = [ C − (cid:104) C p (cid:105) ] /σ p , (12) W n = [ C − (cid:104) C n (cid:105) ] /σ n , (13)we can introduce two scaling functions f p ( W p ) = σ p · ρ p ( C ) (14)for positive correlations and f n ( W n ) = σ n · ρ n ( C ) . (15)for negative correlations. As shown in Fig.3 (c) and (d),the scaling distribution functions for positive and nega-tive correlations in four seasons collapse together. Thisindicates that there is the same mechanism behind thecorrelation of P M . concentration.The different characters of positive and negative corre-lations can be demonstrated further by their PDF of dis-tance r and time lag τ ∗ , which are shown in Fig.4. ρ n ( r )of negative correlations has its peak at a the PDF of r and τ ∗ are shown in Fig.4 (a) and (c). The PDF of neg-ative correlations are presented in Fig.4 (b) for distanceand (d) for time lag. At the peaks of PDF, the distance ofnegative correlations is obviously larger than that of pos-itive correlations. The PDF of time lag has maximum at τ ∗ = 0 for positive correlations and τ ∗ (cid:54) = 0 for negativecorrelations. Negative correlations take on the characterof larger distance and longer time lag.The average positive correlation ¯ C p ( r ) at fixed distance r is shown in Fig. 4 (e). In winter and autumn, the decayof ¯ C p ( r ) follows a power law in some range of r . This couldbe related to the transport of P M . by atmospheric cur-rents. This trend will be weakened in spring and summer[27]. The average negative correlation ¯ C n ( r ) demonstratesquite different behaviors, which are shown in Fig. 4 (f).At large distance, ¯ C n ( r ) becomes nearly constant. Wesuppose that negative correlations are resulted by someexternal factors existing in large scale of distance.To define the network of correlation, the threshold ∆ ofcorrelation is determined from the shuffled data obtainedby permuting randomly the real data in a season. PDF ofcorrelation from shuffle data is compared with that fromreal data in Fig. 5. We define the average of absolute (a) r (100km) r p (r) DJFMAMJJASON (c) t * (hours) r p ( t * ) (e) −1 r (100km) C p (r) (b) r (100km) r n (r) (d) t * (hours) r n ( t * ) (f) −1 r (100km) − C n (r) Fig. 4: PDF of distance r is shown in (a) for positive and(b) for negative correlations. PDF of time lag τ ∗ is shown in(c) for positive and (d) for negative correlations. Averages ofpositive and negative correlations at distance r are plotted in(e) and (f). values of correlations from shuffled data as the threshold∆. We obtain ∆ = 0 .
017 and the adjacency matrix of thenetwork for correlation C according to Eq. (2).The weighted degree of a site, which characterizes itstotal correlation with surrounds, can be calculated usingEq. (3). The distribution of weighted degree for positivecorrelations are shown in Fig. 6. In comparison with Fig.1 of mean P M . concentration, the relevance of weighteddegree to mean P M . concentration can be found. Inthe regions with larger mean P M . concentration, thesites there have also larger weighted degree. There arethe largest weighted degrees in winter as the mean P M . concentration.For negative correlations, distributions of weighted de-gree in different seasons are shown in Fig. 7. On the con-trary, there are the largest weighted degrees in summerand the smallest weighted degrees in winter.The directional degree of a site, which is calculated ac-cording to Eq. (4), characterizes its net influence to sur-roundings. We present the distribution of directional de-gree for positive correlations in Fig. 8. In winter, thereare the strongest directional degrees in the most sites. Thesites of the north-west China, such as Xinjiang, Sichuanand Guizhou, have directional degrees in the directionfrom west to east. The directional degrees indicate netinfluences of P M . concentration along Gobi and InnerMongolia plateau, the North China Plain, Central China ,p-4orrelation and scaling behaviors of P M . concentration in China −2 −0.5 0.0 0.5 1.0 C r ( C ) Real dataShuffle data
Fig. 5: (Color online) PDF of correlations from real data andshuffle data in all seasons .
