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$This is the author s version of a work that was submitted/accepted for publication in the following source: Anh, Vo, Lam, K, Leung, Y, & Yu, Zu-Guo (5) Multifractal Characterization of Hong Kong Air$

1 This is the author s version of a work that was submitted/accepted for publication in the following source: Anh, Vo, Lam, K, Leung, Y, & Yu, Zu-Guo (5) Multifractal Characterization of Hong Kong Air Quality Data. Environmetrics, 6(), pp. -. This file was downloaded from: Notice: Changes introduced as a result of publishing processes such as copy-editing and formatting may not be reflected in this document. For a definitive version of this work, please refer to the published source:

2 Multifractal characterisation of Hong Kong air quality data V.V. Anh, Y. Leung, K.C. Lam and Z.G.Yu Abstract This paper develops a theory for characterisation of air quality data based on their measure representation. The measures are shown to be random cascades generated by an innitely divisible distribution. This probability distribution is uniquely determined by the exponent K (q) q in the multifractal analysis of the cascade. The theory is applied to the SO NO and NO time series at seven locations of the Hong Kong Electric Co. monitoring network. The Gamma density function is demonstrated to give an excellent ttothek (q) curve ofeach time series. This precise characterisation therefore provides a needed tool for modelling pollution episodes as well as classication of the monitoring network. Introduction An air quality management scheme requires an understanding of the trends in monitoring data and patterns of high pollution episodes. The former problem (trend analysis) has been covered extensively in the literature (see, for example, Anh, Duc and Azzi [], Anh, Duc and Tieng [] for the correspondence between anthropogenic trends and the long-range dependence (LRD) component in air quality data). In a recent study, Anhet al. [4] considered the second problem, namely, modelling the intermittency of air quality data. Existing works on air pollution mainly paid attention to the second-order statistics of the time series (such as their covariance structure or spectral density). It is known in many recent studies that turbulent processes display multiple scaling (Meneveau and Sreenivasan [3], Frisch [6]) and a description of this behaviour requires Date: 8 October School of Mathematical Sciences, Queensland University of Technology, GPO Box 434, Brisbane QLD 4, Australia Department of Geography, Chinese University of Hong Kong, Shatin, NT, Hong Kong This work was partially supported by the ARC Large grant A47 and the HKRGC grant CUHK 437/97H

3 the consideration of their higher-order moments. A suitable framework for this description is the multifractal formalism. An example of multifractal models to represent air quality data was given in [4]. In this paper, we go one step further and provide a characterisation of these data based on their measure representation. This is given in the form of the probability density function of the measure. We rst show that the given measure is in fact a multiplicative cascade generated by an innitely divisible distribution. This probability distribution is uniquely determined by the exponent K (q) q in the multifractal analysis of the cascade. This theory will be detailed in the next section. We then apply the theory to the SO NO and NO time series at seven locations of the Hong Kong Electric Co. monitoring network. It will be seen that the Gamma density function provides an excellent t to the K (q) curve of each time series. This precise characterisation therefore provides a needed tool for the classication and optimisation of the monitoring network as well as the prediction of pollution episodes. Methodology Let " (t) be a positive stationary stochastic process on a bounded interval of R assumed to be the unit interval [ ] for convenience, with E"(t) =: The smoothing of " (t) at scale r>is dened as " r (t) = r For <r<u<v we consider the processes Z t+r= t;r= X r v (t) = " r (t) t [ ] : " v (t) " (s) ds: (.) Following Novikov [4], we assume the following scale invariance conditions: (i) The random variables X r u and X u v are independent (ii) The probability distribution of each random variable X u v depends only on the ratio u=v of the corresponding scales. These conditions imply the power-law form for the moments of the processes X u v if they exist. In fact, we may write u E (X u v (t)) q = g q q (.) v from condition (ii) for some function g which also depends on q. From the identity X r v (t) =X r u (t) X u v (t)

4 and condition (i) we get Since u is arbitrary, we then have g q r v g q r v = g q r u u g q : (.3) v r ;K(q) = (.4) v for some function K (q) withk () = : It follows that Writing Y for X r v we obtain K (q) = ln E (X r v (t)) q : ln (v=r) K (q) = E (Y q ln Y ) ln (v=r) E (Y q ) Since K (q) = (EY q ) E Y q (ln Y ) ; (E (Y q ln Y )) : ln (v=r) (EY q ) (E (Y q ln Y )) = E Y q= Y q= ln Y (EY q ) E Y q (ln Y ) (.5) by Schwarz's inequality and v=r > we get K (q) that is, K (q) is a convex function. It is noted that equality holds in (.5) only if K (q) is a linear function of q other than this, K (q) is a strictly convex function. For <q< we assume that K (q) < which reects the fact that, in this range, taking a qth-power necessarily reduces the singularity of X u v : Also, we assume that the probability density function of X u v is skewed in the positive direction. This yields that K (q) > for q>: These assumptions, in conjunction with the strict convexity of K (q) suggest the assumption that This implies that K () = : (.6) EX u v =for arbitrary <u<v: (.7) In this paper, we will consider smoothing at discrete scales r j = ;j+ j = 3 ::: Then the smoothed process at scale r j is X j (t) =" rj (t) = Z t+;j " (s) ds: (.8) ;j+ t;;j 3

