Exploring the fluctuant transmission characteristics of Air Quality Index based on time series network model

Ecological Indicators 108 (2020) 105681

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Original Articles

Exploring the fluctuant transmission characteristics of Air Quality Index based on time series network model

T

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Chaoping Zhua, Ruguo Fana, , Jiaqin Suna, Ming Luob, Yingqing Zhangc a

Economics and Management School, Wuhan University, Wuhan 430072, Hubei, China School of Economics & Management, Guangxi Normal University, Guilin 541004, Guangxi, China c School of Management Science, Guizhou University of Finance and Economics, Guiyang 550025, Guizhou, China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Environmental management Air Quality Index Complex network Phase space reconstruction

Although many works have focused on the issues related to Air Quality Index (AQI) from the perspective of statistics or economics, the dynamic and temporal characteristics of AQI time series remain unclear. The main reason is that AQI time series has both non-linearity and complexity, which cannot be characterized by conventional ways. The inherent transmission laws of AQI volatility are of great importance for environment monitoring and management system. Thus, this paper has endeavored to address the issue from the interdisciplinary perspective of complex network theory. First, we collect the annual and the total AQI data of three cities (Shanghai, Wuhan and Guangzhou) in China from January 1, 2015 to December 31, 2017. Then the coarse graining method is employed in combination with the theory of phase space reconstruction to convert AQI time series into complex network. The coarse graining method is used to transform yearly and total AQI time series into the corresponding symbol sequences. In accordance with the theory of phase space reconstruction, we adopt CeC method and false nearest neighbor algorithm to estimate the optimal time delay and embedding dimension of AQI time series, respectively. The optimal time delay is integrated with the embedding dimension to map symbol sequences into AQI complex network. Based on complex network theory, we investigate the transmission of AQI fluctuation by the temporal characteristics and dynamics of yearly AQI networks (YAN) and total AQI networks (TAN), which include node strength, betweenness centrality and community structure. The empirical results show: (1) the AQI networks of various periods display small-world effect; (2) network evolution tends to be more complex and the modality in YAN has less alternative transformation paths than that in TAN; (3) the node strength and betweenness centrality in measuring the importance of nodes achieve a relative consistency; (4) the three cities have the same dominant fluctuation modalities both in YAN and in TAN; (5) the uneven distribution of transmission ability between clusters implies a preferential transmission whether in YAN or TAN. These findings not only help us to interpret the frequent volatility of AQI and control some extreme short-term AQI fluctuation, but also contribute to the reliability assessment of AQI and the prediction of AQI fluctuation. In summary, this paper not only reveals the temporal dynamic properties of AQI time series, but also provides some policy recommendations for environmental monitoring and management.

1. Introduction As people realized that poor air quality may have adverse effects on human health (Afroz et al., 2003), issues on environmental regulation have aroused extensive attention. In order to improve the quality of environment and provide health guidance for the public, environmental management departments around the world thus set out to monitor the environment quality, in which AQI has been one of the most widely used systems for the measurement of air quality. AQI simplifies the concentration of several air pollutants into a single conceptual index,

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which is based on ambient air quality standards and the impact of various pollutants on human health, ecology and environment. The value of AQI ranges from 0 to 500 and is divided into six grades. Generally, it hardly causes any bad effects to human if AQI is in the second grade (AQI is smaller than 100), which is the ideal interval for the variation of AQI. However, irritative symptoms are common in healthy people when AQI is greater than 200. As a matter of fact, AQI always indicates a trend of frequent fluctuations even at a relatively short time period. For example, the hourly AQI data of three cities (Shanghai, Wuhan and Guangzhou) in China from January 1, 2015 to

Corresponding author. E-mail address: [email protected] (R. Fan).

https://doi.org/10.1016/j.ecolind.2019.105681 Received 28 December 2018; Received in revised form 2 May 2019; Accepted 27 August 2019 1470-160X/ © 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. Hourly AQI time series of three cities in China from 2015 to 2017. “SH”, “WH” and “GZ” are respectively short for Shanghai, Wuhan and Guangzhou (similarly hereinafter).

abundant literatures on AQI, the dynamic and temporal characteristics of AQI time series remain unclear. Therefore, it is imperative to employ a new way to discover the evolution law of AQI and the transmission mechanism behind the fluctuation of AQI time series. Actually, the novel complex network method bridges the gap between time series and nonlinear analysis. In the context of network theory, a complex network is a network with non-trivial topological features, which do not occur in simple networks such as lattices or random graphs but often occur in graphs modeling of real systems. Such non-trivial topological features, for example, a high clustering coefficient, assortativity or disassortativity among vertices, community structure and hierarchical structure, enable us to acquire a deep understanding on the nonlinear characteristics of time series. As an effective method for characterization of time series, the complex network theory provides a range of powerful tools to describe the underlying dynamic of network mapped from time series. Then the problem rests with how to transform a time series into complex network, and ensure that the information embedded in network topology is essentially the same as that in the time series (Sun et al., 2014). At present, scholars have proposed several available approaches for converting a time series into complex network, in which the visibility graph method (Hloupis, 2017; Lacasa et al., 2008; Long, 2013; Wang et al., 2012; Zou et al., 2014; Zhou et al., 2017), coarse-graining method (An et al., 2014, 2018; Jia et al., 2018), sliding window (Gao et al., 2014, 2017; Sun et al., 2014), wavelet-based method (Jia et al., 2017), correlations coefficient method (Xia et al., 2018; Zhang et al., 2018) and phase space reconstruction (Gao and Jin, 2009; Tang et al., 2016; Wang and Tian, 2016) have been widely used. Generally, each method will inevitably exhibit some advantages and deficiencies. For example, visibility graph is one of the simplest ways for this conversion, but its practical significance may be not very clear. As for correlation coefficient method, a short time series may be insufficient to obtain a compelling result. Although it is a little more complicated than other methods, the phase space reconstruction algorithm has its own practical physics significance, as described in Takens theorem (Takens, 1981). In addition, Zhao et al. (2014) validated the dynamically equivalent transformation between time series and complex networks through mathematical testimony. The above researches provide a solid foundation for the conversion from time series to complex network. In this paper, we integrate the coarse-graining method with phase space reconstruction theory to transform a time series into a complex network. Previous studies on complex network mapped from time series have been extended to domains of economy, traffic, construction, astronomy, and so on. However, little attention has been paid to the issue on AQI time series from the perspective of complex network, even though AQI fluctuation may well reflect the situation of air quality that closely

