- Email: [email protected]
Long-range correlation analysis among non-stationary passive scalar series in the turbulent boundary layer
Physica A 517 (2019) 290–296
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Long-range correlation analysis among non-stationary passive scalar series in the turbulent boundary layer ∗
Chen Dongwei a,b , Hu Fei a,b , , Xu Jingjing a,b , Liu lei a a
State Key Laboratory of Atmospheric Boundary Layer Physics and Atmospheric Chemistry, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China b University of Chinese Academy of Sciences, Beijing 100049, China
highlights • Diffusion conditions drive the concerted variability of passive scalars in turbulent boundary layer. • There exist long-range persistent correlations among observed passive scalars in the turbulent boundary layer during air pollution. • The existence of long-range correlations is conducive to understand the interdependent mechanism of air pollutants and serves as an effective way of evaluating numerical air pollution models.
article
info
Article history: Received 8 April 2018 Received in revised form 8 August 2018 Available online 9 October 2018 Keywords: Detrended cross-correlation analysis Long-range correlation Passive scalars Turbulent boundary layer
a b s t r a c t In this paper, we investigated the cross correlations among non-stationary passive scalars series. Based on the detrended cross correlation analysis (DCCA), we found that there existed long-range cross correlations among these series. The DCCA exponents for each pair of passive scalars are all greater than 0.5. This implies that the cross correlation is persistent. Moreover, the degree of cross correlation between H2 O and PM2.5 is greater compared to CO2 and PM2.5 as well as CO2 and H2 O. The same analysis is also performed on randomly shuffled series. The calculated DCCA exponents fluctuate slightly over 0.5, which verifies that there indeed exist cross correlations among original series. © 2018 Published by Elsevier B.V.
1. Introduction Long memory is also called long-range dependence, which refers to dependence structures that decay slowly with increasing distance [1]. A phenomenon is usually considered to be long-memory if the correlation decays slowly in power law form rather than exponential form. Long-memory processes are known to play an important role in many scientific disciplines and applied fields [2]. The traditional way to analyze correlation behaviors is calculating the autocorrelation function of time series or computing the corresponding power spectral density function by Wiener–Khinchin theorem. And this theorem claims that the autocorrelation function and the power spectral density function are a Fourier-transform pair, which requires the stationarity of the time series. But in the real world, due to unexpected factors, time series are often non-stationary and they usually fluctuate violently. Hence parameters such as mean and variance of the series change with time. Non-stationarity can result in spurious correlations so the methods mentioned above become invalid. Actually there was no effective method to quantify the cross correlations exponents between two correlated time series in the presence of ∗ Corresponding author at: State Key Laboratory of Atmospheric Boundary Layer Physics and Atmospheric Chemistry, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China. E-mail address: [email protected] (H. Fei). https://doi.org/10.1016/j.physa.2018.09.094 0378-4371/© 2018 Published by Elsevier B.V.
C. Dongwei et al. / Physica A 517 (2019) 290–296
291
nonstationarity until Podobnik and Stanley [3] first proposed the detrended cross-correlation analysis (DCCA), to detect and quantify the long-range correlations in non-stationary time series. Within this framework, it is possible to separate trends and fluctuations and then quantify the correlation between the fluctuations of two records in different length scales by the DCCA exponents [4]. The method has been applied in many fields and been extended to many forms. Ikeda [5] used DCCA to analyze the cross correlation for stock markets of United States, Japan and Europe while Igor [6] analyzed 1/f behaviors in cross correlations between absolute returns in a US market. Pal and Mitra [7] explored the interdependence between crude oil and world food prices. The same procedure was also performed on the cross-correlations between volume change and price change [8] and the power-law cross correlations between mathematical constants [9]. In terms of extension of the method, Qian et al. [10] consider the influence of common external forces on the detrended partial cross-correlation analysis. Horvatic et al. [11] took periodic trends into account when dealing with detrended cross-correlation analysis. Local and global detrending approaches were also tested to quantify the cross correlations [12]. Besides, there has been a significant increase of studies analyzing correlation properties of observed and simulated meteorological data. Fraedrich and Blender [13] analyzed the scaling characteristics of atmosphere and ocean temperature correlations in observations and climate models. H. D. He [14] tried to enhance the understanding of cross correlations between meteorological factors and pollutants. Analysis were also implemented on the dynamical mechanism in meteorological factors [15], long-range cross-correlation behaviors of different particulate matters [16] and long-term correlations in hourly wind speed records [17]. Studies on the long-term correlations analysis of observed and simulated data not only indicate the complexity of atmospheric motions, but also reveal the difficulty of numerical weather prediction. However, the long-range correlation is seldom analyzed during air pollution period in the turbulent boundary layer, which is one of the top environmental concerns and leads to many health problems. Hence in this paper we concentrate on the cross-correlations and scaling behaviors among observed air pollutants series during air pollution. It contributes to validating the existence of long-range correlations and understanding the interdependent mechanism of these pollutants, which is significant to improve the prediction of numerical air pollution model. A satisfying numerical model not only