5 Under the condition E"(t) = it is reasonable to assume that Then, at generation J X (t) = t [ ] : (.9) X J (t) = X (t) X (t) X (t) X (t) X (t) ::: X J (t) X J; (t) = X (t) X (t) X (t) X (t) ::: X J (t) X J; (t) : (.) Under the scale invariance conditions (i) and (ii), the random variables X j =X j; of (.) are independent and have the same probability distribution. Let W denote a generic member of this family. Note that EW = from (.7). Then (.) can be rewritten as X J (t) = X J; (t) X J (t) X J; (t) = W (t) W (t) :::W J (t) t [ ] : (.) In other words, X J (t) is a multiplicative cascade process (see Holley and Waymire [9], Gupta and Waymire [8]). Denote by J the sequence of random measures dened by the density X J (t) that is, J (dt) =X J (t) dt J = 3 ::: It can be checked that J a.s. has a weak* limit since for each bounded continuous function f on [ ] the sequence R [ ] fd J is an L -bounded martingale (see Holley and Waymire [9], Mandelbrot [], Kahane and Peyriere []). We denote the density corresponding to by X (t) : Then it is seen from (.8) that Summarising, we have established that X (t) =" (t) t [ ] : (.) The positive stationary process " (t) is the limit of a multiplicative cascade with generator W. We next want to characterise this random cascade. We rst note that, for j = 3 ::: from the positivity of" (t) : Thus, X j X j; = R t+ ;j t;;j R t+ ;(j;) " (s) ds t;;(j;) " (s) ds (.3) E X j X j;! q q : 4

6 This inequality together with (.4) imply K (q) q q : (.4) We then have X E! q ; q X j A X j; = X q= K(q) q = : In other words, the Carleman condition is satised (see Feller [5], p. 4). As a result, we get T he probability density function f W of the generator W is uniquely determined by the set fk (q) q = :::g : As recognised by Novikov [4], if the function K (q) has analytic continuation into the complex plane, then the characteristic function of ln W has the form (x) =E e ix W ;K(ix) ln = : (.5) Dene n (x) = = =n ;K(ix) for an arbitrary integer n: Then n is the characteristic function of the probability distribution corresponding to smoothing with scales =n ;j+ : Also, it holds that (x) =( n (x)) n : Thus (x) is innitely divisible (see Feller [5], p. 53) in other words, It is noted from (.3) that ; ln W ' (x) of positive random variables is given by ln W has an innitely divisible distribution. (.6) ' (x) =exp : The most general form for the characteristic function (Z ; e ixs s ) P (ds)+iax (.7) where a and P is a measure on the open interval ( ) such that R ( + s) ; P (ds) < (see Feller [5], p. 539). On the other hand, it follows from (.) and (.4) that the characteristic function of ; ln W is given by E e ;ix ln W = ix E (W ) ;ix Using q = ;ix and equating (.7) with (.8) then yields K (q) = ; a Z ; e ;qs q ; ln s = ix ;K(;ix) : (.8) 5 P (ds) ln : (.9)

7 As constrained by (.6), the following condition must be satised by the measure P (ds) : Z ; e ;s s P (ds) ln =; a : (.) ln Equations (.9) and (.) provide the most general form for the K (q) curve of the positive random process f" (t) t g : In practice, tting this K (q) curve to data requires a proper choice of the measure P (ds) : Novikov [4] suggests the use of the Gamma density function, that is, f (x) =Ax ; exp (;x=) (.) where P (dx) =f (x) dx and A are positive constants. From (.9), (.) and condition (.) we get 8 < q ; (q+); ; (+) K (q) = : ; ; 6= (.) q ; ln(q+) ln(+) =: where =; a= ln, and from (.) we have A = ln ; ;( ; ) ( ; ( +); ) ; : The form (.) will be used for data tting in this paper. It is seen from ((.) and (.4) that the data for the K (q) curve is provided by ln E (X K (q) = lim J) q (.3) J! ;J+ ; ln where it should be noted from (.) that X (t) =" (t) the given positive random process. Since each smoothed process X J may possess long-range dependence (see Anh et al. [3]), the ergodic theorem may not hold for these processes. As a result, the computation of E (X q J) as sample averages may not be suciently accurate. There is an alternative form of the ergodic theorem developed by Holley and Waymire [9] for random cascades which we now summarise. For random cascades with density " (t) limit measure branching number b and generator W dene M J (q) = X k J k q (.4) (q) = lim J! ln M J (q) J ln b (.5) b (q) =log b E (W q ) ; (q ; ) (.6) where the prime in (.4) indicates a sum over those subintervals J k meet the support of : of generation J which 6