December 31, 2017 are presented in Fig. 1. As can be seen from this figure, hourly AQI time series exhibits the characteristic of real-time variation and sharp fluctuation. In reality, the fluctuation of AQI is critical to environmental monitoring and management, thus it is an issue that deserves special interest. The above features of AQI have motivated researchers to conduct massive studies on the issues related to AQI. Liu (2002) analyzed the impact of the PM2.5 on AQI and concluded the influence of fine particles in AQI was overestimated. The optimal method for air quality was discussed by a comparison of the revised Air Quality Index (RAQI) with Pollution Standards Index (PSI) and AQI indices (Cheng et al., 2007), where RAQI was a background arithmetic mean index and mean entropy index that derived from the AQI (Cheng et al., 2004). By comparing the ground based indices with satellite based aerosol products, Zheng et al. (2014) argued the current AQI system in China was a better reflection of actual air quality than the past air pollution index (API) did. API is a simplified concentration of several air pollutants for the characterization of air pollution level by using SO2, NO2, and PM10 concentrations. Dai et al. (2018) applied the econometrics method to measure the recurrence statistics of extreme air pollution events. Meanwhile, there are some literatures about the impacts of air pollution (i.e., high AQI) on human health. For example, some scholars proposed that extreme AQI corresponded to certain environmental situations may cause undesirable effects such as an increasing risk of mortality (Dimitriou et al., 2013; Yang et al., 2004). Kyrkilis et al. (2007) developed an aggregate Air Quality Index to analyze the exposure level of Athenian citizens to polluted air. They found that the exposure had reached high levels and there was a gradual increasing trend. Banerjee and Srivastava Rajeev (2011) adopted the AQI and exceedance factor (i.e., the ratio of the annual average concentration of a particular pollutant with a respective standard for the particular area class) to measure the influence of rapid industrialization and commercial activities on health. The results showed that moderate or severe pollution was harmful to human health. Eleven indicators about AQI were evaluated by fuzzy inference system and c-mean clustering, which showed the criteria pollutants between 2011 and 2012 in Tehran did not have as much effect on the public health as the other pollutants (Hamedian et al., 2016). Besides, there were some other relevant works on AQI (Liu et al., 2017a,b; She et al., 2017; Ye et al., 2018). In a word, extensive attentions were paid to the applicability of AQI system, the impacts of major pollutants on AQI and the adverse effects of poor air quality measured by AQI on human health. Previous papers were usually on the basis of traditional ways, for instance, the statistics or econometrics methods. Nevertheless, the AQI time series is characterized by nonlinear and chaos, which often makes the conventional approaches impossible to capture the properties or essential laws of AQI. Despite 2

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related to our daily life. Why AQI fluctuate frequently? Is it possible to control the extreme fluctuation of AQI? How to assess the reliability of AQI data? How to predict the AQI fluctuation? These problems need to be further studied and discussed. Thus, this paper has endeavored to address these issues from a interdisciplinary perspective. First, we convert the AQI time series to complex network with the coarsegraining method and phase space reconstruction theory. Such a combination method makes our research more reasonable and convinced. Then the universal evolution characteristics of AQI fluctuation are explored by complex network method, which further enrich our understanding on the inherent law of AQI volatility. According to the results of complex network analysis, some policy recommendations for environmental monitoring and management are presented. The main contributions of this research lie in three aspects. (1) Hourly AQI data of three cities with broad representativeness in China is used to explore the universal evolution characteristics, which is an extension of complex network to environmental science. (2) Based on the novel complex network theory, we have modeled the environmental management issue from an interdisciplinary perspective, which provides a new insight for better understanding on the environmental change, especially the variations of air quality. (3) The policy recommendations on environmental monitoring and management have taken the complexity of AQI into consideration, which can strengthen the scientificity and feasibility of current environmental management system. The rest of this work is organized as follows: Section 2 illustrates the process of data collection and describes the detailed procedure of algorithms. In Section 3, the empirical analysis results are presented and discussed at length. Finally, in Section 4, policy recommendations for environmental monitoring and management are provided, the conclusions of this work are summarized and a prospect of further study is proposed.