has minor prediction errors but reveals the existence of long-range correlations, so distinguishing the difference on correlation behaviors between simulated data and observed data is also an effective way to evaluate numerical air pollution models compared to the traditional statistical methods. Air pollutants are passive scalars so they have a negligible back effect on the flow [18]. Due to complicated atmospheric motions ranging from large planetary waves to turbulence [19,20], time series in the turbulent boundary layer are always non-stationary, so we use the detrended cross-correlation analysis on passive scalars (CO2 , H2 O and PM2.5 ) in this paper. PM2.5 , which is the primary air pollutants in China, refers to ambient particulate matter that is less than 2.5 µm in aerodynamic diameter. H2 O is related to the hygroscopic growth of PM2.5 and CO2 is related to greenhouse effect and climate change. The paper is organized as follows. In part 2 we introduce the data set and briefly describe the methodology used here. Then in part 3 some results are presented. We show that there exist long-range cross correlations among passive scalars in the turbulent boundary layer. In part 4 we draw some conclusions. 2. Data and methodology 2.1. Data set CO2 and H2 O datasets are obtained from the eddy covariance systems mounted at seven levels (8 m, 16 m, 47 m, 80 m, 140 m, 220 m and 280 m separately) on a 325 m meteorological tower. The system consists of two parts. One is for measuring wind speed and temperature. The other is for water vapor and CO2 . Actual instantaneous values of velocity and temperature are measured by three-dimensional sonic anemometer-thermometers (Wind Master, Gill, USA). Water vapor and CO2 are sampled by LI-7500 open-path gas analyzers (Li-Cor. Inc., USA). The meteorological tower is located in north-central Beijing (39.967 ◦ N, 116.367 ◦ E) and was set up in 1978 by the Institute of Atmospheric Physics, Chinese Academic of Sciences. The underlying surface is a typical complex urban area including trees, rivers, roads and buildings of various heights [21]. PM2.5 datasets of Beijing are downloaded from the website StateAir [22], which belongs to the U.S. Department of State Air Quality Monitoring Program. PM2.5 data are sampled by beta attenuation mass monitor. Beta rays having kinetic energies less than 1 million electron volts collide with atoms they encounter, causing incremental losses in electron energy which is proportional to the number of collisions [23]. Then the concentration of PM2.5 is obtained through calculations. In Beijing, PM2.5 data are measured at the U.S. Embassy at No.55 An Jia Lou Rd., Chaoyang District (39.95 ◦ N, 116.47 ◦ E). This place locates in the southeast direction of the meteorological tower, with distance of 8 kilometers in straight line. The difference in monitoring places of passive scalars excludes the influence of position on correlations among passive scalars series and guarantees the reliability of our analysis. CO2 and H2 O signals (10 Hz) were collected in 47 m of the tower from 0000 LST (Local Standard Time) 26 Nov. 2015 to 2400 LST 27 Dec. 2015. And PM2.5 series was collected during the same time. But PM2.5 is sampled once per hour, so CO2 and H2 O signals were performed by one-hour average. Due to unexpected factors, 3% of values in CO2 and H2 O datasets are missed and 1% in PM2.5 series. These values are replaced by fitted values through linear or secondary regression. It is found that these passive scalars fluctuate in phase during heavy air pollution. The correlation coefficients of CO2 /PM2.5 , PM2.5 /H2 O and CO2 /H2 O are 0.90, 0.62 and 0.57, respectively, where the notation H2 O/PM2.5 stands for H2 O and PM2.5 series, and the notations CO2 /PM2.5 and CO2 /H2 O also present corresponding meanings.
292
C. Dongwei et al. / Physica A 517 (2019) 290–296
Fig. 1. Figure (a), (b), (c), (d) represents H2 O, CO2 , PM2.5 and horizontal wind speed signals respectively. The time is from 2015-11-26 00:00 LST to 201512-27 24:00 LST. During this period, air pollution appeared four times. Every time the mass concentration of PM2.5 was beyond 300 µgm−3 .
The high correlations of these passive scalars series are related to diffusion conditions. Passive scalars accumulate to a high concentration under poor diffusion conditions and drop to a low concentration under good diffusion conditions. This can be seen in Fig. 1. The first three panels are H2 O, CO2 and PM2.5 signals. The last panel is horizontal wind speed signals. When the wind speed is small, indicating a poor diffusion condition, the concentration of H2 O, CO2 and PM2.5 all start to increase. On the contrary, when the wind speed becomes large, indicating a good diffusion condition, the concentration of H2 O, CO2 and PM2.5 drop to a low concentration. The correlation coefficients between the horizontal wind speed and CO2 , PM2.5 and H2 O are −0.56, −0.49 and −0.53, respectively. It is notable that in this paper we focus on analyzing the cross-correlation behaviors among observed passive scalar series, which is an effective way to evaluate and improve numerical air pollution models. In such models wind speed only serves as diffusion factors providing the initial and boundary diffusion conditions, hence we only analyze the cross correlations among passive scalars. 2.2. Methodology The detrended cross correlation analysis (DCCA) is a generalization of detrended fluctuation analysis (DFA). At first DFA method was used to quantifying the correlation property for non-stationary physiological time series [24]. Then DCCA method was introduced to quantify the cross correlation between two non-stationary time series. The DCCA method will turn to the DFA method if the two series of DCCA are identical. Here we describe the DCCA method briefly. Consider two independent time series xi and yi which are observed simultaneously. The integrated series Xk and Yk are defined as Xk =
k ∑
(xi − x)
i=1
Yk =
k ∑
(yi − y)
i=1
where x and y are arithmetic averages and k = 1, 2, . . . , n. Here we call integrated series as profile. Then we divide Xk and Yk into Nv (⌊n/v⌋) non-overlapping segments or windows with the same length v . For each segment, we get the local trend
C. Dongwei et al. / Physica A 517 (2019) 290–296
293
