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An analysis of multifractal characteristics of API time series in Nanjing, China
Physica A 451 (2016) 171–179
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An analysis of multifractal characteristics of API time series in Nanjing, China Shen Chen-hua a,b,c,∗ , Huang Yi a , Yan Ya-ni a a
College of Geographical Science, Nanjing Normal University, Nanjing 210046, China
b
Jiangsu Center for Collaborative Innovation in Geographical Information Resource, Nanjing 210046, China
c
Key Laboratory of Virtual Geographic Environment of Ministry of Education, Nanjing 210046, China
highlights • Multifractal characteristic of API time series is studied based on MF-DFA and singularity spectrum. • Strength of distribution multifractality is stronger than that of correlation multifractality. • Temporal variation in structure of API time series is mainly related to long-range correlations.
article
info
Article history: Received 7 October 2014 Received in revised form 21 November 2015 Available online 1 February 2016 Keywords: Multifractality Multifractal detrended fluctuation analysis Multifractal characteristics Long-range correlations A broad probability density function API
abstract This paper describes multifractal characteristics of daily air pollution index (API) records in Nanjing from 2001 to 2012. The entire daily API time series is first divided into 12 parts that serve as research objects, and the generalized Hurst exponent is calculated for each series. And then, the multifractal sources are analyzed and singularity spectra are shown. Next, based on a singularity spectrum, the multifractal-characteristics parameters (maximum exponent α 0 , spectrum width ∆α , and asymmetry ∆α as ) are introduced. The results show that the fractality of daily API for each year is multifractal. The multifractal sources originate from both a broad probability density function and different long-range correlations with small and large fluctuations. The strength of the distribution multifractality is stronger than that of the correlation multifractality. The variation in the structure of API time series with increasing years is mainly related to long-range correlations. The structure of API time series in some years is richer. These findings can provide a scientific basis for further probing into the complexity of API. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Air pollution in China is a severe problem with social, economic and political consequences [1]. A World Bank study reported that 16 out of the top 20 most polluted cities in the world were located in China [2]. Air pollution index (API) is a referential parameter frequently used for reporting the levels of ambient air pollution. Three major pollutants, including respirable particulate matter (PM10 ), sulfur dioxide (SO2 ) and nitrogen dioxide (NO2 ), have been selected to report daily API in China [3]. Since June 2000, the State Environment Protection Agency of China has required daily API reports, helping the public to understand local air quality. The higher the API, the more serious the air pollution.
∗
Correspondence to: College of Geographical Science, Nanjing Normal University, Jiangsu, China. E-mail address: [email protected] (C.-h. Shen).
http://dx.doi.org/10.1016/j.physa.2016.01.061 0378-4371/© 2016 Elsevier B.V. All rights reserved.
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Air pollution system is very complex. The structure of API time series and its temporal variation are rather complicated [4]. The structural complexity leads us to adopt multifractal formalism to analyze its structural characteristics [5]. This formalism is a widely used technique for quantitatively delineating the nonlinear evolution of a complex system and the multiscale characteristics of physical quantity, and aids in an understanding of the intrinsic regularity and the mechanism of physical changes [6]. Multifractal detrended fluctuation analysis (MF-DFA) [7], an important approach in the study of the multifractal properties of a non-stationary time series, has been a favored research tool. Based on MF-DFA, a singular spectrum is obtained, and the important information of multifractal characteristics can be extracted from it. The time-series structural complexity is inferred through this information [7]. MF-DFA and its related methods [8–11] have been widely used in air pollution studies [12,13] and in other relevant fields [14]. For instance, Shi et al. found that APIs from July 2000 to June 2006 in Shanghai were characterized by scale invariance, long-range dependence and multifractal scaling [13]. Diosdado et al. found that time series of air pollutant concentration, such as concentrations of ozone, sulfur dioxide, carbon monoxide, nitrogen dioxide and PM10 particles, from 1990 to 2005 in the metropolitan zone of Mexico city, are multifractal [5]. Zhu et al. and Tong et al. investigated the characteristics of air pollutants and the impact of haze in Nanjing, respectively [15,16]. And Shen et al. studied the detrended cross correlation between API and meteorological elements in Nanjing [17,18]. Although the fractal characteristics of API in some Chinese cities and the detrended cross-correlation between API and its influencing factors in Nanjing have been investigated, there is little direct evidence of the yearly fractality, multifractal sources and the variation in fractality with time in Nanjing. Furthermore, the mechanism of variation in fractality has not been fully clarified and developed. The nonlinear mechanism of temporal variation of API in other cities has rarely been reported in the literature [3,19], and it is not clear that the conclusions drawn from other studies are applicable to Nanjing. In this paper, we analyzed API records for the period 2001–2012 in Nanjing, China, choosing Nanjing because it is the capital city of Jiangsu Province, and an important transportation and industrial center within the province. Moreover, Nanjing’s economy has grown rapidly over the past two decades. Expansions in urban population, industrial production, transportation and traffic infrastructure have accelerated the emission levels of various air pollutants. Because of their role in the prevention and control of atmospheric pollution, local government requirements should be taken into account in API forecasts. However, to date, it is not clear whether the API in Nanjing can be fully predicted. If an API time series is long-range correlated, predictability becomes possible. Our objective is thus to investigate the multifractal characteristics of API in Nanjing in order to find a pattern of long-range correlations. Our main contribution is that our focus is not only on empirical evidence of API multifractality in Nanjing, but also on investigating the multifractal sources and the variations in multifractal strengths with increasing year. Our findings can contribute to an understanding of the structural complexity of API time series. The structure of this paper is as follows. Section 2 briefly describes methods and data sources. Section 3 provides detailed empirical results. Section 4 provides a discussion of the results, and the last section provides our conclusions. 2. Methodology and data sources 2.1. Methodology As opposed to simple fractals described by a single scaling exponent, multifractal time series are characterized by a hierarchy of scaling exponents that describe the different scaling behavior of many interwoven subsets of the series. The MF-DFA procedure is briefly described as follows [7]. The original time series in the presence of nonstationarity, x(k), k = 1, 2, . . . , N, is integrated to produce the profile i N ¯ ), i = 1, 2, . . . , N, where x¯ = X (i) = k=1 (x(k) − x k=1 x(k)/N is the average. Next, the entire time series X (i), i = 1, 2, . . . , N, is divided into Ns = N − s overlapping boxes with window size s [20], each containing s + 1 values. The detrended variance f 2 (s, i) of the residuals in a box of size s that starts at i and ends at i + s is calculated as in Eq. (1) [20]
i +s 2 f (s, i) = (X (k) − X˜ (k, i)) (s + 1) 2
(1)
k=i
where the local trend, X˜ (k, i), is the ordinate of a least-squares fit with a straight line or higher order polynomial [20]. Here, we adopted ‘‘overlapping boxes’’ [20] instead of ‘‘non-overlapping boxes’’ [21] to calculate detrended variance. Enough boxes can thus be calculated so that the detrended variance becomes stable. Finally, a qth order detrended fluctuation function is calculated as Eq. (2)
Fq (s) =
Ns 1
Ns i=1
F0 (s) = exp
[f (s, i)]
Ns 1
2Ns i=1
q 2
1q for q ̸= 0
ln f (s, i)
for q = 0.