such as power outage, equipment failure often makes it impossible to acquire a complete dataset. Regarding the stations with missing data, a missing measure rate of 5.00% or less is acceptable on account of the particularity of hourly AQI. Theoretically, each city should have 26,304 hourly AQI observations during the above time period. In practice, we eventually get 25,508 hourly observations for each city, that is, the missing AQI accounts for about 3.03% of the total observations during the selected time period. Such a low missing measure rate will not lead to the trend variation of original time series, instead, we can avoid the trend smoothing caused by the differentiated compensation data. 2.2. The algorithm for converting time series to complex network There are many available ways for the conversion from time series to complex network, such as visibility graph (Lacasa et al., 2008), phase space reconstruction (Gao and Jin, 2009), cross correlation coefficient method (Xia et al., 2018), to name but a few. As described before, each approach has its own advantages and disadvantages. In this paper, we transform time series to complex network by phase space reconstruction theory and coarse graining method. 2.2.1. Theory of phase space reconstruction From the perspective of mathematics and physics, phase space is commonly regarded as a multidimensional space aimed at describing all the possible states of certain dynamic system. The kernel of phase space reconstruction is to map time series of low dimension into phase space of high dimension, and the dynamics properties of time series remain constant after this transformation. Therefore, we can explore the characteristics of time series by phase space reconstruction. The basis of phase space reconstruction theory is Takens theorem (Takens, 1981). According to this theorem, the properties of attractor in the reconstructed phase space are equivalent to that in original time series, thereby a reconstructed attractor consists of several variables with time delay coordinates can be obtained in a dynamic system. Specifically, if we denote the time series as x1, x2, x3…xN, where N is the length of the series, then the reconstructed phase space vectors with time delay coordinates are given by

2. Material and methods 2.1. Data collection and pretreatment This paper selects the hourly AQI of three cities (Shanghai, Wuhan and Guangzhou) from January 1, 2015 to December 31, 2017 as samples. The three cities are respectively located in Eastern China, Central China and Southern China. The dataset is collected from Qingyue Open Environmental Data Center (http://data.epmap.org/), which is originated from National Urban Air Quality Real-time Release Platform (http://106.37.208.233:20035/). Cities located in other areas, for example, cities in Northern China, are not included mainly due to that AQI of these areas is frequently beyond the maximum range, which will bring undesirable disturbances to our research. In view of the data processing, a city that contains too much missing AQI is not considered in the dataset. The AQI data is not available prior to May 13, 2014 because it is not open to the public. Meanwhile, the daily AQI value is calculated in accordance with the average of six major pollutants in a whole day. However, the information embedded in daily AQI data is insufficient to investigate the dynamic characteristics of AQI volatility. Compared with hourly observations, the daily data has a larger time scale which inevitably hides some salient characteristics of AQI fluctuation. In order to better explore the transmission laws of AQI volatility and obtain more convinced results, we simply collect hourly AQI data of the mentioned three cities from January 1, 2015 to December 31, 2017 for study. There are several AQI monitoring stations in each city, for example, the monitoring stations of Shanghai are located in different areas of this city. In general, monitoring stations in distinct area always return different contaminant concentration related to AQI. In order to make full use of monitoring data from the whole stations, the hourly AQI we finally get is calculated on the average density of major pollutants from all stations. The overall process of calculation is based on the Technical Regulation on Ambient Air Quality Index (on trial) (HJ633-2012). However, the occurrence of uncontrollable factors

Yi = (x i , x i + τ , ⋯, x i + (m − 1) τ ),

i = 1, 2, 3, ⋯,

N − (m − 1) τ

(1)

where m is the optimal embedding dimension, τ is the time delay for phase space reconstruction, Yi is the ith reconstructed vector point with embedding dimension m in the reconstructed phase space. Thus, a time series with length N can be reconstructed and N-(m-1)τ vector points can be obtained. For the sake of probing the dynamic properties of original time series by phase space reconstruction theory, we should first ascertain the optimal embedding dimension and time delay for phase space reconstruction. In this paper, we apply CeC method (Kim et al., 1999) to pursue the optimal time delay due to its excellent anti-noise ability. Central to the CeC method is to make full use of the statistical difference exists in two correlations integral. The process of CeC method is as follows: let S(m, N, r, t) be the similar statistics of the disjoint time series via dividing the original time series {xn} into t disjoint segments, then the time delay t (here we denote time delay τ as t) can be estimated by:

S (m , N , r , t ) = where

1 t

t

∑ [Cs (m,

N / t , r , t ) − Csm (1, N / t , r , t )]

s=1

C (m , N , r , t ) =

2 M (M − 1)

∑1 ≤ i ≤ j Θ(r − ∥Yi − Yj∥ )

(2) and

0, p < 0 Θ(p) = ⎧ . 1, ⎨ ⎩ p≥0 In formula (2), C(m, N, r, t) is correlation integral, N denotes the length of time series, M = N − (m − 1)t is the vector point in m-dimensional space, Θ(p) is Heaviside function and r is the pre-set threshold value. Given an infinite series, the limit of S(m, N, r, t) is zero for each threshold value r when N approaches infinity. In reality, the S 3

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accordance with the specific goal we pursue. The AQI data we collect belongs to hourly observations, thus the value of Δt is set to an hour. Then the volatility series v(t) is transformed to the corresponding symbol sequences. Here, the symbol sequences si which consists of three grades can be denoted as

(m, N, r, t) cannot always reach zero because most actual sequences are not infinite and correlation relationship is very common among variables of sequences. The maximum variation of S(m, N, r, t) with respect to r is introduced to describe the above situation, which is defined as:

ΔS (m , t ) = max{S (m , r j, t , N )} − min{S (m , r j, t , N )}

(3)

Additionally, the following formula is generally used to calculate the optimal time delay t: −

si =

−

(4)