8 Theorem (Holley and Waymire [9]) Assume that W > a for some a > and W < b with probability, and that E (W q ) = (EW q ) <b:then, with probability, (q) =; b (q) : (.7) In our case as developed above, b = and (.3) gives W. In fact the scale r j = ;j+ used in (.8) is arbitrary it can be b ;j+ and the inequality W b still holds by denition of the smoothing and the positivity of" (t) : In our development, Consequently, ;K (q) = lim J! ln E (X q J) ln ;J+ = lim J! J ln E (W q ) (J ; ) ln ; using (.) = ; ln E (W q ) : ln K (q) =; (q)+q ; : (.8) The above formula then provides a way to compute K (q) via (.5) and (.7) using sums of q-th powers of the limit measure instead of (.3) using expectations. In fact, the ergodic theorem takes the following form ln E (X q lim J) (J ; ) ln = lim J! J! ln P k J k q J ln + q ; : 3 Experimental results We will now apply the above theory to characterise the air quality data available from the Hong Kong Electric Co. monitoring network. These consist of seven SO series, three NO series and three NO series. The SO series, denoted QMH SO VPK SO VIC SO ABD SO ALC SO CHK SO and WFE SO record the average daily concentrations of sulphur dioxide at Queen Mary Hospital, Victoria Peak, Victoria Road, Aberdeen, Ap Lei Chan, Chung Hom Kok and Wah Fu Estate respectively. The NO series, denoted QMH NO VPK NO VIC NO and the NO series, denoted QMH NO VPK NO VIC NO give the average daily concentrations of nitrogen oxide and nitrogen dioxide at Queen Mary Hospital, Victoria Peak and Victoria Road respectively. The SO series cover the period (consisting of 365 observations), while the NO and NO series only extend from May 993 to end of 995 (consisting of 97 observations). As examples, QMH SO QMH NO QMH NO are plotted in Figures - 3 respectively. It is seen that all three series display intermittency, which is more pronounced in the SO series. This intermittency is quite distinct from the yearly cycle. 7

9 In our previous work (Anh et al. [4]), we established the long-range dependence and multifractality in these data via spectral and multifractal analyses. We now want to obtain the probability distributions of these data. We denote by K d (q) the value of K(q) computed from the data using its denition and dene JX error = (q j ; (q! j +) ; ; ( +) ; ; ) ; K d(q j ) : j= Then the values of and can be estimated through minimising error. In this minimisation, we assume : After obtaining the values of, and, we then get the estimated K(q) curve from (.). The data are assumed to be generated from a multiplicative cascade process (see (.) and (.)). It is known that the (q) function, hence the K (q) function, of these processes are differentiable (Lau and Ngai [, Corollary C]). We then used the subroutine NCONF/DNCONF of Chapter 8 of Imsl Math/Library, Vols and. This subroutine solves a general nonlinear optimisation problem using successive quadratic appriximation algorithm and a nite dierence gradient method. This subroutine is generally reliable and has a fast convergence property of exact derivative quasi-newton methods (Gill et al. [7, Sections 4.6 and 8.6], Seber and Will [5, p. 6]). The K d (q) curves for the SO series were shown in Figure 4 and those for the NO and NO series were shown in Figure 6. Since the relative position of the K(q) curve indicates the extent of intermittency in the data, it can be used to cluster the data series. It is seen that the SO activities can be grouped into three clusters: (VPK, CHK), (VIC, ABD, QMH, WFE) and (ALC). Also the activities of NO and NO are quite distinct (see Figure 6). It can be seen from Figure 4 that the intermittency of the pollution data is highest at VPK and CHK because they are furthest away from the pollution source and hence more aected by dispersion rather than by pollution source strength. The former is more variable than the latter. The ALC SO series exhibits the lowest intermittency because of its proximity to the source. The cluster comprising of VIC, ABD, QMH and WFE lies in between the two and hence exhibits intermediate intermittency. The dierence in NO and NO is related to their origins in the urban environment. NO is the primary pollutant from automobiles and high temperature combustion processes such as power generation. Once emitted, NO is gradually oxidized in the atmosphere and converted to NO and other nitrogen oxide compounds whose concentrations are hence less intermittent. Data tting based on the form (.) was performed on all the data series and shown in Table : We give the tting of SO series in Figure 5 and that of NO and NO series in Figure 7. It is clear that the form (.) gives a perfect t to the data. We also calculated the values of (q) using its denition (.5). We found that the values of K d (q) coincide with those obtained from (.8). Hence we indeed can use (.8) to calculate K(q). 8