(2)
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If time series x(k), k = 1, 2, . . . , N, is long-range power-law correlated, the detrended fluctuation function Fq (s) increases asymptotically with q varying as the power law, defined as Eq. (3) Fq (s) ∝ sh(q)
(3)
where h(q) is the generalized Hurst exponent. For a stationary time series, h(2) is identical to the well-known Hurst exponent H. As is well established, h(q) describes the scaling behavior of the boxes with large fluctuations for q > 0; whereas h(q) depicts the scaling behavior of boxes with small fluctuations for q < 0. Usually, h(q) for large fluctuations is smaller than h(q) for small fluctuations. The traditional method to characterize a multifractal time series is to calculate the singularity spectrum f (α), which can be related to h(q) via a Legendre transformation [7]:
α(q) = h(q) + q
dh(q)
(4)
dq
f (α) = q(α(q) − h(q)) + 1 = q2
dh(q) dq
+1
(5)
where α(q) is the fractal exponent and f (α) denotes the dimension of the subset of the series that is characterized by α(q). Eqs. (4) and (5) give a parametric representation of the f (α) curve [7,22]. For a monofractal time series, the singularity spectrum produces a single point in the f (α) plane, whereas the multifractal process yields a single humped function. Based on Eqs. (3) and (5), three parameters in the multifractal spectrum are designated to describe the complexity of the signal: (1) Maximum exponent α0 ∈ [αmin , αmax ], where f (α0 ) has its maximum, is designated as the singularity strength with maximum spectrum. A small value of α0 means that the underlying process ‘‘loses fine structure’’, that is, becomes more regular in appearance [22]. (2) Singularity spectrum width 1α is denoted as (αq− − αq+ ). 1α provides information on the diversity of the scaling exponents of the measure, and reflects multifractal strength. The wider the range of 1α , the stronger the multifractality strength [22]. (3) Asymmetry. The asymmetrical parameter of the multifractal spectrum can be obtained by 1αas = 1αright − 1αleft = (αq− − α0 ) − (α0 − αq+ ), where 1αas = 0 for symmetric curves, 1αas > 0 for left-skewed curves and 1αas < 0 for right-skewed curves. A positive 1αas means a relative dominance of lower fractal exponents corresponding to smoothlooking structures, while a negative 1αas indicates a relative dominance of higher fractal exponents corresponding to fine structures [22]. 2.2. Data sources API records from January 2001 to December 2012 in Nanjing, China, are the primary data sources for our analysis. Original daily API records were obtained from the Ministry of Environmental Protection of the People’s Republic of China (http://datacenter.mep.gov.cn). The quality of all the daily API records was controlled, and there are no more than 67-days missing values. Since the missing value is specified as one single day rather than a few continuous days, the mean value of API before and after the specified day was calculated as the missing API value. Prior to our analysis, we divided the entire original daily API records over the past 12 years into 12 parts as research objects. Each research object has equal time-series length and represents the evolution of API for each year. A single-sample Kolmogorov–Smirnov test was applied to confirm that the probability density distribution function of API for each year is non-Gaussian, as expected [16]. Fig. 1(a) shows the original daily API records versus time. Fig. 1(b) shows the contour plots of wavelet coefficients for original daily API time series from 2001 to 2012 by means of a Morlet wavelet transformation [23,24]. In this figure, coarse black curves indicate that the original daily API time series contains short periodicity during specific time periods, for instance, there is a short period of approximate 150 days in the range of about 1400–4000 days. Earlier studies indicated that a time series with periodicity has an effect on the results of MF-DFA, sometimes leading to spurious findings of multifractality [8,25–28]. In order to reduce an effect of periodicity on the result of MF-DFA, we employed the Fourier analysis [29] to eliminate the periodicity of a time series. First, a Morlet wavelet transformation was utilized to obtain periodic size. The Fourier-expansion coefficients that are connected to periodic sizes were then truncated. In the end, an inverse Fourier transformation was performed, and the periodic trends were removed. Hereafter, an original daily API time series with its periodicity removed through Fourier analysis is denoted as a processed daily API time series. Fig. 1(c) shows the contour plots of wavelet coefficients for the original daily API time series in 2006. Fig. 1(d) shows the contour plots of wavelet coefficients for the processed daily API time series in 2006. Comparing Fig. 1(c) with Fig. 1(d), the periodic trend in the original daily API time series is found to be partially removed. Fig. 2(a) shows the yearly average of original daily API records. The average value reached a peak in 2002, decreased in 2003, and then remained almost unchanged overall. This pattern clearly shows that air pollution in Nanjing was the most severe in 2002. Fig. 2(b) shows magnitude-frequency distribution for original daily API records. In this figure, N
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Fig. 1. (a) Evolution of original daily API records from January 2001 to December 2012. (b, c and d) Contour plots of wavelet coefficients. (b) For original daily API time series from January 2001 to December 2012. (c) For original daily API time series in 2006. (d) For processed daily API time series in 2006. The thick black contour designates the 5% significance level against red noise [23]; cone of influence (COI) where edge effects might distort the picture is shown as a lighter shade. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
a 120
Yearly average API
b
5 4
)
80
3 ln (
Yearly average API
100
60
2
40
1
20 0 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Years
0
0
1
2
3 ln (C)
4
5
6
Fig. 2. (a) Yearly average tendency. (b) Cumulative magnitude frequency distribution. Daily API records are original.