Scor (t ) = ΔS (t ) + |S (t )| −

3, ⋯,

2.3. Construction of complex networks from time series Firstly, the coarse graining method is used to transform AQI time series to symbol sequences by formula (9) and (10). For example, the AQI of Shanghai is 104 at seven p.m. on October 11, 2015, which is greater than the AQI value of 94 an hour earlier, thus the corresponding symbol is “R”. Similarly, the rest is disposed in the same way. Therefore, we obtain a symbol sequence ‘SDDDRRDDDDRRRR…’, which represents the fluctuation of hourly AQI. Then, the optimal time delay and embedding dimension of yearly (2015, 2016 and 2017) and total (from 2015 to 2017) AQI are calculated, and the results are respectively shown in Figs. 2 and 3. Here we divide the total AQI time series into yearly ones due to that some transmission characteristics of AQI may be overlapped in the former series, thus a short (yearly) series can help us dig out more useful information about the dynamic properties of AQI fluctuation. The time delay and embedding dimension for phase space reconstruction are jointly used to determine the similarity of phase space and the size of chaotic attractor. As shown in Fig. 2, “τ” means the optimal time delay, which represents the best time delay for a time series to reconstruct a phase space. For instance, the original series should move forward six hours to construct a new series when “τ” equals 6. The optimal time delay τ is 6 for Shanghai hourly AQI time series in 2015, while the time delay τ in 2016, 2017 and total are 7, 7 and 9, respectively. The optimal time delay (and embedding dimension) of Wuhan and Guangzhou at various periods is also calculated, as shown in Table 1. The results show that the time delay exist differences in AQI time series at various periods, which can be attributed to that each time series has its own nonlinear dynamic characteristics. Similarly, “m” in Table 1 means the optimal embedding dimension, which indicates the best dimension of series for phase space reconstruction. For example, if “m” equals 4, then we should generate four series including the original one to reconstruct the phase space. Also, Fig. 3 displays that the optimal embedding dimension is 4 for yearly AQI time series and 5 for the whole AQI time series. In summary, when the optimal time delay equals 6 and embedding dimension is equivalent to 5, we should reconstruct a five dimensions series with a time delay of 6 h, that is, four new series with a consecutive time delay of six hours and the original one should be combined to construct a phase space. According to Table 1, we can conclude the time delay of the whole AQI is no less than that of yearly ones for each city, and there is a similar regularity in embedding dimension. The larger time delay and embedding dimension of total AQI time series often mean more possible combinations for vector points, which will result in more complicated transmission characteristics. Thirdly, the optimal time delay is integrated with embedding dimension at various periods to convert symbol sequences to complex networks. As the time delay τ = 6 and embedding dimension m = 4 for Shanghai hourly AQI in 2015, the symbol sequences are transformed

n = 1, 2, (5)

N − (m − 1) t (r)

Let y (n) be the rth nearest neighbors of y(n), the square of the Euclidian distance between the point y(n) and its neighbor y(r)(n) is presented as m−1

Rm2 (n, r ) =

∑

[(x (n + kt ) − x (r ) (n + kt )]2 (6)

k=0

The above distance may change when the dimension of space increases from m to m + 1, and the new distance is given by

Rm2 + 1 (n, r ) = Rm2 (n, r ) + [x (n + mt ) − x (r ) (n + mt )]2

(7)

If the distance obtained by formula (7) is much larger than formula (6), then the false neighbor points are caused by projecting two nonadjacent points of high dimensional chaotic attractors into low dimensional coordinates. The embedding errors designated as a false neighbor is described as 1/2

2 2 ⎡ Rm + 1 (n, r ) − Rm (n, r ) ⎤ 2 ⎥ ⎢ Rm (n, r ) ⎦ ⎣

> Rtol

(8)

where Rtol is a tunable threshold ranging from 10 to 50. Generally, y(r)(n) is deemed as a false nearest neighbor of y(n) if formula (8) is satisfied. Given an actual chaotic time series, the proportions of the false nearest points are calculated under different dimensions. The dimension m can be identified as the optimal embedded dimension when the proportion of false nearest points is not greater than 5.00% or remain constant. 2.2.2. Method of coarse graining We denote the AQI time series as x(t), where t = 1, 2, 3, …, n. Its volatility series v(t) is given by

v (t ) =

x (t + Δt ) − x (t ) Δt

(10)

where the symbol R means “rising”, S means “stable” and D means “descending”, respectively. Thereafter, the time series x(t) can be replaced by symbol sequences S={s1, s2, s3}, si ∈ {R, S, D}. Based on coarse graining method and phase space reconstruction theory, the topological property of time series can be well reflected by the dynamic characteristics of the symbol sequences.

−

where ΔS (t ) is the average of △S(m, t), S (t ) is the average of S(m, N, r, t). The criterion for the best time delay t is when S reaches its first zero point, △S reaches its first local minimum or Scor reaches its global minimum. Aforementioned process is used to solve the optimal time delay for phase space reconstruction, then we apply the false nearest neighbor (FNN) algorithm (Kennel et al., 1992) to seek optimal embedding dimension. The FNN method is useful to determine in an intuitively convincing way the embedding parameters of a system, and one has good reason to assume that it is deterministic (Hegger and Kantz, 1999). First, we introduce the concept of false neighbor. Two non-adjacent points in multi-dimension space may evolve into adjacent points after being projecting into one-dimension space, which is the so called false neighbor. The basic idea of FNN algorithm is that the orbit of chaotic motion will be unfolded and the false neighbor will be gradually eliminated with the increase of embedding dimension, thus the trajectory of the whole chaotic motion is restored. For a m dimension space, the vector points can be denoted as

y (n) = (x (n), x (n + t ), ⋯, x (n + (m − 1) t )),

⎧ R, v (t ) > 0 S, v (t ) = 0 ⎨ D ⎩ , v (t ) < 0

(9)

where Δt is step size. In general, the step size can be determined in 4

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Fig. 2. The optimal time delay for Shanghai hourly AQI time series in (a) 2015, (b) 2016, (c) 2017 and (d) total.

