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A study on daily PM2.5 concentrations in Hong Kong using the EMD-based MFDFA method
Physica A 530 (2019) 121182
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A study on daily PM2.5 concentrations in Hong Kong using the EMD-based MFDFA method ∗
Chen Zhang, Xiaofeng Wang , Shengbing Chen, Le Zou, Xin Zhang, Chao Tang School of computer science and technology, Hefei University, Hefei, 230009, China
highlights • • • •
The multifractal scaling properties of PM2.5 time series are discussed by EMD-based MFDFA. Time series of PM2.5 concentrations for MK, TW and TM are all multifractal and anti-persistence. The multifractality is caused not only by long-range correlation but also by fat-tailed distribution. The roadside site (MK) possesses the highest degree of multifractality, followed by the urban (TW) and the rural one (TM).
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Article history: Received 16 November 2018 Received in revised form 7 April 2019 Available online 8 May 2019 Keywords: PM2.5 EMD MF-DFA Multifractal Hong kong
a b s t r a c t The daily average PM2.5 concentrations of Mong Kok (MK), Tsuen Wan (TW) and Tap Mun (TM) in Hong Kong were obtained from Hong Kong Environmental Protection Department and were analyzed. Without any spatial differentiation, PM2.5 concentrations all were higher in winter and lower in summer. In order to more accurately explore the multifractal characteristic of PM2.5 time series, the empirical mode decomposition (EMD) based multifractal detrended fluctuation analysis (MF-DFA) method is used in this paper. The results show that the PM2.5 time series for MK, TW and TM are all anti-persistence, that is, whenever the PM2.5 concentrations has been up (down), it is more likely that it will be down (up) in the close future. The multifractality is caused not only by long-range correlation but also by fat-tailed distribution. Moreover, the roadside site MK displays the highest degree of multifractality, followed by the urban (TW) and the rural (TM) one. The proposed method can more accurately analyze the multifractal characteristic of PM2.5 time series, which provides a solid foundation for further PM2.5 pollution research. © 2019 Elsevier B.V. All rights reserved.
1. Introduction In recent years, PM2.5 (particulate matter less or equal than 2.5 µm) pollution has been becoming serious. Epidemiological evidences suggest that the increased PM2.5 levels are related to mortality and a number of pulmonary effects [1–3]. The existing researches on PM2.5 mainly focus on analyzing its chemical composition and its spatial distribution [4–6], discussing its association with the meteorological factors [7–9] and simulating its values with the numerical mode [10–12]. Although there are many studies on PM2.5, relatively few researches have focused on the multifractal scaling behavior of PM2.5 time series. Shi [13] used the MF-DFA method [14] to explore the scaling and multifractal properties of the hourly PM2.5 average concentration series at the four air monitoring sites of Chengdu in China. This is the one and only paper ∗ Corresponding author. E-mail address: [email protected] (X. Wang). https://doi.org/10.1016/j.physa.2019.121182 0378-4371/© 2019 Elsevier B.V. All rights reserved.
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Fig. 1. Locations of three sampling sites: Mong Kok (22.321N, 114.11E), Tsuen Wan (22.381N, 114.11E), and Tap Mun (22.473N, 114.371E).
that studies the multifractal scaling behavior of PM2.5 time series. As we all known, the MF-DFA method can unveil the multifractal nature hidden in nonstationary time series, which has been widely used in diverse fields including automatic epileptic seizure detection [15], stock market [16–18], groundwater dynamics [19], daily air temperature [20]. However, in the detrending part of MF-DFA, some shortcomings still remain; Linear, quadratic, cubic, or higher order polynomials may be used in the trend fitting using the least squares (LS) method, but it is not easy to determine which order is the most appropriate. This will result in the inaccurate results when the series are too long or their trends are not of the polynomial type. Fortunately, Huang et al. [21] proposed empirical mode decomposition (EMD), which decomposes raw signals into a limited number of intrinsic mode functions (IMFs) and a residual term. The EMD can resolve the above problem that existed in the MF-DFA method [22]. In the process of detrending, EMD performs better than LS and it does not require any pretesting process, which makes it generally more convenient, more widely applicable, and more accurate. The EMD-based MF-DFA method has been used in the surface electromyography signals [23], epilepsy detection [24], and precipitation complexity measurement [25], futures markets [26]. Compared with the traditional MF-DFA method, the EMD-based MF-DFA method can not only detect multifractal scaling properties of PM2.5 time series, but can give a more accurate result and do not require any pretesting process. Therefore, in order to more accurately explore the multifractal characteristic of PM2.5 time series, the proposed method is used to analyze the multifractal scaling properties of PM 2.5 time series in this paper. This study provides a solid foundation for further PM2.5 pollution research. This paper is organized as follows. Section 2 introduces data source and the EMD-based MFDFA method. Section 3 is the experimental results, which provides an empirical analysis of the proposed method. Section 4 draws the conclusions. 2. Materials and methods 2.1. Study area Hong Kong is situated in the southern tip of the Pearl River Delta region in China, which has one of the highest population densities in the world. Under influence of the Asian monsoon, Hong Kong’s climate is sub-tropical with four distinct seasons. In warm season of summer, the prevalent wind is the southwesterly wind. Warm and damp marine air masses originate from the South China Sea are carried to Hong Kong. In cold season of winter, the most prevailing wind direction is a northeasterly wind. Therefore, in winter, pollutant emitted from the Asian continent is transported to Hong Kong. In addition, winter has much less rainfall in comparison with summer. Three sampling sites were selected based on their different land-use categories, namely roadside, urban and rural environments. Three sampling sites were Mong Kok (MK), Tsuen Wan (TW) and Tap Mun (TM), which are illustrated in Fig. 1. The MK site is a roadside site located in a mixed commercial and residential area with heavy traffic and surrounded by many tall buildings. The urban site TW, located on the rooftop of a government building, represents a densely populated residential area mixed with some commercial and industrial areas. The rural site HT is located at the northeast coast of Hong Kong Island, which experiences the least impact of anthropogenic activities among these three sites.