is the cumulative number of events with size greater than magnitude C and τ is the scaling exponent. It is regarded as dN /dC ∝ C −τ within a certain range [30]. 3. Results 3.1. Fractality of API records for each year The analysis of the multifractal detrended fluctuation for the 12-year processed API records was performed. Fig. 3(a) shows the detrended fluctuation functions Fq (s) of processed API time series in 2006 against time scale s for q = −5, 0 and 5, respectively. In this figure, the time-scale crossover in Fq (s) is detected for q = −5. The size of this time-scale crossover is approximately equal to the periodic size which was obtained by means of a Morlet wavelet transformation [23]. The slopes of Fq (s) against time scales, h(q) = 1Ln(Fq (s))/1Ln(s)(generalized Hurst exponent), differ in different regiments. Periodic trends become dominant at small time scales and large time scales while at medium time scales, the intrinsic fluctuations tend to dominate. At 16 < s < 32, the slopes, h(q), therefore suggest that the local effects, such as meteorological elements and human activities, become dominant [31]. Further analysis shows that time-scale crossovers in Fq (s) against time scales s also exist in other years, and that their results are analogous to the result in 2006. The detrended fluctuation function, Fq (s), thus can develop approximately unique scaling in the range of appropriate time scales. Consequently, there is fractality in the processed API time series in 2006 for the considered value of q in the
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h(q)
3.5 Ln[Fq(s)]
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0.6
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q=5
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q
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0 q
1
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5
0.3 –5
–4
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–2
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0 q
1
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Fig. 3. (a) Detrended fluctuation function Fq (s) of the processed API time series in 2006 versus time scale s for q = −5, 0 and 5, respectively. There exists time-scale crossover for q = −5, clearly. (b) Time series is uncorrelated and random, and the value of green dotted line is always 0.5. (c and d) Blue curves (filled triangle) are the evolution of h(q) against q for the processed time series. Red curves (filled circle) represent the evolution of hshuf (q) against q for shuffled time series. Black curves (filled square) represent the evolution of hcor (q) against q for long-range correlations. (c) Processed API time series in 2004. (d) Processed API time series in 2012. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
range of appropriate time scales. The analogous results in other years are also found, and the typical results in 2004 and 2012 are shown in Fig. 3(c) and (d). Fig. 3(c) and (d) shows the evolution of generalized Hurst exponent h(q) in terms of q in 2004 and 2012, respectively. h(q) is greater than 0.5 and less than 1, and is dependent on q. This indicates that the fractality is multifractal rather than monofractal. This is in agreement with recent results [13], which revealed the multifractal behavior of API time series over Shanghai in 2008. On the whole, the results in other years are similar to the results in 2004 and 2012. Thus, the fractality of API records for each year behaves much the same as multifractality in the range of appropriate time scales. 3.2. Multifractal sources of API records for each year Generally, the multifractal sources originate from both a broad probability density function and different long-range correlations with small and large fluctuations [7]. The shuffle procedure can destroy long-range correlations [7]. For convenience, it is denoted as h(q) for processed API time series, hshuf (q) for shuffled API time series and hcor (q) = h(q) − hshuf (q) + 0.5 for long-range correlations, respectively. If h(q) = hshuf (q) depends on q and hcor (q) = 0.5, multifractality is due to a broad probability density function alone. On the other hand, if hshuf (q) = 0.5 and hcor (q) = h(q), multifractality is due to different long-range correlations alone. If hshuf (q) and hcor (q) depend on q, multifractality is due to both a broad probability density function and different long-range correlations. If both kinds of multifractality are present, the shuffled time series will show weaker multifractality than the original time series [7]. In order to distinguish two types of multifractal behaviors, an uncorrelated random time series and the processed API time series are shuffled. The generalized Hurst exponent hu (q) is denoted as an uncorrelated random time series. After two kinds of time series are reshuffled more than 2500 times, the averages of hshuf (q) and hu (q) are calculated with a determination coefficient >0.99, respectively, and the error bars corresponding to standard deviations are given. Fig. 3(b) shows hshuf u (q) and hu (q) for the uncorrelated random time series versus q, and they are found to be approximately equal to 0.5, and to be independent of q. Fig. 3(c) and (d) shows the q-dependence of h(q), hshuf (q) and hcor (q) in 2004 and 2012, respectively. In Fig. 3(c) and (d), shuf h (q) is not equal to 0.5, and strongly depends on q. Fractality due to a broad probability density function can therefore be preliminarily viewed as multifractality in 2004 and 2012. Comparing the results in other years with the results in 2004 and 2012, and considering that the error from the algorithm and the statistics, hshuf (q) in each year is found to be strongly dependent on q. We can thus conclude that the fractality due to a broad probability density function in Nanjing is multifractal for each year.
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h(2)=H Hurst exponent
Maximum exponent α0
0.9
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b
1
Spectrum width Δα
a
Shuffled series Long-range correlations
0.8 0.7 0.6 0.5 0.4 2000
2002
2004
2006 Year
2008
2010
2012
Fig. 4. (a) Maximum exponent α0 versus year for the processed API time series, shuffled time series and long-range correlations, respectively. (b) Spectrum width 1α against year for the processed API time series, shuffled time series and long-range correlations, respectively. (c) Asymmetry 1αas versus year for the processed API time series, shuffled time series and long-range correlations, respectively. (d) Hurst exponent H versus year for the processed API time series, shuffled time series and long-range correlations, respectively. (a–d) Blue curves (filled square) represent the parameters for the processed API time series, black curves (filled triangle) represent the parameters for long-range correlations, and red curves (filled circle) represent the parameters for shuffled time series (b and c) green line is zero line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Meanwhile, hcor (q) in each year is also obtained. Considering the error from the algorithm, when hcor (−5)–hcor (+5) < 0.1, the fractality due to long-range correlations is judged to be monofractal [17]. Consequently, fractality due to long-range correlations is found to be multifractal in 2004, 2008, and 2010, but, to be monofractal in other years. As expected, API time series over a period of 12 years does not exhibit a simple random behavior. The fractality in each year is multifractal. hcor (q) is not equal to 0.5, indicating the presence of long-range correlations. That is to say, API time series contains long-range correlation persistence. Multifractal sources stem from both a broad probability density function and different long-range correlations with small and large fluctuations. Fractality due to a broad probability density function is multifractal. Fractality due to long-range correlations is monofractal in some years, but is multifractal in other years. 3.3. Multifractal characteristics of API records for each year In order to detect multifractal characteristics of API time series in Nanjing, the singularity spectra are calculated through h(q), hshuf (q) and hcor (q), respectively, based on Eqs. (4) and (5). Fig. 4(a–c) shows that relevant spectrum-characteristic parameters, such as α0 , 1α , and 1αas [22], which are extracted from the singularity spectra, vary with increasing years. h shuf cor These parameters are α0h , α0shuf , α0cor , 1α h , 1α shuf , 1α cor , 1αas , 1αas , and 1αas , respectively, and differ by year. Based on h shuf cor Eqs. (4) and (5), we obtained formulae as follows: α0h = α0shuf + α0cor , 1α h = 1α shuf + 1α cor , 1αas = 1αas + 1αas . Fig. 4(d) shows that Hurst exponent H varies with increasing year. According to Eq. (4), α0h = h(0) is generalized Hurst exponent for q = 0. α0h reflects the basic level of either long-range correlations or a broad probability density function. Fig. 4(a) shows the respective α0h , α0shuf , and α0cor vary with year. In this figure, α0h (blue curve) for the processed API time series is approximately equal to 0.60–0.80 from 2001 to 2012. With years, the variation in α0h is fluctuant. α0shuf (red curve) for the shuffled time series is approximately equal to 0.55 for the period of 2001–2012. This means that shuffling procedures have removed the structures related to long-range correlations, and that the shuffled API time series are not white noise. With increasing years, the series-structure evolution related to a broad probability density function is comparatively small, but the series-structure evolution due to long-range correlations with small and large fluctuations is fluctuant. The seriesstructure evolution of the processed API time series is thus attributed to different long-range correlations. Moreover, α0h is relatively high in 2002, 2004 and 2009, indicating that API time series in these years provide rich series-structure information. A singularity spectrum width, 1α , characterizes the multifractal strength. The gradual decrease in 1α over time indicates a reduction in the degree of multifractality, while an increase in 1α indicates the opposite.