2 ˆ Xˆ k (v) or Yk (v) by linear regression where k = 1, 2, . . . , Nv . The covariance of residuals fDCCA for each segment is computed as
2 = fDCCA
v )( ) 1 ∑( Xk (v, w) − Xkˆ (v, w ) Yk (v, w) − Ykˆ (v, w )
v
w=1
Finally, the detrended covariance FDCCA (v) is considered as
Nv 1 ∑ 2 fDCCA (v, i) FDCCA (v) = √ Nv
i=1
If the series have scale properties related to cross correlation, FDCCA (v) is expected to obey power law as follows FDCCA (v) ∼ v λ Marinho, et al. [25] thinks that one will observe a reduction of fluctuation function caused by the negative contributions 2 due to possible different signs of terms when using fDCCA to get the covariance of residuals. If so, clear scaling behaviors will 2 . f|2DCCA| is defined as follow not be obtained. We use absolute values of fluctuations f|2DCCA| to replace fDCCA f|2DCCA| =
v ⏐ 1 ∑⏐
v
w=1
⏐ ⏐⏐ ⏐ ⏐⏐ (v, w )⏐ (v, w )⏐ ⏐Yk (v, w) − Ykˆ ⏐Xk (v, w) − Xkˆ
So the corresponding fluctuation function F|DCCA| (v ) is presented as
Nv 1 ∑ F|DCCA| (v ) = √ f|2DCCA| (v, i) Nv
i=1
The power law is F|DCCA| (v ) ∼ v λ The DCCA exponent λ quantifies the cross correlation of two time series. If λ > 0.5, there is a long-term cross correlation between two series, which means that the cross correlation is persistent and an increase of one series is likely to be followed by an increase of the other series in the future. The correlation of the variables never disappears, and therefore has a permanent effect on current cross relationship of the variables [5]. And the greater λ is, the stronger the cross correlation will be. On the contrary, if λ < 0.5, the cross correlation is anti-persistent and an increase of one series is likely to be followed by a decrease of the other series in the future. If λ = 0.5, the series is not cross-correlated with the other and the change of one series cannot affect the behavior of the other series [26]. If the two series are identical, the detrended cross correlation analysis (DCCA) will degrade into the detrended fluctuation analysis (DFA). F|DCCA| will change into FDFA
Nv v ( )2 1 1∑ ∑ FDFA (v ) = √ Xk (v, w) − Xkˆ (v, w ) Nv v i=1 w=1
The power law relationship of DCCA will change into the DFA power law relationship FDFA (v ) ∼ v λ The DFA exponent λ also quantifies the long-range correlation for one time series. But if the DFA exponent is greater than 1, correlation exists but ceases to be of a power law form. In addition, the DFA exponent λ can also be viewed as an indicator that describes the ‘‘roughness’’ of the original time series: the larger the DFA exponent is, the smoother the time series [24]. 3. Results and discussion Firstly, we perform the detrended fluctuation analysis (DFA) on CO2 , PM2.5 and H2 O signals. As shown in Table 1, the DFA exponents are 1.10, 1.12 and 1.27 respectively. The DFA exponents are all greater than 1, so correlation exists but ceases to be of a power law form and H2 O series is the smoothest while CO2 and PM2.5 series have similar smoothness.
294
C. Dongwei et al. / Physica A 517 (2019) 290–296 Table 1 DFA Exponents. Time series
Scaling exponent
PM2.5 CO2 H2 O
1.12 1.10 1.27
Fig. 2. The detrended cross correlation analysis between PM2.5 and H2 O series. The determination coefficients of linear regression for original series and randomly shuffled series are 0.978 and 0.991 respectively, which shows clear linear relationship between ln (v) and ln (F (v)).
In this paper, the minimum time scale v of detrended cross correlation analysis is 10 h and the maximum 350 h. As shown in Fig. 2, the DCCA exponent between H2 O and PM2.5 series is 1.207. This indicates that there exists a long-term cross correlation between H2 O and PM2.5 . And the cross correlation is persistent. An increase of PM2.5 is likely to be followed by an increase of H2 O in the future. To verify that the scaling law indeed reflects the cross correlation of two time series, the same analysis is carried out on the randomly shuffled series which destroys any temporal correlation in the data, while the shuffled data still remain exactly the same fluctuation distribution [26]. If the shuffled time series follow random noise, then the persistence found above does not come from the data themselves, but from their time evolution relations. The calculated DCCA exponent λ for shuffled series of H2 O and PM2.5 is 0.509 presented in Fig. 2. λ is very close to 0.5, which indicates that the obvious randomness and non-correlation in shuffled series are different from the original time series. The same procedure is also performed for other passive scalars. In Fig. 3, the DCCA exponent λ is 1.125 for CO2 and PM2.5 . And for shuffled series, λ is 0.495. In Fig. 4, the DCCA exponent for CO2 and H2 O is 1.184 while for the randomly shuffled series, λ equals 0.496 and is still close to 0.5. It is notable that the DCCA exponents for all randomly shuffled series will change if the original series are shuffled again. But the values of λ fluctuate slightly around 0.5. This indicates that there indeed exists obvious randomness and non-correlation in shuffled series. The results mentioned above suggest that CO2 and PM2.5 series are cross-correlated persistently, and CO2 and H2 O series also have the same relationship. The scaling exponent of H2 O/PM2.5 is greater compared to CO2 /PM2.5 and CO2 /H2 O. This implies that the degree of cross correlation between H2 O and PM2.5 is greater than that of CO2 /PM2.5 and CO2 /H2 O. It is understandable since H2 O is related to the hygroscopic growth of PM2.5 , and water vapor has an obvious effect on PM2.5 concentration. 4. Conclusions In this paper, we find that CO2 , H2 O and PM2.5 signals fluctuate in phase during air pollution. And based on the detrended cross correlation analysis (DCCA), we find that there exist long-range cross correlations among these series. The DCCA exponents for each pair of passive scalars are all greater than 0.5, which implies that the cross correlation is persistent. Moreover, the degree of cross correlation between H2 O and PM2.5 is greater than other pairs (CO2 /PM2.5 and CO2 /H2 O). To verify the effectiveness of the method, the analysis is also performed on randomly shuffled series. And the calculated DCCA exponents for shuffled series fluctuate slightly around 0.5. This conforms that there indeed exist cross correlations among original series.