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2.2. Methodology 2.2.1. Data We choose daily average PM2.5 concentrations (µg/m3 ) of the above three sampling sites, recorded from 1 March 2016 to 28 February 2017. All data obtained from Hong Kong Environmental Protection Department. 2.2.2. The EMD algorithm The empirical mode decomposition (EMD) first proposed by Huang et al. [17] is an innovative data processing algorithm for nonlinear and non-stationary time series. The EMD method decomposed the time series x (t ) into a number of intrinsic mode functions (IMFs), which satisfy the following two conditions: (1) in the whole time series, the numbers of local extrema and the numbers of zero crossings must be equal or differ by 1 at most; and (2) at any time point, the mean value of the ‘‘upper envelope’’ and the ‘‘lower envelope’’ must be zero. The decomposing process is called a sifting process, which can be describes as follows: (1) Identify all extrema of x(t); (2) Interpolate the local maxima to form an upper envelope U(x); (3) Interpolate the local minima to form a lower envelope L(x); (4) Calculate the mean envelope: e (t ) = [U (x) + L (x)]/2 (5) Extract the mean from the signal g (t ) = x (t ) − e(t); (6) Check whether g (t) satisfies the IMF∑conditions. If YES, g (t) is an IMF, stop sifting; If NO, let x (t ) = g (t ) and keep sifting. Finally, we obtain n rn (t ) = x (t ) − i=1 gi (t ), where rn is a residue representing the trend of the time series. 2.2.3. The EMD-based MFDFA algorithms Let {x (t )} be time series, t = 1, . . . , N, where N is the length of time series. The EMD-based MFDFA method [22] can be described as follows. Step 1: Construct the cumulative sum u (t ) =
t ∑
x(i), t = 1, 2, . . . , N.
(1)
i=1
Step 2: The series u (t ) is partitioned into Ns disjoint segments of the same size s, where Ns = [N /s]. Each segment can be denoted by uv such that uv (i) = u(l + i) for 1 ≤ i ≤ s, where l = (v − 1)s. Step 3. For each segment uv , we obtain the EMD-based local trend rn (i) by the EMD method. We can obtain the residual sequence
εv (i) = uv (i) − rn (i) ,
(2)
Step 4. The detrended fluctuation function F (v, s) of the segment uv is defined as the root of the mean squares of the sample residuals εv (i)
[F (v, s)]2 =
s 1∑
s
[εv (i)]2
(3)
i=1
The qth order overall detrended fluctuation is calculated as follows,
{ Fq (s) =
}1/q
Ns ] 1 ∑[ F (v, s)q Ns
(4)
v=1
where q can take any real value except for q = 0. When q = 0, we have
{ F0 (s) = exp
Ns 1 ∑
Ns
} ln[F (v, s)]
(5)
v=1
Step 5. Varying the value of s, we can determine the power-law dependence of the detrended fluctuation function Fq (s) on the size scale s, which reads Fq (s) ∼ sh(q) ,
(6)
where h(q) is the generalized Hurst index. If h(q) depends on q, then the correlation is multifractal, otherwise, the correlation is monofractal. In general, when h (2) > 0.5, the kinds of fluctuations related to q are persistent, that is, an increase (decrease) is likely to be followed by another increase (decrease). When h (2) < 0.5, the kinds of fluctuations related toq are anti-persistent, that is, an increase (decrease) is likely to be followed by another decrease (increase). However, if h (2) = 0.5, then the kinds of fluctuations displays a random walk behavior. The Renyi exponent τ (q), which is related to the general Hurst exponent h(q) obtained from MFDFA, can be obtain by Eq (7):
τ (q) = qh (q) − 1
(7)
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Fig. 2. The annual average PM2.5 concentrations at the three sites from 1 March 2016 to 28 February 2017.
Fig. 3. The monthly average PM2.5 mass concentrations at the three sites from 1 March 2016 to 28 February 2017.