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Fig. 4(b) shows the variation in 1α h , 1α shuf , and 1α cor versus year, respectively. In this figure, 1α h (blue curve) is fluctuant with increasing years, and is the highest in 2008 and 2010, and the lowest in 2003. This suggests that the multifractal strengths are also fluctuant with increasing years, and the structure of the processed API time series is the most complex in 2008 and 2010, and is the simplest in 2003. Comparing 1α shuf with 1α cor , it can be seen that 1α shuf is greater than 1α cor ; that is to say, the strength of distribution multifractality is stronger than that of the correlation multifractality. Fig. 4(b) shows that 1α cor is fluctuant with increasing years. In 2004, 2008 and 2010, 1α cor is comparatively large, but is approximately equal to zero in other years. This confirms that fractality due to long-range correlations in 2004, 2008, and 2010 is multifractal, and that it is monofractal in other years. Moreover, 1α h = 1α shuf + 1α cor , meaning that 1α h is tightly connected with the multifractal strength of the distribution and correlation. The ratio of 1α shuf to 1α h is defined, and represents the contribution of the distribution’s multifractal strength to total multifractal strength. In Nanjing, total multifractal strength can thus be primarily attributed to the multifractal strength due to distribution, while the contribution of the multifractal strength due to correlation to total multifractal strength is limited. h An asymmetry of a singularity spectrum, 1αas , indicates the relative importance between the high and low fractal exponents, and reflects the different range sizes of scaling exponents related to small and large probability or small and h shuf cor large fluctuations. Moreover, 1αas = 1αas + 1αas , indicates that the relative importance between high and lower fractal exponents is tightly connected with distribution multifractality and correlation multifractality. When long-range correlation fractality is monofractal, the relative importance between high and lower fractal exponents is closely related to distribution h multifractality. A positive 1αas , meaning that the lower fractal exponent is dominant, exhibits some asymmetry with consistently left-skewed spectra, and indicates that a wider range of scaling exponent spectrum is required to characterize h shuf the variation in series’ structure. A negative 1αas is exactly the opposite. 1αas reflects different ranges of scaling exponents cor due to large and small probability. 1αas reflects different ranges of scaling exponents due to large and small fluctuations. h shuf cor h Fig. 4(c) illustrates that 1αas , 1αas , and 1αas vary with year. The figure shows that the evolution of 1αas is h comparatively large in 2004, 2008, 2009, and 2012. The positive 1αas in 2004 and 2012 means that singularity spectrum shape is left-skewed, lower fractal exponent is dominant, and the structure appears relatively smooth. But, the negative 1α =has in 2008 and 2009 means that singularity spectrum shape is right-skewed, higher fractal exponent is dominant, and the structure is relatively fine. cor h Since the fractality in 2004, 2008, and 2010 is multifractal, the corresponding 1αas is not equal to zero. 1αas in 2004, shuf 2008, and 2010 is thus attributed to distribution multifractality and correlation multifractality. In particular, 1αas > 0 and cor shuf cor 1αas > 0 in 2004, and 1αas < 0 and 1αas < 0 in 2008, indicate that the small and large fluctuations due to long-range correlations are consistent with the low and high probability due to distribution. But, in 2010, the situation is exactly the opposite, indicating that there is competition between small probability due to distribution and large fluctuations due to different long-range correlations. There is an indirect evidence to support this result. There were more than 1158 construction sites in 2010, and more than 2000 construction sites in 2012 in Nanjing downtown. On the one hand, an increasing number of construction sites causes the increasing amount of bare ground to result in higher levels of dust and a rising API. On the other hand, in order to effectively control these urban fugitive dust sources, the municipal government took measures to curb severe pollution, for instance, the focused management of urban green land. As a result, the public green area per capita reached 13.6 m2 in 2010, and 14.3 m2 in 2012 in Nanjing. An increase in green land gradually declined the dominant levels of high API. This result is consistent with the variation in multifractality with years. Fig. 4(d) shows that Hurst exponent varies with years. Hurst exponent (red curve) for the shuffled time series is approximately equal to 0.52 for the period of 2001–2012. This means that the series persistence’s evolution with increasing years is thus related to different long-range correlations. Hurst exponent for long-range correlation >0.5 thus implies that API time series in Nanjing can be predictable. Since the level of positive persistence due to long-range correlations is not very strong (the average of Hurst exponent due to long-range correlations over 12 year is about 0.6), predictable reliability merits further investigation. As can be seen in Fig. 4, the distribution-multifractality strength is stronger than that of the correlation multifractality, the temporal variation in the structure of API time series is mainly related to long-range correlations, and the time-series structure in some years is comparatively fine. 4. Discussions 4.1. Effect of time-series length on MF-DFA The length of the time series has an effect on DFA. Longer time series will lead to more reliable result of multifractal analysis based on MF-DFA [32]. However, it is of importance how to ensure the stability of Fq (s). In MF-DFA, profiles X (i), i = 1, 2, . . . , N, are divided into Ns = int(N /s), which are non-overlapping segments of equal length s [21], and Fq (s) can be obtained through the summation of detrended variance in different windows. The total summation is Ns . When N is relatively small, Ns is relatively small, too. For instance, when N = 365 and s = 10, only int (N /s) = 36 data groups will be used to calculate Fq (s). However, we modified calculation method of Fq (s) according to Ref. [20]. The profiles X (i) are
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divided into Ns = N − s, which are overlapping windows, each containing equal length s. For the above example, 355 data groups will be applied to calculate Fq (s). Calculating enough data can ensure the stability of Fq (s). In Fig. 1(c), the very short period is found in the original API time series in 2006. A short period in a shorter time series has an effect on the result of MF-DFA. When the timepoint range of a short period is relatively large, Fq (s) will contain a contribution from the intrinsic fluctuation due to long-range correlations and periodic trends. Thus, the removal of a short period is important to ensure the reliability of Fq (s) related to the long-range correlations. Similar to Ref. [29], Fourier analysis was performed to remove high frequency signals. However, we found that it is rather difficult to fully remove these high frequency signals this way. Thus, more optimized method to remove high frequency signals is needed to be investigated in the future.
4.2. Multifractal analysis based on multifractal sources Generally, the multifractal sources originate from both a broad probability density function and from different long-range correlations with small and large fluctuations. When a multifractal analysis is performed, a multifractal source should be distinguished to ensure reliable results. Thus, reshuffled processes of time series can be first applied to detect multifractal sources, and then multifractalities due to long-range correlation and a broad probability density function must be separated. Next, a corresponding analysis due to a broad probability density function or different long-range correlations with small and large fluctuations is performed, which might produce reliable results.
4.3. Long-range correlation of API Regarding complexity, the API time series fluctuations are macro effects, caused by local interactions and correlations between various factors. On the one hand, an air pollution system contains many components, such as pollutant sources, atmospheric pollutant components, solar radiation, atmospheric self-purification, topographical features, and meteorological factors. Each component has a certain influence on the average daily air pollution concentration. When all the components are considered together, they interact and correlate with each other on vastly different timescales. Under this circumstance, the API fluctuations have characteristics of certainty to some extent. On the other hand, because the atmospheric system is open and dissipative, API evolution is uncertain. Hence, API evolution with time is irregular and nonlinear, and it has the basic characteristics of a complex system. The various interactions and correlations between a variety of complicated factors have cumulative effects on API fluctuation, and create a fluctuation of API that becomes a ‘macro order’ structure, characterized by certainty. The micromechanisms of API evolution are very complex. A self-organized criticality (SOC) model of air pollution is adopted to explain emergent complex behavior in Chengdu [30]. It is characteristic of SOC behavior that the distributional function of API satisfies the requirement of a power-law distribution, and that an API time series has long-range persistence. Fig. 2(b) also shows that cumulative magnitude-frequency distributions exhibit power-law scaling. This is regarded as the typical ‘‘critical’’ dynamical behavior of SOC systems. The SOC model seems suitable for explaining the long-range correlation of API in Nanjing. However, an analysis in detail of the long-range correlation can be performed in the future.