modality represents a node in network. The directed edges which indicate the transformation order of combined symbol are used to connect the nodes. And the transformation frequency from one node to another is equal to the weight of the directed edges between two nodes. Thus, a directed and weighted complex network is established (see Fig. 4). For clarity, we draw the schematic for mapping time series into complex network, as presented in Fig. 5. 3. Results and analysis 3.1. Statistics and properties of AQI networks To explore the network properties and evolution characteristics of AQI networks, we calculate network diameter, average path length, average degree and network clustering coefficient, respectively. The network diameter is the maximum distance between any two nodes in network, which can be defined as Fig. 3. The optimal embedding dimension for Shanghai hourly AQI time series at various periods.

d = max{dij}

(11)

Average path length is one of the most robust measures of network topology. In some real networks, a small average path length is apt to facilitate the dissemination of information. For a directed network, the average path length between all pairs of nodes is given by

into vector points with the dimension of 4 and time-lag of 6 h. Finally, we obtain 8218 vector points in the reconstructed phase space, and each vector point can be denoted as a combined symbol like ‘RSDR’ (modality of AQI fluctuation every six hours). Each combined symbol is regarded as a fluctuation modality, and every unique

L=

1 N (N − 1)

∑ dij (12)

i≥j

Table 1 The optimal time delay and embedding dimension of hourly AQI time series for three cities at various periods. Shanghai

τ m

Wuhan

Guangzhou

2015

2016

2017

total

2015

2016

2017

total

2015

2016

2017

Total

6 4

7 4

7 4

9 5

7 4

11 4

9 4

18 5

12 4

15 4

8 4

15 5

5

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Fig. 4. Shanghai hourly AQI network in 2015.

calculated by (Barrat et al., 2004):

Ci =

1 si (ki − 1)

∑ j, k

wij + wik 2

aij ajk aik (13)

where si is the strength of node i, ki denotes the degree of node i, wij is the weight from node i to node j, aijajkaki denotes a triangle from nodes (i, j, k). If nodes (i, j, k) forms a triangle, then aijajkaki equals 1, otherwise it equals 0. Network clustering coefficient is the average of clustering coefficients for all nodes. The calculated results of network properties are listed in Table 2. The network diameter is smaller than 4 even though the number of nodes in each network is greater than 80, indicating the transformation between any two nodes takes no more than four hours. And the average shortest-path length is less than 2, which means the average conversion time between modalities is shorter than 2 h. Meanwhile, the average path length of YAN is smaller than that of TAN for each city. This confirms that the simpler the network topology is, the shorter time it takes to the transformation between modalities. Besides, both the YAN and the TAN have large average clustering coefficients, which indicate nodes in AQI networks form compact connections. Thus, the AQI networks at various periods are characterized by small average shortestpath lengths and large clustering coefficients, proving they have smallworld effect. This effect indicates “shortcuts” exist among modalities, which contribute to the rapid spread of information. Thus a slight change in connections between nodes may lead to drastic shift on the performance of network. Also, it shows the relationships between AQI fluctuations are close and extensive, and the small-world effect facilitates the transmission of fluctuation between modalities, which in turn explains why AQI volatility is so frequent. As the connections between nodes in AQI networks are established in accordance with chronological order, the in-degree is identical to out-degree except the initial node and the terminal node, so here we just choose the out-degree for study (similarly hereinafter). As can be seen from Table 2, the average out-degree in YAN is smaller than that in TAN for each city. The finding suggests that each modality in YAN has

Fig. 5. Schematic for mapping time series into complex network.

where dij in un-weighted network is the distance between node i and node j, while in weighted network it denotes the sum of the weights on the shortest path between two nodes. Clustering coefficient is another robust measure of network topology. Its value represents the aggregation degree of nodes in network. For a weighted network, the weighted clustering coefficient is 6

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Table 2 Properties of AQI networks at various periods. Shanghai

2015 2016 2017 total

Wuhan

Guangzhou

d

L



C

d

L



C

d

L



C

3 4 3 4

1.69 1.71 1.63 1.95

28.3 27.3 31.4 41.1

0.756 0.862 0.835 0.658

3 4 3 4

1.73 1.75 1.65 1.97

25.1 26.3 29.6 40.2

0.871 0.876 0.779 0.733

3 4 3 4

1.65 1.65 1.56 1.88

30.7 31.3 36.4 48.3

0.800 0.853 0.766 0.729

probability than the “descending” of AQI. Furthermore, there is an increasing trend for the average NS in YAN from 2015 to 2017, which means average control ability of the nodes is gradually rising and network evolution tends to be more complex. Also, NS distributions and their cumulative probability distributions at various periods are shown in Fig. 6. As can be seen from Fig. 6, the three cities present a similar result of NS distribution. It suggests that the majority of nodes have a low NS, while only a small amount of nodes hold a high NS. At first glimpse, there is a “heavy tail” effect in the inset figure of NS distribution, which means a potential power-law characteristic in AQI networks. But the double logarithmic curve between NS and their cumulative probability does not follow a linear relationship (see Fig. 6), indicating AQI networks are not scale-free networks. However, some salient features still can be found in NS distributions and their cumulative probability distributions. For example, the top 15 nodes of NS in YAN occupy no less than 50.00% of the overall NS and the top 80 nodes of NS in TAN burden more than 80.00% of the overall NS. It can be concluded that the top nodes play a decisive role in network transmission, though they account for a comparatively low proportions in AQI networks. This is in line with the truth of Pareto’s Principle, that is, the most important tends to account for a small proportion, for instance, 20.00%, while the rest is less important and possesses a large proportion. This result may contribute to the prediction of AQI fluctuation. Meanwhile, such a result implies that nodes do not always have the same connectivity, the behind mechanism is that the fluctuant modalities in AQI networks play a dominant role in the transmission of AQI time series. In other words, the fluctuation of AQI does not change irregularly, to some extent, it is partly attracted, restrained and influenced by dominant volatility. Besides, there is no difference in the modalities whose cumulative NS exceed 50.00% of the overall NS in YAN. For example, the top 16 (about 19.75%) modalities ‘RDRD’, ‘DRDR’, ‘RRDD’, ‘DRRD’, ‘RRDR’, ‘RDDR’, ‘RRRD’, ‘RDRR’, ‘DRRR’, ‘DDRR’, ‘DDRD’, ‘DRDD’, ‘RDDD’, ‘DDDR’, ‘RRRR’ and ‘DDDD’ account for more than 55.00% of the overall NS in YAN for Shanghai, which is the same as that in YAN for Wuhan and Guangzhou. In addition, the top 32 (about 13.22%) modalities shoulder more than 51.00% of the overall NS in TAN for each city. Thus, the three cities have the same dominant fluctuation modalities whether in YAN or TAN. The above results suggest that the AQI fluctuation do have some evolutionary laws in common. Furthermore, the remarkable evolution patterns cannot be discovered simply in traditional research ways.