Through the Legendre transform, the Hölder exponent α (q) and singularity spectrum f (α) can be calculated as follows:
α (q) = h (q) + qh′ (q)
(8)
f (α) = q [α (q) − h (q)] + 1
(9)
where h (q) represents the derivative of h (q) with respect to q. The Hölder exponent α (q) characterizes the strength of the singularity, and f (α) represents the Hausdorff dimension of the fractal subset with the exponent α . ′
3. Results and discussion 3.1. PM2.5 mass concentrations In order to better understand the tendency of PM2.5 mass concentrations, the annual average PM2.5 concentrations at these three sites was shown in Fig. 2 the monthly average PM2.5 concentration at these three sites was shown in Fig. 3 and the seasonal average PM2.5 concentration at these three sites was shown in Fig. 4. In Fig. 3, PM2.5 concentrations have an apparent monthly variation. The highest monthly average concentrations were 31.96 µg/m3 , 27.54 µg/m3 and 24.3214 µg/m3 in December 2016 for MK, TW and TM, respectively. Nevertheless, the lowest monthly average concentrations were 21.68 µg/m3 , 20.52 µg/m3 , 16.23 µg/m3 in June 2016 for MK, TW and TM, respectively. Fig. 3 showed its increasing trend from summer to winter. Fig. 4 showed the PM2.5 concentrations were higher in winter and lower in summer for the three sites, which is consistent with the result of Cheng [27]. This suggested that the variations of PM2.5 concentrations in Hong Kong were
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Fig. 4. The seasonal average PM2.5 concentrations at the three sites from 1 March 2016 to 28 February 2017. (Spring: 3/2016-4/2016; Summer: 5/2016-8/2016; Autumn: 9/2016-10/2016; Winter: 11/2016-2/2017).
closely associated with Asian monsoon circulations. In the summer, the prevailing southeastern winds brought the clean air masses that originated in the South Chinese Sea to Hong Kong, which resulted in the lower PM2.5 concentrations. In addition, the frequent rains in summer also resulted in the lower PM2.5 concentrations. However, in winter, the prevailing northeast winds brought the pollutant emitted from the Asian continent to Hong Kong. Beyond that, cold weather and less rainfall both leaded to the higher concentration lever in winter. 3.2. Validating the method through experiment The synthetic multifractal signals are applied to assess the performance of the EMD-based MFDFA method. The p model [28] as the simplest multiplicative cascading process is used to generate multifractal signals in this paper. We first start from a line and partition it into two segments of the same length and assign two given proportions of measure p1 = 0.3 and p2 = 1 − p1 . Then each segment is divided into two smaller segments and the measure is redistributed in the same multiplicative way. This procedure is repeated for 16 times, then a multifractal signal of size 216 = 65536 is generated. If the multiplicative cascade process goes to infinity, the mass exponent function has an analytic expression as follows [22]:
τ (q) = −ln(pq1 + pq2 )/ln2
(10)
Eq. (10) can approximate the empirical mass exponent function of the constructed multifractal signal. The EMD-based MFDFA is used to determine the empirical mass exponent function τ (q), which is illustrated in Fig. 5 [22]. For comparison, we also draw the theoretical line, Eq. (10) and the empirical τ (q) function obtained by the MF-DFA method in Fig. 5. It is clear that the empirical τ (q) curve extracted based on the EMD-based MFDFA method and the theoretical curve overlap with each other. Whereas, a marked discrepancy is found between the empirical τ (q) curve extracted based on the classical MF-DFA method and the theoretical curve when q > 2, and the classical MF-DFA method systematically underestimates the τ (q) values when q > 2. This test shows that the EMD-based MFDFA method is able to extract the multifractal nature of signals more accurate than the classical MFDFA method at least in certain situations. Based on this, in order to more accurately explore the multifractal characteristic of PM2.5 time series, the EMD-based MFDFA method is used to analyze the multifractal scaling properties of PM2.5 time series in this paper. 3.3. The analyze of PM2.5 levels using the EMD-based MFDFA method The EMD-based MFDFA method is used to analyze the multifractal properties of the PM2.5 time series for MK, TW and TM in Hong Kong. Fig. 6 shows the generalized Hurst exponents h (q) of MK, TW and TM in Hong Kong with q varying from −6 to 6. The corresponding Rényi exponent and multifractal spectra f (α) are shown in Figs. 7 and 8, respectively. From Fig. 6, we find that the generalized Hurst exponent h (q) depends on q and it decreases with the increase of q. This finding means the PM2.5 time series for MK, TW and TM all possess the multifractal characters, which cannot be described fully using traditional monofractal theory. Fig. 7 shows that the corresponding Rényi exponent τ (q) is a nonlinear function depended on q, which also confirms multifractal characters of PM2.5 time series. When q = 2, the generalized Hurst exponent h (q) is exactly the well-known Hurst exponent H. The Hurst exponent H of MK, TW and TM are presented in Table 1. We can find that the Hurst exponents of MK, TW and TM are 0.1992 (0.1991,
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Fig. 5. Plots of τ (q) extracted from the EMD-based MFDFA and the classical MFDFA as a function of q. The continuous line is the theoretical formula.