DJF MAMJJA SON (cid:176) N30 (cid:176)
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DJF MAMJJA SON (cid:176) N30 (cid:176)
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E 120 (cid:176) E Fig. 8: (Color online) Distribution of directional degree in net-work of positive correlations. and Yangtze River Delta. This phenomena can be relatedto the the east Asia winter monsoon [27, 28], which hasbeen shown by numerous studies. In other seasons, thedirectional degrees are smaller and less directional than inwinter. In summer especially, only the sites around PearlRiver Delta have visible directional degrees in the direc-tion from south to north. The distribution of directionaldegree for negative correlations are shown in Fig. 9. Nosignificant directional influence can be found for negativecorrelations.According to Eq. 5, G ij between sites i and j can becalculated. The threshold Θ = 3 .
25 of G can be obtainedby averaging absolute values of the shuffled data of G . Itis found that Θ is larger than all absolute values of nega-tive G ij . Therefore, only a network of positive G ij can bedefined by the adjacency matrix B of Eq. 6 . The distri-bution of weighted degree in this G network is shown inFig. 10. The weighted degrees in summer and autumn arenearly zero. In winter and spring, there are large weighteddegrees in the eastern part of China, especially around Bei-jing. The weighted degrees of G network demonstrate dif-ferent properties from that of C network, which is shownin Fig. 6. This is because of that G ij characterizes actu-ally the significance of the correlation C ij among ˆ C ij ( τ ) ofdifferent time lag τ . The distribution of directional degreesof G network is shown in Fig. 11. We can see that thereare large directional degrees only in winter and spring andin the eastern part of China, as the weighted degrees. Thedirection of directional degrees is from north to south. Wethink that the large weighted and directional degrees of G network are resulted in by anthropogenic emissions in theregion and the east Asia winter monsoon. With the C and G networks, different properties of P M . concentration inChina have been characterized.p-5ongwen. Zhang et al. DJF MAMJJA SON (cid:176) N30 (cid:176)
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DJF MAMJJA SON (cid:176) N30 (cid:176)
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We have studied the correlations of
P M . concentrations in different sites of China. Usingthe hourly P M . concentration data in 754 monitoringsites over China from Dec. 2014 to Nov. 2015, we cancalculate the correlations between different sites in fourseasons. The probability distribution functions of positiveand negative correlations depend on season. With aver-ages and standard deviations of correlation, the differentprobability distribution functions of different seasons canbe scaled into one scaling function. This indicates thatthere is maybe the same mechanism related to correlationof P M . concentration in different seasons. The positivecorrelations are resulted by the transport of P M . . Forpositive correlations, there are the largest average in win-ter and the smallest average in summer. The negativecorrelations are caused probably by large scale oscillatingclimate conditions. In opposite to positive correlations,there are the largest average in summer and the smallestaverage in winter for negative correlations. DJF MAMJJA SON (cid:176) N30 (cid:176)
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P M . concentrations in different sites of Chinaare studied from the aspect of complex network. Networksof P M . concentration can be defined either by correla-tions or by their significances. From weighted and direc-tional degrees of network, different properties of P M . concentration can be studied. In the networks of posi-tive correlations, the largest weighted degrees appear inwinter and in the North China plain as far as location isconcerned. The location distribution of weighted degreeand its seasonal dependence are in accord with that ofmean P M . concentration. In the networks of negativecorrelations, the largest weighted degrees appear in sum-mer. This indicates further that the origins of positiveand negative correlations are different. Significant direc-tional degrees are found for positive correlations in winter.They demonstrate the existence of net influences of P M . concentrations along Gobi and inner Mongolia plateau,the North China Plain, Central China, and Yangtze RiverDelta. From significances of positive correlation, we candefine a network which has large weighted and directionaldegrees only in winter and spring and in the eastern partof China. The directional degrees are in the direction fromnorth to south. These properties of P M . concentrationscould be related to anthropogenic emissions in the regionand the Asia winter monsoon. ∗ ∗ ∗ We are grateful to the financial support by Key ResearchProgram of Frontier Sciences, CAS (Grant No. QYZD-SSW-SYS019) and the fellowship program of the Planningand Budgeting Committee of the Council for Higher Ed-ucation of Israel, the Israel Ministry of Science and Tech-nology (MOST) with the Italy Ministry of Foreign Affairs,MOST with the Japan Science and Technology Agency,the BSF-NSF foundation, the Israel Science Foundation,ONR and DTRA. Yongwen Zhang thanks the postdoctoralfellowship funded by the Kunming University of Sciencesp-6orrelation and scaling behaviors of
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