5. Conclusions In this paper, the multifractal characteristics of API time series in Nanjing, China, from 2001 to 2012, are studied based on the MF-DFA and singularity spectrum method. By studying three multifractal parameters (α0 , 1α , and 1αas ) of a singularity spectrum, multifractal characteristics of daily API records in Nanjing are found to be clearly in agreement with recent results obtained in other cities, China. The source of API multifractality in Nanjing originates from a broad probability density function and long-range correlations. Multifractal strength due to a broad probability density function is stronger than multifractal strength due to long-range correlations with small and large fluctuations. The temporal variation in the structure of API time series is mainly related to long-range correlations. Multifractality may be fluctuant with increasing years. The three multifractal parameters describe the long-range correlations from different views, and a specified combination of the parameters represents a particular long-range correlation of the corresponding processes. These results aid in a full understanding of the long-range correlations of API records and may provide theoretical support in API prediction.
Acknowledgment The authors are grateful for a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
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An analysis of multifractal characteristics of API time series in Nanjing, China Shen Chen-hua a,b,c,∗ , Huang Yi a , Yan Ya-ni a a
College of Geographical Science, Nanjing Normal University, Nanjing 210046, China
b
Jiangsu Center for Collaborative Innovation in Geographical Information Resource, Nanjing 210046, China
c
Key Laboratory of Virtual Geographic Environment of Ministry of Education, Nanjing 210046, China
highlights • Multifractal characteristic of API time series is studied based on MF-DFA and singularity spectrum. • Strength of distribution multifractality is stronger than that of correlation multifractality. • Temporal variation in structure of API time series is mainly related to long-range correlations.
article
info
Article history: Received 7 October 2014 Received in revised form 21 November 2015 Available online 1 February 2016 Keywords: Multifractality Multifractal detrended fluctuation analysis Multifractal characteristics Long-range correlations A broad probability density function API
abstract This paper describes multifractal characteristics of daily air pollution index (API) records in Nanjing from 2001 to 2012. The entire daily API time series is first divided into 12 parts that serve as research objects, and the generalized Hurst exponent is calculated for each series. And then, the multifractal sources are analyzed and singularity spectra are shown. Next, based on a singularity spectrum, the multifractal-characteristics parameters (maximum exponent α 0 , spectrum width ∆α , and asymmetry ∆α as ) are introduced. The results show that the fractality of daily API for each year is multifractal. The multifractal sources originate from both a broad probability density function and different long-range correlations with small and large fluctuations. The strength of the distribution multifractality is stronger than that of the correlation multifractality. The variation in the structure of API time series with increasing years is mainly related to long-range correlations. The structure of API time series in some years is richer. These findings can provide a scientific basis for further probing into the complexity of API. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Air pollution in China is a severe problem with social, economic and political consequences [1]. A World Bank study reported that 16 out of the top 20 most polluted cities in the world were located in China [2]. Air pollution index (API) is a referential parameter frequently used for reporting the levels of ambient air pollution. Three major pollutants, including respirable particulate matter (PM10 ), sulfur dioxide (SO2 ) and nitrogen dioxide (NO2 ), have been selected to report daily API in China [3]. Since June 2000, the State Environment Protection Agency of China has required daily API reports, helping the public to understand local air quality. The higher the API, the more serious the air pollution.
∗
Correspondence to: College of Geographical Science, Nanjing Normal University, Jiangsu, China. E-mail address: [email protected] (C.-h. Shen).
http://dx.doi.org/10.1016/j.physa.2016.01.061 0378-4371/© 2016 Elsevier B.V. All rights reserved.
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Air pollution system is very complex. The structure of API time series and its temporal variation are rather complicated [4]. The structural complexity leads us to adopt multifractal formalism to analyze its structural characteristics [5]. This formalism is a widely used technique for quantitatively delineating the nonlinear evolution of a complex system and the multiscale characteristics of physical quantity, and aids in an understanding of the intrinsic regularity and the mechanism of physical changes [6]. Multifractal detrended fluctuation analysis (MF-DFA) [7], an important approach in the study of the multifractal properties of a non-stationary time series, has been a favored research tool. Based on MF-DFA, a singular spectrum is obtained, and the important information of multifractal characteristics can be extracted from it. The time-series structural complexity is inferred through this information [7]. MF-DFA and its related methods [8–11] have been widely used in air pollution studies [12,13] and in other relevant fields [14]. For instance, Shi et al. found that APIs from July 2000 to June 2006 in Shanghai were characterized by scale invariance, long-range dependence and multifractal scaling [13]. Diosdado et al. found that time series of air pollutant concentration, such as concentrations of ozone, sulfur dioxide, carbon monoxide, nitrogen dioxide and PM10 particles, from 1990 to 2005 in the metropolitan zone of Mexico city, are multifractal [5]. Zhu et al. and Tong et al. investigated the characteristics of air pollutants and the impact of haze in Nanjing, respectively [15,16]. And Shen et al. studied the detrended cross correlation between API and meteorological elements in Nanjing [17,18]. Although the fractal characteristics of API in some Chinese cities and the detrended cross-correlation between API and its influencing factors in Nanjing have been investigated, there is little direct evidence of the yearly fractality, multifractal sources and the variation in fractality with time in Nanjing. Furthermore, the mechanism of variation in fractality has not been fully clarified and developed. The nonlinear mechanism of temporal variation of API in other cities has rarely been reported in the literature [3,19], and it is not clear that the conclusions drawn from other studies are applicable to Nanjing. In this paper, we analyzed API records for the period 2001–2012 in Nanjing, China, choosing Nanjing because it is the capital city of Jiangsu Province, and an important transportation and industrial center within the province. Moreover, Nanjing’s economy has grown rapidly over the past two decades. Expansions in urban population, industrial production, transportation and traffic infrastructure have accelerated the emission levels of various air pollutants. Because of their role in the prevention and control of atmospheric pollution, local government requirements should be taken into account in API forecasts. However, to date, it is not clear whether the API in Nanjing can be fully predicted. If an API time series is long-range correlated, predictability becomes possible. Our objective is thus to investigate the multifractal characteristics of API in Nanjing in order to find a pattern of long-range correlations. Our main contribution is that our focus is not only on empirical evidence of API multifractality in Nanjing, but also on investigating the multifractal sources and the variations in multifractal strengths with increasing year. Our findings can contribute to an understanding of the structural complexity of API time series. The structure of this paper is as follows. Section 2 briefly describes methods and data sources. Section 3 provides detailed empirical results. Section 4 provides a discussion of the results, and the last section provides our conclusions. 2. Methodology and data sources 2.1. Methodology As opposed to simple fractals described by a single scaling exponent, multifractal time series are characterized by a hierarchy of scaling exponents that describe the different scaling behavior of many interwoven subsets of the series. The MF-DFA procedure is briefly described as follows [7]. The original time series in the presence of nonstationarity, x(k), k = 1, 2, . . . , N, is integrated to produce the profile i N ¯ ), i = 1, 2, . . . , N, where x¯ = X (i) = k=1 (x(k) − x k=1 x(k)/N is the average. Next, the entire time series X (i), i = 1, 2, . . . , N, is divided into Ns = N − s overlapping boxes with window size s [20], each containing s + 1 values. The detrended variance f 2 (s, i) of the residuals in a box of size s that starts at i and ends at i + s is calculated as in Eq. (1) [20]
i +s 2 f (s, i) = (X (k) − X˜ (k, i)) (s + 1) 2
(1)
k=i
where the local trend, X˜ (k, i), is the ordinate of a least-squares fit with a straight line or higher order polynomial [20]. Here, we adopted ‘‘overlapping boxes’’ [20] instead of ‘‘non-overlapping boxes’’ [21] to calculate detrended variance. Enough boxes can thus be calculated so that the detrended variance becomes stable. Finally, a qth order detrended fluctuation function is calculated as Eq. (2)
Fq (s) =
Ns 1
Ns i=1
F0 (s) = exp
[f (s, i)]
Ns 1
2Ns i=1
q 2
1q for q ̸= 0
ln f (s, i)
for q = 0.