Fig. 6. The generalized Hurst exponents h (q) of MK, TW and TM in Hong Kong. Table 1 The H, ∆h and ∆α of MK, TW and TM in Hong Kong. H
∆h ∆α
MK
TW
TM
0.1992 (0.1991, 0.1993) 0.7087 1.0735
0.1789 (0.1788, 0.1790) 0.4938 0.7245
0.2119 (0.2118,0.2121) 0.3665 0.5521
0.1993), 0.1789 (0.1788, 0.1790) and 0.2119 (0.2118, 0.2121), respectively, where the numbers in parentheses denote the 95% confidence interval. All the Hurst exponents H are less than 0.5, which means these PM2.5 time series do not obey random walk and present anti-persistent properties. In other words, whenever the PM2.5 level has been up (down), it is more likely that it will be down (up) in the close future. This can forecast the variation tendency of PM2.5 level. Fig. 8 shows the inverse parabolic shapes of spectra, which is the other pieces of empirical evidence for the fact of multifractality in these PM2.5 time series. The degree of multifractality can be quantified by Cao [29]:
∆h = h (qmin ) − h (qmax )
(11)
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Fig. 7. The Rényi exponent τ (q) of MK, TW and TM in Hong Kong.
Fig. 8. The multifractal spectrum f (α) of MK, TW and TM in Hong Kong.
∆α = αmax − αmin .
(12)
The larger∆h and ∆α , the stronger the multifractality degree. It can reveals the probability distributions of the fluctuation. Fig. 8 shows the multifractal spectrum f (α) of MK, TW and TM in Hong Kong. From Figs. 8 and 6, we can get the ∆h and ∆α of MK, TW and TM, Which are shown in Table 1. The∆h of MK, TW and TM are 0.7087, 0.4938 and 0.3665, respectively. The ∆α of MK, TW and TM are 1.0735, 0.7245 and 0.5521, respectively. The roadside site MK displays the highest degree of multifractality, followed by the urban (TW), and the rural one (TM). The major reason may be that the MK site is a roadside site located in a mixed commercial and residential area with heavy traffic and surrounded by many tall buildings. Complex environment leads to a more complicated change of PM2.5 concentrations, therefore, the PM2.5 time series of MK displays a stronger multifractality degree. By contrast, the rural site TM is located at the northeast coast of Hong Kong Island, which experiences the least impact of anthropogenic activities among the three sites. This make the PM2.5 time series of TM have a smaller fluctuation and a weaker multifractality degree. In addition, the multifractality degree of the urban site TW is between the roadside site MK and the rural site TM. In the previous section, we have confirmed that PM2.5 time series are multifractal. Therefore, we further explore the origin of multifractality in this section. As we all known, the long-range correlation and fat-tail distribution are the two major sources of multifractality. Shuffling and phase randomization are two popular methods used to quantify the
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Fig. 9. The generalized Hurst exponents h (q) of MK, TW and TM in Hong Kong. ‘‘Original’’ is the original series, ‘‘Surrogate’’ and ‘‘Shuffled’’ represent the specified series that use phase randomization and shuffling procedures, respectively.
contributions of these two sources. First, the shuffling procedure can destroy all correlations of time series, while the distributions remain unchanged. For this reason, we can obtain the contribution of long-range correlations through the shuffling procedure. Second, the phase randomization procedure can eliminate the fat-tailed distribution, but the linear properties are preserved. Based on this, we can investigate the contribution of the fat-tailed distribution through the phase randomization procedure. Fig. 9 shows the generalized scaling exponents h(q) for three cities. As shown in Fig. 9, the multifractality is caused not only by long-range correlation but also by fat-tailed distribution. Moreover, the deviation of surrogated series from the original curve is smaller than that of shuffled series obviously for three cities. These results mean that the multifractality is more attributed to long-range correlation. 4. Conclusions The daily average PM2.5 concentrations at three different sites in Hong Kong have been analyzed in this paper. The PM 2.5 concentrations of these three sites all were higher in the winter and lower in the summer, which was related with the meteorological conditions under different seasons. In addition, the EMD-based MFDFA method is used to analyze the multifractal properties of the PM2.5 time series. The experimental results are summarized as follows. First of all, the PM2.5 time series for MK, TW and TM are all anti-persistence, that is, whenever the PM2.5 concentrations has been up (down), it is more likely that it will be down (up) in the close future. Second, the roadside site (MK) possesses the highest degree of multifractality, followed by the urban (TW) and the rural one (TM). This is closely related to their geographical environment. Finally, the multifractality is caused not only by long-range correlation but also by fat-tailed distribution. Moreover, the multifractality is more attributed to long-range correlation. Compared with the traditional MF-DFA method, the EMD-based MFDFA method exhibits superior performance. In this work, the proposed method was applied to analyze the multifractal properties of the PM2.5 time series, which is a new approach to research in this field. The EMD-based MFDFA method could be helpful in promoting multifractal studies.
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Acknowledgments This work was supported by the National Nature Science Foundation of China (Grant No. 61806068, No. 61672204), the natural science research key project of Anhui university, China (Grant No. KJ2018A0556, No. KJ2018A0555), the grant of Natural Science Foundation of Hefei University, China (Grant No. 16-17RC19, No. 0391648022, No. 18ZR07ZDA), the grant of Major Science and Technology Project of Anhui Province, China (Grant No. 17030901026), key Technologies R&D Program of Anhui Province, China (Grant No. 1804a09020058), the grant of Anhui Provincial Natural Science Foundation, China, (Grant No. 1908085MF184). References [1] Q. Cao, G. Rui, Y. Liang, Study on PM2.5 pollution and the mortality due to lung cancer in China based on geographic weighted regression model, Bmc Public Health 18 (1) (2018) 925. [2] S.J. Yong, M.N. Lim, Y.J. 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A study on daily PM2.5 concentrations in Hong Kong using the EMD-based MFDFA method ∗
Chen Zhang, Xiaofeng Wang , Shengbing Chen, Le Zou, Xin Zhang, Chao Tang School of computer science and technology, Hefei University, Hefei, 230009, China
highlights • • • •
The multifractal scaling properties of PM2.5 time series are discussed by EMD-based MFDFA. Time series of PM2.5 concentrations for MK, TW and TM are all multifractal and anti-persistence. The multifractality is caused not only by long-range correlation but also by fat-tailed distribution. The roadside site (MK) possesses the highest degree of multifractality, followed by the urban (TW) and the rural one (TM).