C. Dongwei et al. / Physica A 517 (2019) 290–296
295
Fig. 3. The detrended cross correlation analysis between CO2 and PM2.5 series. The determination coefficients of linear regression for original series and randomly shuffled series are 0.983 and 0.969 respectively, which shows clear linear relationship between ln (v) and ln (F (v)).
Fig. 4. The detrended cross correlation analysis between CO2 and H2 O series. The determination coefficients of linear regression for original series and randomly shuffled series are 0.980 and 0.990 respectively, which shows clear linear relationship between ln (v) and ln (F (v)).
Acknowledgments This research is supported by the National Key R&D Program of China (Grant No 2016YFC0208802) and the National Natural Science Foundation of China (Grant Nos 11472272, 41605010 and 41675012). References [1] J. Beran, Long-range dependence, in: Wiley Interdisciplinary Reviews Computational Statistics, vol. 2, 2010, pp. 26–35. [2] J. Beran, Y. Feng, S. Ghosh, R. Kulik, Long-Memory Processes, Wiley Statsref Statistics Reference Online, 2013, pp. 107–208. [3] B. Podobnik, H.E. Stanley, Detrended cross-correlation analysis: a new method for analyzing two nonstationary time series, Phys. Rev. Lett. 100 (2008) 084102. [4] A.S.S. Paiva, M.A.R. Castro, R.F.S. Andrade, DCCA analysis of renewable and conventional energy prices, Physica A 490 (2018) 1408–1414. [5] T. Ikeda, A detrended cross correlation analysis for stock markets of the united states japan,and the europe, Physica A 484 (2017) 194–198. [6] Igor Gvozdanovic, Boris Podobnik, Duan Wang, H.Eugene Stanley, 1/f behavior in cross-correlations between absolute returns in a US market, Physica A 391 (2012) 2860–2866.
296
C. Dongwei et al. / Physica A 517 (2019) 290–296
[7] D. Pal, S.K. Mitra, Interdependence between crude oil and world food prices: a detrended cross correlation analysis, Physica A 492 (2018) 1032–1044. [8] Boris. Podobnik, Davor. Horvatic, Alexander. Petersen, H. Eugene Stanley, Cross-Correlations between Volume Change and Price Change, in: Proceedings of the National Academy of Sciences of the USA, 106, 2009, pp. 22079–22084. [9] A. Machado, S. Filho, Detrended Cross-Correlation Analysis: Measuring cross-correlation between mathematical constants, in: XXXII Encontro Nacional de Física da Matéeria Condensada, 2009.. [10] Xi-Yuan. Qian, Ya-Min. Liu, Zhi-Qiang. Jiang, Boris. Podobnik, Wei-Xing. Zhou, H. Eugene Stanley, Eugene stanley detrended partial cross-correlation analysis of two time series influenced by common external forces, Phys. Rev. E 91 (2015) 062816. [11] Davor. Horvatic, H.Eugene Stanley, Boris. Podobnik, Detrended cross-correlation analysis for non-stationary time series with periodic trends, Europhys. Lett. 94 (2011) 18007. [12] B. Podobnik, I. Grosse, D. Horvatic, S. Ilic, P. Ch. Ivanov, H.E. Stanley, Quantifying cross-correlations using local and global detrending approaches, Eur. Phys. J. B 71 (2009) 243–250. [13] K. Fraedrich, R. Blender, Scaling of atmosphere and ocean temperature correlations in observations and climate models, Phys. Rev. Lett. 90 (2003) 108501. [14] H.D. He, Multifractal analysis of interactive patterns between meteorological factors and pollutants in urban and rural areas, Atmos. Environ. 149 (2017) 47–54. [15] D.D. Kang, I.L. Dong, K. Kim, G. Lim, D.H. Ha, Dynamical mechanism in meteorological factors using detrended cross-correlation analysis, J. Korean Phys. Soc. 65 (2014) 577–590. [16] H.D. He, W. Pan, W.Z. Lu, Y. Xue, G.H. Peng, Multifractal property and long-range cross-correlation behavior of particulate matters at urban traffic intersection in Shanghai, Stoch. Environ. Res. Risk Assess. 30 (2016) 1515–1525. [17] M.D.O. Santos, T. Stosic, B.D. Stosic, Long-term correlations in hourly wind speed records in pernambuco, brazil, Physica A 391 (2012) 1546–1552. [18] B.I. Shraiman, E.D. Siggia, Scalar turbulence, Nature 405 (2000) 639. [19] Y. Kang, D. Belušić, K. Smith-Miles, Detecting and classifying events in noisy time series, J. Atmos. Sci. 71 (2014) 1090–1104. [20] D. Belušić, L. Mahrt, Is geometry more universal than physics in atmospheric boundary layer flow? J. Geophys. Res. Atmospheres 117 (2012) 9115. [21] Z.P. Song, F. Hu, J.J. Xu, X.L. Cheng, Self-Similar characteristic for the ramp structures of wind speed, Atmospheric Ocean. Sci. Lett. 7 (2014) 320–323. [22] http://www.stateair.net/web/historical/1/1.html (accessed 26 Feb. 2018). [23] Guidance for using continuous monitors in PM2.5 monitoring networks. http://www.epa.gov/ttnamti1/files/ambient/pm25/r-98-012.pdf (accessed 30 June, 2018). [24] C.K. Peng, S. Havlin, H.E. Stanley, A.L. Goldberger, Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series, Chaos 5 (1995) 82. [25] E.B.S. Marinho, A.M.Y.R. Sousa, R.F.S. Andrade, Using detrended cross correlation analysis in geophysical data, Physica A 392 (2013) 2195–2201. [26] K. Shi, Detrended cross-correlation analysis of temperature, rainfall, pm 10 and ambient dioxins in hong kong, Atmos. Environ. 97 (2014) 130–135.