(2)
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If time series x(k), k = 1, 2, . . . , N, is long-range power-law correlated, the detrended fluctuation function Fq (s) increases asymptotically with q varying as the power law, defined as Eq. (3) Fq (s) ∝ sh(q)
(3)
where h(q) is the generalized Hurst exponent. For a stationary time series, h(2) is identical to the well-known Hurst exponent H. As is well established, h(q) describes the scaling behavior of the boxes with large fluctuations for q > 0; whereas h(q) depicts the scaling behavior of boxes with small fluctuations for q < 0. Usually, h(q) for large fluctuations is smaller than h(q) for small fluctuations. The traditional method to characterize a multifractal time series is to calculate the singularity spectrum f (α), which can be related to h(q) via a Legendre transformation [7]:
α(q) = h(q) + q
dh(q)
(4)
dq
f (α) = q(α(q) − h(q)) + 1 = q2
dh(q) dq
+1
(5)
where α(q) is the fractal exponent and f (α) denotes the dimension of the subset of the series that is characterized by α(q). Eqs. (4) and (5) give a parametric representation of the f (α) curve [7,22]. For a monofractal time series, the singularity spectrum produces a single point in the f (α) plane, whereas the multifractal process yields a single humped function. Based on Eqs. (3) and (5), three parameters in the multifractal spectrum are designated to describe the complexity of the signal: (1) Maximum exponent α0 ∈ [αmin , αmax ], where f (α0 ) has its maximum, is designated as the singularity strength with maximum spectrum. A small value of α0 means that the underlying process ‘‘loses fine structure’’, that is, becomes more regular in appearance [22]. (2) Singularity spectrum width 1α is denoted as (αq− − αq+ ). 1α provides information on the diversity of the scaling exponents of the measure, and reflects multifractal strength. The wider the range of 1α , the stronger the multifractality strength [22]. (3) Asymmetry. The asymmetrical parameter of the multifractal spectrum can be obtained by 1αas = 1αright − 1αleft = (αq− − α0 ) − (α0 − αq+ ), where 1αas = 0 for symmetric curves, 1αas > 0 for left-skewed curves and 1αas < 0 for right-skewed curves. A positive 1αas means a relative dominance of lower fractal exponents corresponding to smoothlooking structures, while a negative 1αas indicates a relative dominance of higher fractal exponents corresponding to fine structures [22]. 2.2. Data sources API records from January 2001 to December 2012 in Nanjing, China, are the primary data sources for our analysis. Original daily API records were obtained from the Ministry of Environmental Protection of the People’s Republic of China (http://datacenter.mep.gov.cn). The quality of all the daily API records was controlled, and there are no more than 67-days missing values. Since the missing value is specified as one single day rather than a few continuous days, the mean value of API before and after the specified day was calculated as the missing API value. Prior to our analysis, we divided the entire original daily API records over the past 12 years into 12 parts as research objects. Each research object has equal time-series length and represents the evolution of API for each year. A single-sample Kolmogorov–Smirnov test was applied to confirm that the probability density distribution function of API for each year is non-Gaussian, as expected [16]. Fig. 1(a) shows the original daily API records versus time. Fig. 1(b) shows the contour plots of wavelet coefficients for original daily API time series from 2001 to 2012 by means of a Morlet wavelet transformation [23,24]. In this figure, coarse black curves indicate that the original daily API time series contains short periodicity during specific time periods, for instance, there is a short period of approximate 150 days in the range of about 1400–4000 days. Earlier studies indicated that a time series with periodicity has an effect on the results of MF-DFA, sometimes leading to spurious findings of multifractality [8,25–28]. In order to reduce an effect of periodicity on the result of MF-DFA, we employed the Fourier analysis [29] to eliminate the periodicity of a time series. First, a Morlet wavelet transformation was utilized to obtain periodic size. The Fourier-expansion coefficients that are connected to periodic sizes were then truncated. In the end, an inverse Fourier transformation was performed, and the periodic trends were removed. Hereafter, an original daily API time series with its periodicity removed through Fourier analysis is denoted as a processed daily API time series. Fig. 1(c) shows the contour plots of wavelet coefficients for the original daily API time series in 2006. Fig. 1(d) shows the contour plots of wavelet coefficients for the processed daily API time series in 2006. Comparing Fig. 1(c) with Fig. 1(d), the periodic trend in the original daily API time series is found to be partially removed. Fig. 2(a) shows the yearly average of original daily API records. The average value reached a peak in 2002, decreased in 2003, and then remained almost unchanged overall. This pattern clearly shows that air pollution in Nanjing was the most severe in 2002. Fig. 2(b) shows magnitude-frequency distribution for original daily API records. In this figure, N
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8
8
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32
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32
API
450
300 250 200 150
1/4
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1/2
128
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1024 0
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1/32 500
730 1095 1460 1825 2190 2555 2920 3285 3650 4015 4380 Days
1000 1500 2000 2500 3000 3500 4000 Days 8
8 4
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Fig. 1. (a) Evolution of original daily API records from January 2001 to December 2012. (b, c and d) Contour plots of wavelet coefficients. (b) For original daily API time series from January 2001 to December 2012. (c) For original daily API time series in 2006. (d) For processed daily API time series in 2006. The thick black contour designates the 5% significance level against red noise [23]; cone of influence (COI) where edge effects might distort the picture is shown as a lighter shade. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
a 120
Yearly average API
b
5 4
)
80
3 ln (
Yearly average API
100
60
2
40
1
20 0 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Years
0
0
1
2
3 ln (C)
4
5
6
Fig. 2. (a) Yearly average tendency. (b) Cumulative magnitude frequency distribution. Daily API records are original.