article
info
Article history: Received 16 November 2018 Received in revised form 7 April 2019 Available online 8 May 2019 Keywords: PM2.5 EMD MF-DFA Multifractal Hong kong
a b s t r a c t The daily average PM2.5 concentrations of Mong Kok (MK), Tsuen Wan (TW) and Tap Mun (TM) in Hong Kong were obtained from Hong Kong Environmental Protection Department and were analyzed. Without any spatial differentiation, PM2.5 concentrations all were higher in winter and lower in summer. In order to more accurately explore the multifractal characteristic of PM2.5 time series, the empirical mode decomposition (EMD) based multifractal detrended fluctuation analysis (MF-DFA) method is used in this paper. The results show that the PM2.5 time series for MK, TW and TM are all anti-persistence, that is, whenever the PM2.5 concentrations has been up (down), it is more likely that it will be down (up) in the close future. The multifractality is caused not only by long-range correlation but also by fat-tailed distribution. Moreover, the roadside site MK displays the highest degree of multifractality, followed by the urban (TW) and the rural (TM) one. The proposed method can more accurately analyze the multifractal characteristic of PM2.5 time series, which provides a solid foundation for further PM2.5 pollution research. © 2019 Elsevier B.V. All rights reserved.
1. Introduction In recent years, PM2.5 (particulate matter less or equal than 2.5 µm) pollution has been becoming serious. Epidemiological evidences suggest that the increased PM2.5 levels are related to mortality and a number of pulmonary effects [1–3]. The existing researches on PM2.5 mainly focus on analyzing its chemical composition and its spatial distribution [4–6], discussing its association with the meteorological factors [7–9] and simulating its values with the numerical mode [10–12]. Although there are many studies on PM2.5, relatively few researches have focused on the multifractal scaling behavior of PM2.5 time series. Shi [13] used the MF-DFA method [14] to explore the scaling and multifractal properties of the hourly PM2.5 average concentration series at the four air monitoring sites of Chengdu in China. This is the one and only paper ∗ Corresponding author. E-mail address: [email protected] (X. Wang). https://doi.org/10.1016/j.physa.2019.121182 0378-4371/© 2019 Elsevier B.V. All rights reserved.
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Fig. 1. Locations of three sampling sites: Mong Kok (22.321N, 114.11E), Tsuen Wan (22.381N, 114.11E), and Tap Mun (22.473N, 114.371E).
that studies the multifractal scaling behavior of PM2.5 time series. As we all known, the MF-DFA method can unveil the multifractal nature hidden in nonstationary time series, which has been widely used in diverse fields including automatic epileptic seizure detection [15], stock market [16–18], groundwater dynamics [19], daily air temperature [20]. However, in the detrending part of MF-DFA, some shortcomings still remain; Linear, quadratic, cubic, or higher order polynomials may be used in the trend fitting using the least squares (LS) method, but it is not easy to determine which order is the most appropriate. This will result in the inaccurate results when the series are too long or their trends are not of the polynomial type. Fortunately, Huang et al. [21] proposed empirical mode decomposition (EMD), which decomposes raw signals into a limited number of intrinsic mode functions (IMFs) and a residual term. The EMD can resolve the above problem that existed in the MF-DFA method [22]. In the process of detrending, EMD performs better than LS and it does not require any pretesting process, which makes it generally more convenient, more widely applicable, and more accurate. The EMD-based MF-DFA method has been used in the surface electromyography signals [23], epilepsy detection [24], and precipitation complexity measurement [25], futures markets [26]. Compared with the traditional MF-DFA method, the EMD-based MF-DFA method can not only detect multifractal scaling properties of PM2.5 time series, but can give a more accurate result and do not require any pretesting process. Therefore, in order to more accurately explore the multifractal characteristic of PM2.5 time series, the proposed method is used to analyze the multifractal scaling properties of PM 2.5 time series in this paper. This study provides a solid foundation for further PM2.5 pollution research. This paper is organized as follows. Section 2 introduces data source and the EMD-based MFDFA method. Section 3 is the experimental results, which provides an empirical analysis of the proposed method. Section 4 draws the conclusions. 2. Materials and methods 2.1. Study area Hong Kong is situated in the southern tip of the Pearl River Delta region in China, which has one of the highest population densities in the world. Under influence of the Asian monsoon, Hong Kong’s climate is sub-tropical with four distinct seasons. In warm season of summer, the prevalent wind is the southwesterly wind. Warm and damp marine air masses originate from the South China Sea are carried to Hong Kong. In cold season of winter, the most prevailing wind direction is a northeasterly wind. Therefore, in winter, pollutant emitted from the Asian continent is transported to Hong Kong. In addition, winter has much less rainfall in comparison with summer. Three sampling sites were selected based on their different land-use categories, namely roadside, urban and rural environments. Three sampling sites were Mong Kok (MK), Tsuen Wan (TW) and Tap Mun (TM), which are illustrated in Fig. 1. The MK site is a roadside site located in a mixed commercial and residential area with heavy traffic and surrounded by many tall buildings. The urban site TW, located on the rooftop of a government building, represents a densely populated residential area mixed with some commercial and industrial areas. The rural site HT is located at the northeast coast of Hong Kong Island, which experiences the least impact of anthropogenic activities among these three sites.