less alternative transformation paths than that in TAN. Accordingly, the AQI fluctuation in YAN presents a greater predictability. Moreover, it signifies that the control of some extreme short-term AQI volatility is expected to become a reality. The reason why the AQI networks at various periods are heterogeneous is that AQI tends to be affected by a series of factors, such as regulation of environmental department, climatic variation and diverse human activities at a specified time period, and so on. 3.2. Node strength and its distribution Node strength (NS) is a comprehensive indicator of the influence of a node in network, which is similar to the node degree in un-weighted network. The indicator takes all the neighbors linked to certain node and the corresponding weight into consideration. In other words, the larger the NS of certain node is, the more frequent this node appears in the network, which in turn implies that the node is of great importance in the network. In a directed and weighted network, the strength of a node generally consists of incoming strength and outgoing strength, which can be calculated by: N

Siin =

∑ aji wji,

N

Siout =

j=1

∑ aij wij (14)

j=1

where aij is the element of adjacency matrix, wij is the weight of edge from node i to node j. Generally, the incoming strength of certain node can manifest the probability from other modalities to the current modality, and a large probability indicates other modalities are more likely to transform to the current modality. Similarly, the outgoing strength of certain node can reflect the probability from current modality to other modalities. Thus, a large indicator often means this node is more likely to transform to other nodes. As earlier mentioned, in this paper we only select the outgoing strength of nodes for research. The maximum NS, modality with maximum NS and average NS in AQI networks are listed in Table 3. The results show that the maximum NS in TAN are greater than that in YAN for each city. At the same time period, the biggest NS is 501 for Guangzhou in 2015, 471 for Wuhan in 2016 and 423 for Shanghai in 2017, respectively. And the most frequent modalities with maximum node strength in various YAN are “DRDR” and “RRDR”. It indicates that the “rising” of AQI has a higher Table 3 The maximum node strength, modality with maximum node strength, and average node strength in YAN and TAN for each city. City

Indicator

2015

2016

2017

Total

Shanghai

Smax Mmax Sm

415 RDRD 101.4

459 RRDR 106.4

423 RRDR 107.4

600 DRRDR 104.8

Wuhan

Smax Mmax Sm

464 DRDR 102.7

471 RRRR 106.5

404 DRRD 107.3

697 RDRDR 106.4

Guangzhou

Smax Mmax Sm

501 DRDR 101.2

377 DRDR 105.0

380 RRDR 107.3

528 DRRDR 104.7

3.3. Characteristics of betweenness centrality The indicator betweenness centrality (BC) reflects the intermediary status of a node, which plays a key role in identifying nodes that have high control ability (An et al., 2018). The more central a node lies in, the larger the number of shortest paths that pass through this node is (Barthélemy, 2011). This indicator is generally estimated by

BCi =

∑ j≠k≠i

Njk (i) Njk

(15)

where Njk(i) denotes the whole number of shortest paths from node j to 7

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Fig. 6. Node strength distributions and their cumulative probability distributions in YAN and TAN for (a) Shanghai, (b) Wuhan and (c) Guangzhou.

than that for the other two cities (see Fig. 7), revealing the nodes in YAN for Wuhan have a greater mediation effect. Thus, in terms of the intermediary effect of nodes on AQI fluctuation, it may be easier for Wuhan to control the transmission of AQI fluctuation compared with Shanghai and Guangzhou. And these nodes with high BC often mean that they have excellent ability to control the transmission of other pairs of nodes that go through them. However, as can be seen from Fig. 7(d), the variation of BC in TAN is nearly the same for three cities. Thus, if we adopt a long period of AQI time series to explore its volatility, some salient dynamic characteristics may be covered by several stochastic factors. This verifies that the division of total AQI series into three yearly ones is of great significance for discovering the transmission properties of AQI fluctuation. Moreover, the dominant modalities

node k that pass through node i, Njk denotes the whole number of shortest path from node j to node k. The scatter relationship between node numbering and the corresponding BC is presented in Fig. 7. This figure indicates that most of the modalities hold a relatively low BC while only a few nodes have a large one. The nodes with large BC, to some extent, are governing the transmission between modalities. It is worth noting that a few nodes have the zero BC, indicating that they play no intermediary roles in AQI fluctuation conduction. The results are consistent with that in Fig. 6. It also suggests that the transmission of AQI fluctuation largely depends on nodes with a high BC, thereby it is possible to effectively control the transmission patterns in the near future. Meanwhile, the top 20 nodes of BC in YAN for Wuhan are higher

Fig. 7. Betweenness centrality of nodes in AQI networks for each city in (a) 2015, (b) 2016, (c) 2017 and (d) total. 8

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Fig. 8. The relationship between node strength and betweenness centrality in AQI networks for each city in (a) 2015, (b) 2016, (c) 2017 and (d) total.