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Long-range correlation analysis among non-stationary passive scalar series in the turbulent boundary layer ∗
Chen Dongwei a,b , Hu Fei a,b , , Xu Jingjing a,b , Liu lei a a
State Key Laboratory of Atmospheric Boundary Layer Physics and Atmospheric Chemistry, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China b University of Chinese Academy of Sciences, Beijing 100049, China
highlights • Diffusion conditions drive the concerted variability of passive scalars in turbulent boundary layer. • There exist long-range persistent correlations among observed passive scalars in the turbulent boundary layer during air pollution. • The existence of long-range correlations is conducive to understand the interdependent mechanism of air pollutants and serves as an effective way of evaluating numerical air pollution models.
article
info
Article history: Received 8 April 2018 Received in revised form 8 August 2018 Available online 9 October 2018 Keywords: Detrended cross-correlation analysis Long-range correlation Passive scalars Turbulent boundary layer
a b s t r a c t In this paper, we investigated the cross correlations among non-stationary passive scalars series. Based on the detrended cross correlation analysis (DCCA), we found that there existed long-range cross correlations among these series. The DCCA exponents for each pair of passive scalars are all greater than 0.5. This implies that the cross correlation is persistent. Moreover, the degree of cross correlation between H2 O and PM2.5 is greater compared to CO2 and PM2.5 as well as CO2 and H2 O. The same analysis is also performed on randomly shuffled series. The calculated DCCA exponents fluctuate slightly over 0.5, which verifies that there indeed exist cross correlations among original series. © 2018 Published by Elsevier B.V.
1. Introduction Long memory is also called long-range dependence, which refers to dependence structures that decay slowly with increasing distance [1]. A phenomenon is usually considered to be long-memory if the correlation decays slowly in power law form rather than exponential form. Long-memory processes are known to play an important role in many scientific disciplines and applied fields [2]. The traditional way to analyze correlation behaviors is calculating the autocorrelation function of time series or computing the corresponding power spectral density function by Wiener–Khinchin theorem. And this theorem claims that the autocorrelation function and the power spectral density function are a Fourier-transform pair, which requires the stationarity of the time series. But in the real world, due to unexpected factors, time series are often non-stationary and they usually fluctuate violently. Hence parameters such as mean and variance of the series change with time. Non-stationarity can result in spurious correlations so the methods mentioned above become invalid. Actually there was no effective method to quantify the cross correlations exponents between two correlated time series in the presence of ∗ Corresponding author at: State Key Laboratory of Atmospheric Boundary Layer Physics and Atmospheric Chemistry, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China. E-mail address: [email protected] (H. Fei). https://doi.org/10.1016/j.physa.2018.09.094 0378-4371/© 2018 Published by Elsevier B.V.
C. Dongwei et al. / Physica A 517 (2019) 290–296
291
nonstationarity until Podobnik and Stanley [3] first proposed the detrended cross-correlation analysis (DCCA), to detect and quantify the long-range correlations in non-stationary time series. Within this framework, it is possible to separate trends and fluctuations and then quantify the correlation between the fluctuations of two records in different length scales by the DCCA exponents [4]. The method has been applied in many fields and been extended to many forms. Ikeda [5] used DCCA to analyze the cross correlation for stock markets of United States, Japan and Europe while Igor [6] analyzed 1/f behaviors in cross correlations between absolute returns in a US market. Pal and Mitra [7] explored the interdependence between crude oil and world food prices. The same procedure was also performed on the cross-correlations between volume change and price change [8] and the power-law cross correlations between mathematical constants [9]. In terms of extension of the method, Qian et al. [10] consider the influence of common external forces on the detrended partial cross-correlation analysis. Horvatic et al. [11] took periodic trends into account when dealing with detrended cross-correlation analysis. Local and global detrending approaches were also tested to quantify the cross correlations [12]. Besides, there has been a significant increase of studies analyzing correlation properties of observed and simulated meteorological data. Fraedrich and Blender [13] analyzed the scaling characteristics of atmosphere and ocean temperature correlations in observations and climate models. H. D. He [14] tried to enhance the understanding of cross correlations between meteorological factors and pollutants. Analysis were also implemented on the dynamical mechanism in meteorological factors [15], long-range cross-correlation behaviors of different particulate matters [16] and long-term correlations in hourly wind speed records [17]. Studies on the long-term correlations analysis of observed and simulated data not only indicate the complexity of atmospheric motions, but also reveal the difficulty of numerical weather prediction. However, the long-range correlation is seldom analyzed during air pollution period in the turbulent boundary layer, which is one of the top environmental concerns and leads to many health problems. Hence in this paper we concentrate on the cross-correlations and scaling behaviors among observed air pollutants series during air pollution. It contributes to validating the existence of long-range correlations and understanding the interdependent mechanism of these pollutants, which is significant to improve the prediction of numerical air pollution model. A satisfying numerical model not only has minor prediction errors but reveals the existence of long-range correlations, so distinguishing the