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2.2. Methodology 2.2.1. Data We choose daily average PM2.5 concentrations (µg/m3 ) of the above three sampling sites, recorded from 1 March 2016 to 28 February 2017. All data obtained from Hong Kong Environmental Protection Department. 2.2.2. The EMD algorithm The empirical mode decomposition (EMD) first proposed by Huang et al. [17] is an innovative data processing algorithm for nonlinear and non-stationary time series. The EMD method decomposed the time series x (t ) into a number of intrinsic mode functions (IMFs), which satisfy the following two conditions: (1) in the whole time series, the numbers of local extrema and the numbers of zero crossings must be equal or differ by 1 at most; and (2) at any time point, the mean value of the ‘‘upper envelope’’ and the ‘‘lower envelope’’ must be zero. The decomposing process is called a sifting process, which can be describes as follows: (1) Identify all extrema of x(t); (2) Interpolate the local maxima to form an upper envelope U(x); (3) Interpolate the local minima to form a lower envelope L(x); (4) Calculate the mean envelope: e (t ) = [U (x) + L (x)]/2 (5) Extract the mean from the signal g (t ) = x (t ) − e(t); (6) Check whether g (t) satisfies the IMF∑conditions. If YES, g (t) is an IMF, stop sifting; If NO, let x (t ) = g (t ) and keep sifting. Finally, we obtain n rn (t ) = x (t ) − i=1 gi (t ), where rn is a residue representing the trend of the time series. 2.2.3. The EMD-based MFDFA algorithms Let {x (t )} be time series, t = 1, . . . , N, where N is the length of time series. The EMD-based MFDFA method [22] can be described as follows. Step 1: Construct the cumulative sum u (t ) =
t ∑
x(i), t = 1, 2, . . . , N.
(1)
i=1
Step 2: The series u (t ) is partitioned into Ns disjoint segments of the same size s, where Ns = [N /s]. Each segment can be denoted by uv such that uv (i) = u(l + i) for 1 ≤ i ≤ s, where l = (v − 1)s. Step 3. For each segment uv , we obtain the EMD-based local trend rn (i) by the EMD method. We can obtain the residual sequence
εv (i) = uv (i) − rn (i) ,
(2)
Step 4. The detrended fluctuation function F (v, s) of the segment uv is defined as the root of the mean squares of the sample residuals εv (i)
[F (v, s)]2 =
s 1∑
s
[εv (i)]2
(3)
i=1
The qth order overall detrended fluctuation is calculated as follows,
{ Fq (s) =
}1/q
Ns ] 1 ∑[ F (v, s)q Ns
(4)
v=1
where q can take any real value except for q = 0. When q = 0, we have
{ F0 (s) = exp
Ns 1 ∑
Ns
} ln[F (v, s)]
(5)
v=1
Step 5. Varying the value of s, we can determine the power-law dependence of the detrended fluctuation function Fq (s) on the size scale s, which reads Fq (s) ∼ sh(q) ,
(6)
where h(q) is the generalized Hurst index. If h(q) depends on q, then the correlation is multifractal, otherwise, the correlation is monofractal. In general, when h (2) > 0.5, the kinds of fluctuations related to q are persistent, that is, an increase (decrease) is likely to be followed by another increase (decrease). When h (2) < 0.5, the kinds of fluctuations related toq are anti-persistent, that is, an increase (decrease) is likely to be followed by another decrease (increase). However, if h (2) = 0.5, then the kinds of fluctuations displays a random walk behavior. The Renyi exponent τ (q), which is related to the general Hurst exponent h(q) obtained from MFDFA, can be obtain by Eq (7):
τ (q) = qh (q) − 1
(7)
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Fig. 2. The annual average PM2.5 concentrations at the three sites from 1 March 2016 to 28 February 2017.
Fig. 3. The monthly average PM2.5 mass concentrations at the three sites from 1 March 2016 to 28 February 2017.