is the cumulative number of events with size greater than magnitude C and τ is the scaling exponent. It is regarded as dN /dC ∝ C −τ within a certain range [30]. 3. Results 3.1. Fractality of API records for each year The analysis of the multifractal detrended fluctuation for the 12-year processed API records was performed. Fig. 3(a) shows the detrended fluctuation functions Fq (s) of processed API time series in 2006 against time scale s for q = −5, 0 and 5, respectively. In this figure, the time-scale crossover in Fq (s) is detected for q = −5. The size of this time-scale crossover is approximately equal to the periodic size which was obtained by means of a Morlet wavelet transformation [23]. The slopes of Fq (s) against time scales, h(q) = 1Ln(Fq (s))/1Ln(s)(generalized Hurst exponent), differ in different regiments. Periodic trends become dominant at small time scales and large time scales while at medium time scales, the intrinsic fluctuations tend to dominate. At 16 < s < 32, the slopes, h(q), therefore suggest that the local effects, such as meteorological elements and human activities, become dominant [31]. Further analysis shows that time-scale crossovers in Fq (s) against time scales s also exist in other years, and that their results are analogous to the result in 2006. The detrended fluctuation function, Fq (s), thus can develop approximately unique scaling in the range of appropriate time scales. Consequently, there is fractality in the processed API time series in 2006 for the considered value of q in the
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b
4
1 Processed series
0.9
h(q)
3.5 Ln[Fq(s)]
175
3 q=-5 2.5 2
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Shuffled series
0.7
Long-range correlations
0.6
q=0
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q=5
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range of appropriate time scales. The analogous results in other years are also found, and the typical results in 2004 and 2012 are shown in Fig. 3(c) and (d). Fig. 3(c) and (d) shows the evolution of generalized Hurst exponent h(q) in terms of q in 2004 and 2012, respectively. h(q) is greater than 0.5 and less than 1, and is dependent on q. This indicates that the fractality is multifractal rather than monofractal. This is in agreement with recent results [13], which revealed the multifractal behavior of API time series over Shanghai in 2008. On the whole, the results in other years are similar to the results in 2004 and 2012. Thus, the fractality of API records for each year behaves much the same as multifractality in the range of appropriate time scales. 3.2. Multifractal sources of API records for each year Generally, the multifractal sources originate from both a broad probability density function and different long-range correlations with small and large fluctuations [7]. The shuffle procedure can destroy long-range correlations [7]. For convenience, it is denoted as h(q) for processed API time series, hshuf (q) for shuffled API time series and hcor (q) = h(q) − hshuf (q) + 0.5 for long-range correlations, respectively. If h(q) = hshuf (q) depends on q and hcor (q) = 0.5, multifractality is due to a broad probability density function alone. On the other hand, if hshuf (q) = 0.5 and hcor (q) = h(q), multifractality is due to different long-range correlations alone. If hshuf (q) and hcor (q) depend on q, multifractality is due to both a broad probability density function and different long-range correlations. If both kinds of multifractality are present, the shuffled time series will show weaker multifractality than the original time series [7]. In order to distinguish two types of multifractal behaviors, an uncorrelated random time series and the processed API time series are shuffled. The generalized Hurst exponent hu (q) is denoted as an uncorrelated random time series. After two kinds of time series are reshuffled more than 2500 times, the averages of hshuf (q) and hu (q) are calculated with a determination coefficient >0.99, respectively, and the error bars corresponding to standard deviations are given. Fig. 3(b) shows hshuf u (q) and hu (q) for the uncorrelated random time series versus q, and they are found to be approximately equal to 0.5, and to be independent of q. Fig. 3(c) and (d) shows the q-dependence of h(q), hshuf (q) and hcor (q) in 2004 and 2012, respectively. In Fig. 3(c) and (d), shuf h (q) is not equal to 0.5, and strongly depends on q. Fractality due to a broad probability density function can therefore be preliminarily viewed as multifractality in 2004 and 2012. Comparing the results in other years with the results in 2004 and 2012, and considering that the error from the algorithm and the statistics, hshuf (q) in each year is found to be strongly dependent on q. We can thus conclude that the fractality due to a broad probability density function in Nanjing is multifractal for each year.
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Meanwhile, hcor (q) in each year is also obtained. Considering the error from the algorithm, when hcor (−5)–hcor (+5) < 0.1, the fractality due to long-range correlations is judged to be monofractal [17]. Consequently, fractality due to long-range correlations is found to be multifractal in 2004, 2008, and 2010, but, to be monofractal in other years. As expected, API time series over a period of 12 years does not exhibit a simple random behavior. The fractality in each year is multifractal. hcor (q) is not equal to 0.5, indicating the presence of long-range correlations. That is to say, API time series contains long-range correlation persistence. Multifractal sources stem from both a broad probability density function and different long-range correlations with small and large fluctuations. Fractality due to a broad probability density function is multifractal. Fractality due to long-range correlations is monofractal in some years, but is multifractal in other years. 3.3. Multifractal characteristics of API records for each year In order to detect multifractal characteristics of API time series in Nanjing, the singularity spectra are calculated through h(q), hshuf (q) and hcor (q), respectively, based on Eqs. (4) and (5). Fig. 4(a–c) shows that relevant spectrum-characteristic parameters, such as α0 , 1α , and 1αas [22], which are extracted from the singularity spectra, vary with increasing years. h shuf cor These parameters are α0h , α0shuf , α0cor , 1α h , 1α shuf , 1α cor , 1αas , 1αas , and 1αas , respectively, and differ by year. Based on h shuf cor Eqs. (4) and (5), we obtained formulae as follows: α0h = α0shuf + α0cor , 1α h = 1α shuf + 1α cor , 1αas = 1αas + 1αas . Fig. 4(d) shows that Hurst exponent H varies with increasing year. According to Eq. (4), α0h = h(0) is generalized Hurst exponent for q = 0. α0h reflects the basic level of either long-range correlations or a broad probability density function. Fig. 4(a) shows the respective α0h , α0shuf , and α0cor vary with year. In this figure, α0h (blue curve) for the processed API time series is approximately equal to 0.60–0.80 from 2001 to 2012. With years, the variation in α0h is fluctuant. α0shuf (red curve) for the shuffled time series is approximately equal to 0.55 for the period of 2001–2012. This means that shuffling procedures have removed the structures related to long-range correlations, and that the shuffled API time series are not white noise. With increasing years, the series-structure evolution related to a broad probability density function is comparatively small, but the series-structure evolution due to long-range correlations with small and large fluctuations is fluctuant. The seriesstructure evolution of the processed API time series is thus attributed to different long-range correlations. Moreover, α0h is relatively high in 2002, 2004 and 2009, indicating that API time series in these years provide rich series-structure information. A singularity spectrum width, 1α , characterizes the multifractal strength. The gradual decrease in 1α over time indicates a reduction in the degree of multifractality, while an increase in 1α indicates the opposite.