difference on correlation behaviors between simulated data and observed data is also an effective way to evaluate numerical air pollution models compared to the traditional statistical methods. Air pollutants are passive scalars so they have a negligible back effect on the flow [18]. Due to complicated atmospheric motions ranging from large planetary waves to turbulence [19,20], time series in the turbulent boundary layer are always non-stationary, so we use the detrended cross-correlation analysis on passive scalars (CO2 , H2 O and PM2.5 ) in this paper. PM2.5 , which is the primary air pollutants in China, refers to ambient particulate matter that is less than 2.5 µm in aerodynamic diameter. H2 O is related to the hygroscopic growth of PM2.5 and CO2 is related to greenhouse effect and climate change. The paper is organized as follows. In part 2 we introduce the data set and briefly describe the methodology used here. Then in part 3 some results are presented. We show that there exist long-range cross correlations among passive scalars in the turbulent boundary layer. In part 4 we draw some conclusions. 2. Data and methodology 2.1. Data set CO2 and H2 O datasets are obtained from the eddy covariance systems mounted at seven levels (8 m, 16 m, 47 m, 80 m, 140 m, 220 m and 280 m separately) on a 325 m meteorological tower. The system consists of two parts. One is for measuring wind speed and temperature. The other is for water vapor and CO2 . Actual instantaneous values of velocity and temperature are measured by three-dimensional sonic anemometer-thermometers (Wind Master, Gill, USA). Water vapor and CO2 are sampled by LI-7500 open-path gas analyzers (Li-Cor. Inc., USA). The meteorological tower is located in north-central Beijing (39.967 ◦ N, 116.367 ◦ E) and was set up in 1978 by the Institute of Atmospheric Physics, Chinese Academic of Sciences. The underlying surface is a typical complex urban area including trees, rivers, roads and buildings of various heights [21]. PM2.5 datasets of Beijing are downloaded from the website StateAir [22], which belongs to the U.S. Department of State Air Quality Monitoring Program. PM2.5 data are sampled by beta attenuation mass monitor. Beta rays having kinetic energies less than 1 million electron volts collide with atoms they encounter, causing incremental losses in electron energy which is proportional to the number of collisions [23]. Then the concentration of PM2.5 is obtained through calculations. In Beijing, PM2.5 data are measured at the U.S. Embassy at No.55 An Jia Lou Rd., Chaoyang District (39.95 ◦ N, 116.47 ◦ E). This place locates in the southeast direction of the meteorological tower, with distance of 8 kilometers in straight line. The difference in monitoring places of passive scalars excludes the influence of position on correlations among passive scalars series and guarantees the reliability of our analysis. CO2 and H2 O signals (10 Hz) were collected in 47 m of the tower from 0000 LST (Local Standard Time) 26 Nov. 2015 to 2400 LST 27 Dec. 2015. And PM2.5 series was collected during the same time. But PM2.5 is sampled once per hour, so CO2 and H2 O signals were performed by one-hour average. Due to unexpected factors, 3% of values in CO2 and H2 O datasets are missed and 1% in PM2.5 series. These values are replaced by fitted values through linear or secondary regression. It is found that these passive scalars fluctuate in phase during heavy air pollution. The correlation coefficients of CO2 /PM2.5 , PM2.5 /H2 O and CO2 /H2 O are 0.90, 0.62 and 0.57, respectively, where the notation H2 O/PM2.5 stands for H2 O and PM2.5 series, and the notations CO2 /PM2.5 and CO2 /H2 O also present corresponding meanings.
292
C. Dongwei et al. / Physica A 517 (2019) 290–296
Fig. 1. Figure (a), (b), (c), (d) represents H2 O, CO2 , PM2.5 and horizontal wind speed signals respectively. The time is from 2015-11-26 00:00 LST to 201512-27 24:00 LST. During this period, air pollution appeared four times. Every time the mass concentration of PM2.5 was beyond 300 µgm−3 .
The high correlations of these passive scalars series are related to diffusion conditions. Passive scalars accumulate to a high concentration under poor diffusion conditions and drop to a low concentration under good diffusion conditions. This can be seen in Fig. 1. The first three panels are H2 O, CO2 and PM2.5 signals. The last panel is horizontal wind speed signals. When the wind speed is small, indicating a poor diffusion condition, the concentration of H2 O, CO2 and PM2.5 all start to increase. On the contrary, when the wind speed becomes large, indicating a good diffusion condition, the concentration of H2 O, CO2 and PM2.5 drop to a low concentration. The correlation coefficients between the horizontal wind speed and CO2 , PM2.5 and H2 O are −0.56, −0.49 and −0.53, respectively. It is notable that in this paper we focus on analyzing the cross-correlation behaviors among observed passive scalar series, which is an effective way to evaluate and improve numerical air pollution models. In such models wind speed only serves as diffusion factors providing the initial and boundary diffusion conditions, hence we only analyze the cross correlations among passive scalars. 2.2. Methodology The detrended cross correlation analysis (DCCA) is a generalization of detrended fluctuation analysis (DFA). At first DFA method was used to quantifying the correlation property for non-stationary physiological time series [24]. Then DCCA method was introduced to quantify the cross correlation between two non-stationary time series. The DCCA method will turn to the DFA method if the two series of DCCA are identical. Here we describe the DCCA method briefly. Consider two independent time series xi and yi which are observed simultaneously. The integrated series Xk and Yk are defined as Xk =
k ∑
(xi − x)
i=1
Yk =
k ∑
(yi − y)
i=1
where x and y are arithmetic averages and k = 1, 2, . . . , n. Here we call integrated series as profile. Then we divide Xk and Yk into Nv (⌊n/v⌋) non-overlapping segments or windows with the same length v . For each segment, we get the local trend
C. Dongwei et al. / Physica A 517 (2019) 290–296
293
2 ˆ Xˆ k (v) or Yk (v) by linear regression where k = 1, 2, . . . , Nv . The covariance of residuals fDCCA for each segment is computed as