10 4 Conclusions This paper presents a theoretical model for the probability distribution of air quality data. The method is applied to the data series of SO NO and NO at seven locations in Hong Kong. The accuracy of the models in tting these data indicates that the probability distribution, particularly the resulting K (q) curve, provides a valuable tool for their characterisation and prediction. It can also be used for clustering the monitoring network. References [] V. V. Anh, H. Duc, and M. Azzi. Modelling anthropgenic trends in air quality data. J. Air & Waste Management Association, 47():66{7, 997. [] V. V. Anh, H. Duc, and Q. Tieng. Modelling persistence and intermittency in air pollution. In H. Power, T. Tirabassi, and C. A. Brebbia, editors, Air Pollution V: Modelling, Monitoring and Management, pages 443{45. Computational Mechanics Publications, Southampton, 997. [3] V. V. Anh, C. C. Heyde, and Q. Tieng. Stochastic models for fractal processes. Journal of Statistical Planning and Inference, 8(/):3{35, 999. [4] V. V. Anh, K. C. Lam, Y. Leung, and Q. Tieng. Multifractal analysis of Hong Kong air quality data. Environmetrics, :39{49, 999. [5] W. Feller. An Introduction to Probability Theory and its Applications, volume II. Wiley, New York, 97. [6] U. Frisch. Turbulence. Cambridge University Press, 995. [7] P. E. Gill, W. Murray, and M. H. Wright. Practical Optimization. Academic Press, New York, 98. [8] V. K. Gupta and E. C. Waymire. A statistical analysis of mesoscale rainfall as a random cascade. Journal of Applied Meteorology, 3:5{67, 993. [9] R. Holley and E. C. Waymire. Multifractal dimensions and scaling exponents for strongly bounded random cascades. The Annals of Applied Probability, (4):89{845, 99. [] J.-P. Kahane and J. Peyriere. Sur certaines martingales de benoit mandelbrot. Advances in Mathematics, :3{45, 976. [] K.-S. Lau and S.-M. Ngai. Multifractal measures and a weak separation condition. Advances in Mathematics, 4:45{96, 999. [] B. B. Mandelbrot. Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mech., 6:33{358, 974. [3] C. Meneveau and K. R. Sreenivasan. The multifractal nature of turbulent energy dissipation. J. Fluid Mech., 4:49{484, 99. [4] E. A. Novikov. Innitely divisible distributions in turbulence. Physical Review E, 5(5), 994. [5] G. A. F. Seber and C. J. Wild. Nonlinear Regression. John Wiley & Sons, New York,

11 Table : The values of quantities,, and error of all organisms selected. pollution error ABD SO E-4 ALC SO E-4 CHK SO E-4 QMH SO E-4 VIC SO E-3 VPK SO E-3 WFE SO E-3 QMH NO E-4 QMH NO E-3 VIC NO E-4 VIC NO E-3 VPK NO E-3 VPK NO E QMH SO Figure : Maximum daily concentrations of SO (parts per billion) at Queen Mary Hospital.

12 9 QMH NO Figure : Maximum daily concentrations of NO (parts per billion) at Queen Mary Hospital. QMH NO Figure 3: Maximum daily concentrations of NO (parts per billion) at Queen Mary Hospital.

13 .5 ABD SO ALC SO CHK SO.5 QMH SO VIC SO K(q) VPK SO WFE SO q Figure 4: The K (q) curves of seven SO series..5 ABD SO ALC SO CHK SO WFE SO.5 K(q) q Figure 5: Fitting of the K (q) curves of SO at the sites ABD, ALC, CHK and WFE.

14 .5.5 QMH NO QMH NO VIC NO VIC NO VPK NO VPK NO K(q) q Figure 6: The K (q) curves of three NO series and three NO series..5.5 QMH NO QMH NO VIC NO VIC NO VPK NO VPK NO K(q) q Figure 7: Fitting of the K (q) curves of three NO series and three NO series. 3

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