method can be used to divide network into clusters, the capacity of transmission between clusters is incapable of being measured. Accordingly, in order to estimate the transmission ability between clusters, we introduce the transmission ability TP,Q from cluster P to cluster Q , which is defined as (Gao et al., 2014):

with the highest BC are totally different for the three cities, even though the YAN are at the same period. We come to the conclusion that AQI fluctuation should be regulated and controlled in line with the characteristics of different periods. The result also reveals that modalities with a high BC in TAN hardly show any difference, while in YAN they have less in common. By identifying the special modalities of AQI fluctuation at specified time period, policy makers can scientifically formulate and implement effective measures to prevent the AQI from sharply rising. As shown in Fig. 8, the relationship between BC and NS exhibits an interesting result. It is worth noting that there are blank areas in Fig. 8(a)–(c) when the node degree ranges from 120 to 250. The blank areas indicate that there are hardly any nodes with both moderate NS and BC in AQI networks. The distributions of NS and BC are naturally divided into two segments (see Fig. 8), in which the bottom left segment is filled with nodes that have small BC and low NS, while the top right segment is concentrated with nodes that have large BC and high NS. It suggests that a low NS often comes with a small BC and a high NS always accompanies by a large BC. Such a meaningful result can serve as a significant reference to the reliability assessment of AQI. This finding may come from the fact that AQI are under more of a fluctuant state than a stable state. Consequently, the two indicators for measuring the importance of nodes achieve a relative consistency.

TP, Q =

∑ i ∈ P, j ∈ Q

wij (16)

where wij is the weight of directed link from node i to node j, node i belongs to cluster P and node j belongs to cluster Q. With the community identification method mentioned above and the formula (16), we obtain the clustering result of each network and the corresponding transmission ability, as demonstrated in Fig. 9. This figure illustrates that there are four or five clusters in each AQI network and the transmission between any two clusters is available. In order to distinguish cluster from each other, each cluster in AQI network is marked with a different shape and color. Meanwhile, we calculate the percentage of modality each cluster holds and discriminate their transmission ability with links of different widths. The transmission clusters for Shanghai AQI network in 2015 are presented in Fig. 9(a1). It can be seen from this figure that the percentages of modality from cluster 1 (C1) to cluster 5 (C5) are 19.34%, 22.22%, 10.70%, 22.64% and 25.10%, respectively. However, their total transmission abilities to other clusters (except themselves) are 1010, 1064, 631, 960 and 748, respectively. Therefore, it can be concluded that the transmission ability of clusters in YAN is not proportional to the number of nodes they occupy. The result reveals that cluster in the YAN with a large number of nodes does not necessarily have high transmission ability, and a similar result can be obtained in the TAN. The above discovery provides a potential way for the transmission process prediction of AQI fluctuation. In order to better depict the transmission features of clusters, we draw the distribution map of transmission ability between clusters, as shown in Fig. 10. This figure shows there are mainly two levels in the distribution of transmission ability within and between clusters. The

3.4. Analysis of transmission ability between clusters For complex network, it is very common to form a community structure, which contributes to the analysis of the clustering effect of network transmission. The fluctuation clustering effect is mainly due to the frequent transmission between modalities. Modalities with high mutual transmission probability tend to concentrate into a sub-network and the cluster gradually forms. A cluster is characterized by compact connections among nodes within the cluster while sparse links in different clusters. And Blondel et al. (2008) proposed an effective algorithm based on modularity for the detection of community. Though this 9

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Fig. 9. Transmission clusters for each city in yearly AQI networks and total AQI networks. In the top left of each subgraph, the letters “a”, “b” and “c” respectively stand for Shanghai, Wuhan and Guangzhou, and the number from 1 to 4 respectively represent the year 2015, 2016, 2017 and total. For example, “b3” refers to the transmission clusters graph of Wuhan in 2017. In order to distinguish cluster from each other, each cluster in the subgraphs is coded with a different color and shape. The proportion along with a cluster is the percentage of modality the cluster holds. The width of arrow is proportional to transmission ability between clusters, which can be observed from the number next to the arrow.

first level is the transmission ability within a cluster is higher than that between clusters both in YAN and TAN, which also indicates the result of clustering is effective. The second level is the distribution of transmission ability between clusters is not uniform, for example, the transmission from cluster 5 to cluster 3 has the lowest transmission ability in YAN, as presented in Fig. 10(a1). However, Fig. 10(a4) shows the transmission ability from cluster 2 to cluster 3 is the highest in TAN. As a result, there exists a preferential transmission between clusters whether in YAN or TAN. Such an uneven distribution of transmission helps us track out more useful information about the transmission of AQI fluctuation. For instance, the uneven transmission characteristic between clusters can be used to predict the AQI fluctuation. Besides, it provides a critical foundation for establishing a more scientific mechanism of AQI monitoring and management.

network. With the methods provided by complex network theory, node strength (NS), betweenness centrality (BC) and community structure and other network properties are investigated to study the dynamic and transmission of AQI fluctuation. Based on the empirical analysis results, we arrive at some heuristic conclusions about the transmission of AQI fluctuation. On the one hand, AQI networks at various periods have both small average shortest-path lengths and large value of clustering coefficients. It suggests a small-world effect in AQI network, which is in agreement with previous study on climate network (Tsonis et al., 2006). This effect means “shortcuts” exist in AQI fluctuation, thereby a slight change in connections between nodes may lead to drastic shift on the performance of network. To a certain degree, the small-world effect enables us to better interpret the frequent volatility of AQI time series. On the other hand, there is an increasing trend for NS in YAN, indicating the average control ability of nodes is gradually rising and network evolution tends to be more complex. Meanwhile, the modality in YAN has less alternative transformation paths than that in TAN, that is, the complexity and diversity of fluctuation in TAN is more notable than that in YAN. Meanwhile, the dominant fluctuation modalities with large NS can control the transmission of AQI fluctuation, which provides powerful evidences for the control of some extreme short-term AQI volatility. Thus, it is observed that some extreme short-term AQI fluctuation still can be controlled, though network evolution exhibits an