2 = fDCCA
v )( ) 1 ∑( Xk (v, w) − Xkˆ (v, w ) Yk (v, w) − Ykˆ (v, w )
v
w=1
Finally, the detrended covariance FDCCA (v) is considered as
Nv 1 ∑ 2 fDCCA (v, i) FDCCA (v) = √ Nv
i=1
If the series have scale properties related to cross correlation, FDCCA (v) is expected to obey power law as follows FDCCA (v) ∼ v λ Marinho, et al. [25] thinks that one will observe a reduction of fluctuation function caused by the negative contributions 2 due to possible different signs of terms when using fDCCA to get the covariance of residuals. If so, clear scaling behaviors will 2 . f|2DCCA| is defined as follow not be obtained. We use absolute values of fluctuations f|2DCCA| to replace fDCCA f|2DCCA| =
v ⏐ 1 ∑⏐
v
w=1
⏐ ⏐⏐ ⏐ ⏐⏐ (v, w )⏐ (v, w )⏐ ⏐Yk (v, w) − Ykˆ ⏐Xk (v, w) − Xkˆ
So the corresponding fluctuation function F|DCCA| (v ) is presented as
Nv 1 ∑ F|DCCA| (v ) = √ f|2DCCA| (v, i) Nv
i=1
The power law is F|DCCA| (v ) ∼ v λ The DCCA exponent λ quantifies the cross correlation of two time series. If λ > 0.5, there is a long-term cross correlation between two series, which means that the cross correlation is persistent and an increase of one series is likely to be followed by an increase of the other series in the future. The correlation of the variables never disappears, and therefore has a permanent effect on current cross relationship of the variables [5]. And the greater λ is, the stronger the cross correlation will be. On the contrary, if λ < 0.5, the cross correlation is anti-persistent and an increase of one series is likely to be followed by a decrease of the other series in the future. If λ = 0.5, the series is not cross-correlated with the other and the change of one series cannot affect the behavior of the other series [26]. If the two series are identical, the detrended cross correlation analysis (DCCA) will degrade into the detrended fluctuation analysis (DFA). F|DCCA| will change into FDFA
Nv v ( )2 1 1∑ ∑ FDFA (v ) = √ Xk (v, w) − Xkˆ (v, w ) Nv v i=1 w=1
The power law relationship of DCCA will change into the DFA power law relationship FDFA (v ) ∼ v λ The DFA exponent λ also quantifies the long-range correlation for one time series. But if the DFA exponent is greater than 1, correlation exists but ceases to be of a power law form. In addition, the DFA exponent λ can also be viewed as an indicator that describes the ‘‘roughness’’ of the original time series: the larger the DFA exponent is, the smoother the time series [24]. 3. Results and discussion Firstly, we perform the detrended fluctuation analysis (DFA) on CO2 , PM2.5 and H2 O signals. As shown in Table 1, the DFA exponents are 1.10, 1.12 and 1.27 respectively. The DFA exponents are all greater than 1, so correlation exists but ceases to be of a power law form and H2 O series is the smoothest while CO2 and PM2.5 series have similar smoothness.
294
C. Dongwei et al. / Physica A 517 (2019) 290–296 Table 1 DFA Exponents. Time series
Scaling exponent
PM2.5 CO2 H2 O
1.12 1.10 1.27
Fig. 2. The detrended cross correlation analysis between PM2.5 and H2 O series. The determination coefficients of linear regression for original series and randomly shuffled series are 0.978 and 0.991 respectively, which shows clear linear relationship between ln (v) and ln (F (v)).
In this paper, the minimum time scale v of detrended cross correlation analysis is 10 h and the maximum 350 h. As shown in Fig. 2, the DCCA exponent between H2 O and PM2.5 series is 1.207. This indicates that there exists a long-term cross correlation between H2 O and PM2.5 . And the cross correlation is persistent. An increase of PM2.5 is likely to be followed by an increase of H2 O in the future. To verify that the scaling law indeed reflects the cross correlation of two time series, the same analysis is carried out on the randomly shuffled series which destroys any temporal correlation in the data, while the shuffled data still remain exactly the same fluctuation distribution [26]. If the shuffled time series follow random noise, then the persistence found above does not come from the data themselves, but from their time evolution relations. The calculated DCCA exponent λ for shuffled series of H2 O and PM2.5 is 0.509 presented in Fig. 2. λ is very close to 0.5, which indicates that the obvious randomness and non-correlation in shuffled series are different from the original time series. The same procedure is also performed for other passive scalars. In Fig. 3, the DCCA exponent λ is 1.125 for CO2 and PM2.5 . And for shuffled series, λ is 0.495. In Fig. 4, the DCCA exponent for CO2 and H2 O is 1.184 while for the randomly shuffled series, λ equals 0.496 and is still close to 0.5. It is notable that the DCCA exponents for all randomly shuffled series will change if the original series are shuffled again. But the values of λ fluctuate slightly around 0.5. This indicates that there indeed exists obvious randomness and non-correlation in shuffled series. The results mentioned above suggest that CO2 and PM2.5 series are cross-correlated persistently, and CO2 and H2 O series also have the same relationship. The scaling exponent of H2 O/PM2.5 is greater compared to CO2 /PM2.5 and CO2 /H2 O. This implies that the degree of cross correlation between H2 O and PM2.5 is greater than that of CO2 /PM2.5 and CO2 /H2 O. It is understandable since H2 O is related to the hygroscopic growth of PM2.5 , and water vapor has an obvious effect on PM2.5 concentration. 4. Conclusions In this paper, we find that CO2 , H2 O and PM2.5 signals fluctuate in phase during air pollution. And based on the detrended cross correlation analysis (DCCA), we find that there exist long-range cross correlations among these series. The DCCA exponents for each pair of passive scalars are all greater than 0.5, which implies that the cross correlation is persistent. Moreover, the degree of cross correlation between H2 O and PM2.5 is greater than other pairs (CO2 /PM2.5 and CO2 /H2 O). To verify the effectiveness of the method, the analysis is also performed on randomly shuffled series. And the calculated DCCA exponents for shuffled series fluctuate slightly around 0.5. This conforms that there indeed exist cross correlations among original series.