Through the Legendre transform, the Hölder exponent α (q) and singularity spectrum f (α) can be calculated as follows:
α (q) = h (q) + qh′ (q)
(8)
f (α) = q [α (q) − h (q)] + 1
(9)
where h (q) represents the derivative of h (q) with respect to q. The Hölder exponent α (q) characterizes the strength of the singularity, and f (α) represents the Hausdorff dimension of the fractal subset with the exponent α . ′
3. Results and discussion 3.1. PM2.5 mass concentrations In order to better understand the tendency of PM2.5 mass concentrations, the annual average PM2.5 concentrations at these three sites was shown in Fig. 2 the monthly average PM2.5 concentration at these three sites was shown in Fig. 3 and the seasonal average PM2.5 concentration at these three sites was shown in Fig. 4. In Fig. 3, PM2.5 concentrations have an apparent monthly variation. The highest monthly average concentrations were 31.96 µg/m3 , 27.54 µg/m3 and 24.3214 µg/m3 in December 2016 for MK, TW and TM, respectively. Nevertheless, the lowest monthly average concentrations were 21.68 µg/m3 , 20.52 µg/m3 , 16.23 µg/m3 in June 2016 for MK, TW and TM, respectively. Fig. 3 showed its increasing trend from summer to winter. Fig. 4 showed the PM2.5 concentrations were higher in winter and lower in summer for the three sites, which is consistent with the result of Cheng [27]. This suggested that the variations of PM2.5 concentrations in Hong Kong were
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Fig. 4. The seasonal average PM2.5 concentrations at the three sites from 1 March 2016 to 28 February 2017. (Spring: 3/2016-4/2016; Summer: 5/2016-8/2016; Autumn: 9/2016-10/2016; Winter: 11/2016-2/2017).
closely associated with Asian monsoon circulations. In the summer, the prevailing southeastern winds brought the clean air masses that originated in the South Chinese Sea to Hong Kong, which resulted in the lower PM2.5 concentrations. In addition, the frequent rains in summer also resulted in the lower PM2.5 concentrations. However, in winter, the prevailing northeast winds brought the pollutant emitted from the Asian continent to Hong Kong. Beyond that, cold weather and less rainfall both leaded to the higher concentration lever in winter. 3.2. Validating the method through experiment The synthetic multifractal signals are applied to assess the performance of the EMD-based MFDFA method. The p model [28] as the simplest multiplicative cascading process is used to generate multifractal signals in this paper. We first start from a line and partition it into two segments of the same length and assign two given proportions of measure p1 = 0.3 and p2 = 1 − p1 . Then each segment is divided into two smaller segments and the measure is redistributed in the same multiplicative way. This procedure is repeated for 16 times, then a multifractal signal of size 216 = 65536 is generated. If the multiplicative cascade process goes to infinity, the mass exponent function has an analytic expression as follows [22]:
τ (q) = −ln(pq1 + pq2 )/ln2
(10)
Eq. (10) can approximate the empirical mass exponent function of the constructed multifractal signal. The EMD-based MFDFA is used to determine the empirical mass exponent function τ (q), which is illustrated in Fig. 5 [22]. For comparison, we also draw the theoretical line, Eq. (10) and the empirical τ (q) function obtained by the MF-DFA method in Fig. 5. It is clear that the empirical τ (q) curve extracted based on the EMD-based MFDFA method and the theoretical curve overlap with each other. Whereas, a marked discrepancy is found between the empirical τ (q) curve extracted based on the classical MF-DFA method and the theoretical curve when q > 2, and the classical MF-DFA method systematically underestimates the τ (q) values when q > 2. This test shows that the EMD-based MFDFA method is able to extract the multifractal nature of signals more accurate than the classical MFDFA method at least in certain situations. Based on this, in order to more accurately explore the multifractal characteristic of PM2.5 time series, the EMD-based MFDFA method is used to analyze the multifractal scaling properties of PM2.5 time series in this paper. 3.3. The analyze of PM2.5 levels using the EMD-based MFDFA method The EMD-based MFDFA method is used to analyze the multifractal properties of the PM2.5 time series for MK, TW and TM in Hong Kong. Fig. 6 shows the generalized Hurst exponents h (q) of MK, TW and TM in Hong Kong with q varying from −6 to 6. The corresponding Rényi exponent and multifractal spectra f (α) are shown in Figs. 7 and 8, respectively. From Fig. 6, we find that the generalized Hurst exponent h (q) depends on q and it decreases with the increase of q. This finding means the PM2.5 time series for MK, TW and TM all possess the multifractal characters, which cannot be described fully using traditional monofractal theory. Fig. 7 shows that the corresponding Rényi exponent τ (q) is a nonlinear function depended on q, which also confirms multifractal characters of PM2.5 time series. When q = 2, the generalized Hurst exponent h (q) is exactly the well-known Hurst exponent H. The Hurst exponent H of MK, TW and TM are presented in Table 1. We can find that the Hurst exponents of MK, TW and TM are 0.1992 (0.1991,
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Fig. 5. Plots of τ (q) extracted from the EMD-based MFDFA and the classical MFDFA as a function of q. The continuous line is the theoretical formula.