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Fig. 4(b) shows the variation in 1α h , 1α shuf , and 1α cor versus year, respectively. In this figure, 1α h (blue curve) is fluctuant with increasing years, and is the highest in 2008 and 2010, and the lowest in 2003. This suggests that the multifractal strengths are also fluctuant with increasing years, and the structure of the processed API time series is the most complex in 2008 and 2010, and is the simplest in 2003. Comparing 1α shuf with 1α cor , it can be seen that 1α shuf is greater than 1α cor ; that is to say, the strength of distribution multifractality is stronger than that of the correlation multifractality. Fig. 4(b) shows that 1α cor is fluctuant with increasing years. In 2004, 2008 and 2010, 1α cor is comparatively large, but is approximately equal to zero in other years. This confirms that fractality due to long-range correlations in 2004, 2008, and 2010 is multifractal, and that it is monofractal in other years. Moreover, 1α h = 1α shuf + 1α cor , meaning that 1α h is tightly connected with the multifractal strength of the distribution and correlation. The ratio of 1α shuf to 1α h is defined, and represents the contribution of the distribution’s multifractal strength to total multifractal strength. In Nanjing, total multifractal strength can thus be primarily attributed to the multifractal strength due to distribution, while the contribution of the multifractal strength due to correlation to total multifractal strength is limited. h An asymmetry of a singularity spectrum, 1αas , indicates the relative importance between the high and low fractal exponents, and reflects the different range sizes of scaling exponents related to small and large probability or small and h shuf cor large fluctuations. Moreover, 1αas = 1αas + 1αas , indicates that the relative importance between high and lower fractal exponents is tightly connected with distribution multifractality and correlation multifractality. When long-range correlation fractality is monofractal, the relative importance between high and lower fractal exponents is closely related to distribution h multifractality. A positive 1αas , meaning that the lower fractal exponent is dominant, exhibits some asymmetry with consistently left-skewed spectra, and indicates that a wider range of scaling exponent spectrum is required to characterize h shuf the variation in series’ structure. A negative 1αas is exactly the opposite. 1αas reflects different ranges of scaling exponents cor due to large and small probability. 1αas reflects different ranges of scaling exponents due to large and small fluctuations. h shuf cor h Fig. 4(c) illustrates that 1αas , 1αas , and 1αas vary with year. The figure shows that the evolution of 1αas is h comparatively large in 2004, 2008, 2009, and 2012. The positive 1αas in 2004 and 2012 means that singularity spectrum shape is left-skewed, lower fractal exponent is dominant, and the structure appears relatively smooth. But, the negative 1α =has in 2008 and 2009 means that singularity spectrum shape is right-skewed, higher fractal exponent is dominant, and the structure is relatively fine. cor h Since the fractality in 2004, 2008, and 2010 is multifractal, the corresponding 1αas is not equal to zero. 1αas in 2004, shuf 2008, and 2010 is thus attributed to distribution multifractality and correlation multifractality. In particular, 1αas > 0 and cor shuf cor 1αas > 0 in 2004, and 1αas < 0 and 1αas < 0 in 2008, indicate that the small and large fluctuations due to long-range correlations are consistent with the low and high probability due to distribution. But, in 2010, the situation is exactly the opposite, indicating that there is competition between small probability due to distribution and large fluctuations due to different long-range correlations. There is an indirect evidence to support this result. There were more than 1158 construction sites in 2010, and more than 2000 construction sites in 2012 in Nanjing downtown. On the one hand, an increasing number of construction sites causes the increasing amount of bare ground to result in higher levels of dust and a rising API. On the other hand, in order to effectively control these urban fugitive dust sources, the municipal government took measures to curb severe pollution, for instance, the focused management of urban green land. As a result, the public green area per capita reached 13.6 m2 in 2010, and 14.3 m2 in 2012 in Nanjing. An increase in green land gradually declined the dominant levels of high API. This result is consistent with the variation in multifractality with years. Fig. 4(d) shows that Hurst exponent varies with years. Hurst exponent (red curve) for the shuffled time series is approximately equal to 0.52 for the period of 2001–2012. This means that the series persistence’s evolution with increasing years is thus related to different long-range correlations. Hurst exponent for long-range correlation >0.5 thus implies that API time series in Nanjing can be predictable. Since the level of positive persistence due to long-range correlations is not very strong (the average of Hurst exponent due to long-range correlations over 12 year is about 0.6), predictable reliability merits further investigation. As can be seen in Fig. 4, the distribution-multifractality strength is stronger than that of the correlation multifractality, the temporal variation in the structure of API time series is mainly related to long-range correlations, and the time-series structure in some years is comparatively fine. 4. Discussions 4.1. Effect of time-series length on MF-DFA The length of the time series has an effect on DFA. Longer time series will lead to more reliable result of multifractal analysis based on MF-DFA [32]. However, it is of importance how to ensure the stability of Fq (s). In MF-DFA, profiles X (i), i = 1, 2, . . . , N, are divided into Ns = int(N /s), which are non-overlapping segments of equal length s [21], and Fq (s) can be obtained through the summation of detrended variance in different windows. The total summation is Ns . When N is relatively small, Ns is relatively small, too. For instance, when N = 365 and s = 10, only int (N /s) = 36 data groups will be used to calculate Fq (s). However, we modified calculation method of Fq (s) according to Ref. [20]. The profiles X (i) are
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divided into Ns = N − s, which are overlapping windows, each containing equal length s. For the above example, 355 data groups will be applied to calculate Fq (s). Calculating enough data can ensure the stability of Fq (s). In Fig. 1(c), the very short period is found in the original API time series in 2006. A short period in a shorter time series has an effect on the result of MF-DFA. When the timepoint range of a short period is relatively large, Fq (s) will contain a contribution from the intrinsic fluctuation due to long-range correlations and periodic trends. Thus, the removal of a short period is important to ensure the reliability of Fq (s) related to the long-range correlations. Similar to Ref. [29], Fourier analysis was performed to remove high frequency signals. However, we found that it is rather difficult to fully remove these high frequency signals this way. Thus, more optimized method to remove high frequency signals is needed to be investigated in the future.
4.2. Multifractal analysis based on multifractal sources Generally, the multifractal sources originate from both a broad probability density function and from different long-range correlations with small and large fluctuations. When a multifractal analysis is performed, a multifractal source should be distinguished to ensure reliable results. Thus, reshuffled processes of time series can be first applied to detect multifractal sources, and then multifractalities due to long-range correlation and a broad probability density function must be separated. Next, a corresponding analysis due to a broad probability density function or different long-range correlations with small and large fluctuations is performed, which might produce reliable results.
4.3. Long-range correlation of API Regarding complexity, the API time series fluctuations are macro effects, caused by local interactions and correlations between various factors. On the one hand, an air pollution system contains many components, such as pollutant sources, atmospheric pollutant components, solar radiation, atmospheric self-purification, topographical features, and meteorological factors. Each component has a certain influence on the average daily air pollution concentration. When all the components are considered together, they interact and correlate with each other on vastly different timescales. Under this circumstance, the API fluctuations have characteristics of certainty to some extent. On the other hand, because the atmospheric system is open and dissipative, API evolution is uncertain. Hence, API evolution with time is irregular and nonlinear, and it has the basic characteristics of a complex system. The various interactions and correlations between a variety of complicated factors have cumulative effects on API fluctuation, and create a fluctuation of API that becomes a ‘macro order’ structure, characterized by certainty. The micromechanisms of API evolution are very complex. A self-organized criticality (SOC) model of air pollution is adopted to explain emergent complex behavior in Chengdu [30]. It is characteristic of SOC behavior that the distributional function of API satisfies the requirement of a power-law distribution, and that an API time series has long-range persistence. Fig. 2(b) also shows that cumulative magnitude-frequency distributions exhibit power-law scaling. This is regarded as the typical ‘‘critical’’ dynamical behavior of SOC systems. The SOC model seems suitable for explaining the long-range correlation of API in Nanjing. However, an analysis in detail of the long-range correlation can be performed in the future.
5. Conclusions In this paper, the multifractal characteristics of API time series in Nanjing, China, from 2001 to 2012, are studied based on the MF-DFA and singularity spectrum method. By studying three multifractal parameters (α0 , 1α , and 1αas ) of a singularity spectrum, multifractal characteristics of daily API records in Nanjing are found to be clearly in agreement with recent results obtained in other cities, China. The source of API multifractality in Nanjing originates from a broad probability density function and long-range correlations. Multifractal strength due to a broad probability density function is stronger than multifractal strength due to long-range correlations with small and large fluctuations. The temporal variation in the structure of API time series is mainly related to long-range correlations. Multifractality may be fluctuant with increasing years. The three multifractal parameters describe the long-range correlations from different views, and a specified combination of the parameters represents a particular long-range correlation of the corresponding processes. These results aid in a full understanding of the long-range correlations of API records and may provide theoretical support in API prediction.
Acknowledgment The authors are grateful for a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
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