C. Dongwei et al. / Physica A 517 (2019) 290–296
295
Fig. 3. The detrended cross correlation analysis between CO2 and PM2.5 series. The determination coefficients of linear regression for original series and randomly shuffled series are 0.983 and 0.969 respectively, which shows clear linear relationship between ln (v) and ln (F (v)).
Fig. 4. The detrended cross correlation analysis between CO2 and H2 O series. The determination coefficients of linear regression for original series and randomly shuffled series are 0.980 and 0.990 respectively, which shows clear linear relationship between ln (v) and ln (F (v)).
Acknowledgments This research is supported by the National Key R&D Program of China (Grant No 2016YFC0208802) and the National Natural Science Foundation of China (Grant Nos 11472272, 41605010 and 41675012). References [1] J. Beran, Long-range dependence, in: Wiley Interdisciplinary Reviews Computational Statistics, vol. 2, 2010, pp. 26–35. [2] J. Beran, Y. Feng, S. Ghosh, R. Kulik, Long-Memory Processes, Wiley Statsref Statistics Reference Online, 2013, pp. 107–208. [3] B. Podobnik, H.E. Stanley, Detrended cross-correlation analysis: a new method for analyzing two nonstationary time series, Phys. Rev. Lett. 100 (2008) 084102. [4] A.S.S. Paiva, M.A.R. Castro, R.F.S. Andrade, DCCA analysis of renewable and conventional energy prices, Physica A 490 (2018) 1408–1414. [5] T. Ikeda, A detrended cross correlation analysis for stock markets of the united states japan,and the europe, Physica A 484 (2017) 194–198. [6] Igor Gvozdanovic, Boris Podobnik, Duan Wang, H.Eugene Stanley, 1/f behavior in cross-correlations between absolute returns in a US market, Physica A 391 (2012) 2860–2866.
296
C. Dongwei et al. / Physica A 517 (2019) 290–296
[7] D. Pal, S.K. Mitra, Interdependence between crude oil and world food prices: a detrended cross correlation analysis, Physica A 492 (2018) 1032–1044. [8] Boris. Podobnik, Davor. Horvatic, Alexander. Petersen, H. Eugene Stanley, Cross-Correlations between Volume Change and Price Change, in: Proceedings of the National Academy of Sciences of the USA, 106, 2009, pp. 22079–22084. [9] A. Machado, S. Filho, Detrended Cross-Correlation Analysis: Measuring cross-correlation between mathematical constants, in: XXXII Encontro Nacional de Física da Matéeria Condensada, 2009.. [10] Xi-Yuan. Qian, Ya-Min. Liu, Zhi-Qiang. Jiang, Boris. Podobnik, Wei-Xing. Zhou, H. Eugene Stanley, Eugene stanley detrended partial cross-correlation analysis of two time series influenced by common external forces, Phys. Rev. E 91 (2015) 062816. [11] Davor. Horvatic, H.Eugene Stanley, Boris. Podobnik, Detrended cross-correlation analysis for non-stationary time series with periodic trends, Europhys. Lett. 94 (2011) 18007. [12] B. Podobnik, I. Grosse, D. Horvatic, S. Ilic, P. Ch. Ivanov, H.E. Stanley, Quantifying cross-correlations using local and global detrending approaches, Eur. Phys. J. B 71 (2009) 243–250. [13] K. Fraedrich, R. Blender, Scaling of atmosphere and ocean temperature correlations in observations and climate models, Phys. Rev. Lett. 90 (2003) 108501. [14] H.D. He, Multifractal analysis of interactive patterns between meteorological factors and pollutants in urban and rural areas, Atmos. Environ. 149 (2017) 47–54. [15] D.D. Kang, I.L. Dong, K. Kim, G. Lim, D.H. Ha, Dynamical mechanism in meteorological factors using detrended cross-correlation analysis, J. Korean Phys. Soc. 65 (2014) 577–590. [16] H.D. He, W. Pan, W.Z. Lu, Y. Xue, G.H. Peng, Multifractal property and long-range cross-correlation behavior of particulate matters at urban traffic intersection in Shanghai, Stoch. Environ. Res. Risk Assess. 30 (2016) 1515–1525. [17] M.D.O. Santos, T. Stosic, B.D. Stosic, Long-term correlations in hourly wind speed records in pernambuco, brazil, Physica A 391 (2012) 1546–1552. [18] B.I. Shraiman, E.D. Siggia, Scalar turbulence, Nature 405 (2000) 639. [19] Y. Kang, D. Belušić, K. Smith-Miles, Detecting and classifying events in noisy time series, J. Atmos. Sci. 71 (2014) 1090–1104. [20] D. Belušić, L. Mahrt, Is geometry more universal than physics in atmospheric boundary layer flow? J. Geophys. Res. Atmospheres 117 (2012) 9115. [21] Z.P. Song, F. Hu, J.J. Xu, X.L. Cheng, Self-Similar characteristic for the ramp structures of wind speed, Atmospheric Ocean. Sci. Lett. 7 (2014) 320–323. [22] http://www.stateair.net/web/historical/1/1.html (accessed 26 Feb. 2018). [23] Guidance for using continuous monitors in PM2.5 monitoring networks. http://www.epa.gov/ttnamti1/files/ambient/pm25/r-98-012.pdf (accessed 30 June, 2018). [24] C.K. Peng, S. Havlin, H.E. Stanley, A.L. Goldberger, Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series, Chaos 5 (1995) 82. [25] E.B.S. Marinho, A.M.Y.R. Sousa, R.F.S. Andrade, Using detrended cross correlation analysis in geophysical data, Physica A 392 (2013) 2195–2201. [26] K. Shi, Detrended cross-correlation analysis of temperature, rainfall, pm 10 and ambient dioxins in hong kong, Atmos. Environ. 97 (2014) 130–135.