Fig. 6. The generalized Hurst exponents h (q) of MK, TW and TM in Hong Kong. Table 1 The H, ∆h and ∆α of MK, TW and TM in Hong Kong. H
∆h ∆α
MK
TW
TM
0.1992 (0.1991, 0.1993) 0.7087 1.0735
0.1789 (0.1788, 0.1790) 0.4938 0.7245
0.2119 (0.2118,0.2121) 0.3665 0.5521
0.1993), 0.1789 (0.1788, 0.1790) and 0.2119 (0.2118, 0.2121), respectively, where the numbers in parentheses denote the 95% confidence interval. All the Hurst exponents H are less than 0.5, which means these PM2.5 time series do not obey random walk and present anti-persistent properties. In other words, whenever the PM2.5 level has been up (down), it is more likely that it will be down (up) in the close future. This can forecast the variation tendency of PM2.5 level. Fig. 8 shows the inverse parabolic shapes of spectra, which is the other pieces of empirical evidence for the fact of multifractality in these PM2.5 time series. The degree of multifractality can be quantified by Cao [29]:
∆h = h (qmin ) − h (qmax )
(11)
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Fig. 7. The Rényi exponent τ (q) of MK, TW and TM in Hong Kong.
Fig. 8. The multifractal spectrum f (α) of MK, TW and TM in Hong Kong.
∆α = αmax − αmin .
(12)
The larger∆h and ∆α , the stronger the multifractality degree. It can reveals the probability distributions of the fluctuation. Fig. 8 shows the multifractal spectrum f (α) of MK, TW and TM in Hong Kong. From Figs. 8 and 6, we can get the ∆h and ∆α of MK, TW and TM, Which are shown in Table 1. The∆h of MK, TW and TM are 0.7087, 0.4938 and 0.3665, respectively. The ∆α of MK, TW and TM are 1.0735, 0.7245 and 0.5521, respectively. The roadside site MK displays the highest degree of multifractality, followed by the urban (TW), and the rural one (TM). The major reason may be that the MK site is a roadside site located in a mixed commercial and residential area with heavy traffic and surrounded by many tall buildings. Complex environment leads to a more complicated change of PM2.5 concentrations, therefore, the PM2.5 time series of MK displays a stronger multifractality degree. By contrast, the rural site TM is located at the northeast coast of Hong Kong Island, which experiences the least impact of anthropogenic activities among the three sites. This make the PM2.5 time series of TM have a smaller fluctuation and a weaker multifractality degree. In addition, the multifractality degree of the urban site TW is between the roadside site MK and the rural site TM. In the previous section, we have confirmed that PM2.5 time series are multifractal. Therefore, we further explore the origin of multifractality in this section. As we all known, the long-range correlation and fat-tail distribution are the two major sources of multifractality. Shuffling and phase randomization are two popular methods used to quantify the
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Fig. 9. The generalized Hurst exponents h (q) of MK, TW and TM in Hong Kong. ‘‘Original’’ is the original series, ‘‘Surrogate’’ and ‘‘Shuffled’’ represent the specified series that use phase randomization and shuffling procedures, respectively.
contributions of these two sources. First, the shuffling procedure can destroy all correlations of time series, while the distributions remain unchanged. For this reason, we can obtain the contribution of long-range correlations through the shuffling procedure. Second, the phase randomization procedure can eliminate the fat-tailed distribution, but the linear properties are preserved. Based on this, we can investigate the contribution of the fat-tailed distribution through the phase randomization procedure. Fig. 9 shows the generalized scaling exponents h(q) for three cities. As shown in Fig. 9, the multifractality is caused not only by long-range correlation but also by fat-tailed distribution. Moreover, the deviation of surrogated series from the original curve is smaller than that of shuffled series obviously for three cities. These results mean that the multifractality is more attributed to long-range correlation. 4. Conclusions The daily average PM2.5 concentrations at three different sites in Hong Kong have been analyzed in this paper. The PM 2.5 concentrations of these three sites all were higher in the winter and lower in the summer, which was related with the meteorological conditions under different seasons. In addition, the EMD-based MFDFA method is used to analyze the multifractal properties of the PM2.5 time series. The experimental results are summarized as follows. First of all, the PM2.5 time series for MK, TW and TM are all anti-persistence, that is, whenever the PM2.5 concentrations has been up (down), it is more likely that it will be down (up) in the close future. Second, the roadside site (MK) possesses the highest degree of multifractality, followed by the urban (TW) and the rural one (TM). This is closely related to their geographical environment. Finally, the multifractality is caused not only by long-range correlation but also by fat-tailed distribution. Moreover, the multifractality is more attributed to long-range correlation. Compared with the traditional MF-DFA method, the EMD-based MFDFA method exhibits superior performance. In this work, the proposed method was applied to analyze the multifractal properties of the PM2.5 time series, which is a new approach to research in this field. The EMD-based MFDFA method could be helpful in promoting multifractal studies.
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Acknowledgments This work was supported by the National Nature Science Foundation of China (Grant No. 61806068, No. 61672204), the natural science research key project of Anhui university, China (Grant No. KJ2018A0556, No. KJ2018A0555), the grant of Natural Science Foundation of Hefei University, China (Grant No. 16-17RC19, No. 0391648022, No. 18ZR07ZDA), the grant of Major Science and Technology Project of Anhui Province, China (Grant No. 17030901026), key Technologies R&D Program of Anhui Province, China (Grant No. 1804a09020058), the grant of Anhui Provincial Natural Science Foundation, China, (Grant No. 1908085MF184). References [1] Q. Cao, G. Rui, Y. Liang, Study on PM2.5 pollution and the mortality due to lung cancer in China based on geographic weighted regression model, Bmc Public Health 18 (1) (2018) 925. [2] S.J. Yong, M.N. Lim, Y.J. 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