4. Conclusion and discussion 4.1. Conclusion The inherent transmission laws of AQI volatility are of great importance for environment monitoring and management system. However, AQI time series has both non-linearity and complexity, which cannot be characterized by traditional methods. This paper has addressed the issue from the interdisciplinary perspective of complex 10

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Fig. 10. Distribution of transmission ability between clusters. The meaning of the alphanumeric code in top left of each subgraph is the same as that in Fig. 9. Different color corresponds to distinct transmission ability between clusters, which can be distinguished by the color bar. The direction of transmission is from columns to rows.

4.2. Policy recommendations

increasingly complex trend. Besides, a low NS often comes with a small BC and a high NS always accompanies by a large BC, which implies the indicators NS and BC for measuring the importance of nodes achieve a relative consistency. In terms of the top modalities that shoulder more than half of the overall NS, the three cities have the same dominant fluctuation modalities both in YAN and in TAN. The transmission ability from one cluster to another is unevenly distributed, indicating a preferential transmission between clusters whether in YAN or TAN. As a result, the consistent relationship between NS and BC may contribute to reliability assessment of AQI, and the undifferentiated dominant fluctuation modalities and the uneven distribution of transmission ability between clusters jointly provide a new approach for the prediction of AQI fluctuation.

Although a great deal of endeavor has been made to explore the properties of AQI, the transmission regularity of AQI fluctuation has yet to be unearthed. One of the most significant reasons is that AQI time series has both non-linearity and complexity, which often makes the traditional time series analysis methods fail to work. Meanwhile, the transmission of AQI fluctuation is impossible to be characterized by traditional methods. Different from conventional time series analysis, this paper employs the nonlinear analysis method based on complex network theory to investigate the dynamic and temporal characteristics of AQI fluctuation. We obtain some valuable results about AQI fluctuation that are by no means discovered by traditional methods. As environment management often involves a series of complex activities, the results of prior linear analysis may be of little practical significance to decision-making. Instead, the results revealed by complex network

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research. Also, this model can serve as a research paradigm for other fields such as the energy consumption, stock market and foreign exchange rates. However, the intricate volatility of AQI time series is affected by a combination of multiple factors, there are still many questions need to be studied in theory and practice. Future works should adopt AQI time series of different time scales and introduce more variables into models.

analysis can provide several practical insights for environmental monitoring and management. (1) A deeper understanding on the temporal dynamic properties of the fluctuation of AQI time series. Many regulators may regard the volatility of AQI as irregular and random, while others think that it possesses periodic characteristics. The above two defective viewpoints often result in distorted cognition on AQI fluctuation. As a matter of fact, the volatility of AQI displays characteristics of nonlinear and chaos dynamics, and the fluctuation of AQI does not change irregularly. For example, the small-world effect indicates there exists “shortcuts” in AQI fluctuation, a slight change of external conditions may lead to an extreme fluctuation of AQI. For environmental department, it provides a fresh way to interpret the frequent volatility of AQI. Thus, without realizing the inherent laws of AQI fluctuation, the actual environment quality may not be well reflected by environmental monitoring and management system. (2) A novel perspective for the comparison of similarities and differences of AQI for different cities. Traditionally, the similarities and differences information of AQI between cities can be obtained simply by descriptive analysis. Now a new method of complex network is available for acquisition of the information. With the tools provided by complex network, we can get a series of temporal dynamic properties of AQI fluctuation and do some contrasts to different cities. Such a comprehensive and diversified comparison undoubtedly enhances our understanding on the essential gaps of air quality between cities. Thus we can learn from the practices of cities that do well in environmental monitoring and management, while draw lessons from cities that achieve poor performance in this respect. (3) A reasonable scheme to control the extreme short-term AQI fluctuation. As an effective indicator for measuring the influence of nodes in AQI network, the NS provides us useful information about network evolution. For example, the result of NS analysis show the fluctuation modality in YAN has less alternative transformation paths than that in TAN, that is, the AQI fluctuation in YAN presents a greater predictability. Meanwhile, the dominant fluctuation modalities with large NS are the so called turning points of fluctuation transmission, which have the power to control the transmission of AQI fluctuation. The results lay the foundation for effective control of some extreme short-term AQI volatility. In summary, the network properties NS enable us to identify the dominant fluctuation modality, thereby controlling the transmission turning points of AQI fluctuation. Based on the results of complex network analysis, environmental managers can find out the dominant modality of AQI fluctuation and formulate a reasonable scheme to control some extreme short-term AQI fluctuation. (4) A scientific approach for reliability assessment of AQI and the prediction of AQI fluctuation. As a low NS often comes with a small BC and a high NS always accompanies by a large BC, the transmission characteristics of AQI fluctuation can be judged by the consistent relationship between NS and BC. For environmental management, this unique relationship can be used for reliability assessment of AQI. Based on the same dominant fluctuation modalities and the preferential transmission ability between clusters in AQI network, environment managers can adopt a novel approach to predict AQI fluctuation. For example, if the AQI time series exhibits a fluctuation modality “RRDD” that belongs to a dominant fluctuation modality, then its transmission probability that represents the possible transmission path can be calculated in accordance with the transmission ability distribution. Given the possible transmission path, the prediction of AQI fluctuation will become available.

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