n this paper we apply the alpha Model to air quality prediction purposes. This model is used to predict the ground level ozone concentration which is influenced by multiple pollutants such as NO2, CO, Pb and particulate matter. The model was previously elaborated and successfully tested on other complex systems. Our main object here is to show that the alpha model is a powerful technique for analyzing performance complexities of implementing input-output relations. Air quality is taken as an example

Boutabba, N.

El Aoun, C.

Eleuch, H.

English

Arab Journals Platform

Appl. Math. Inf. Sci.9, No. 1, 27-30 (2015) 27Applied Mathematics & Information SciencesAn International Journalhttp://dx.doi.org/10.12785/amis/090104Alpha Model: A Mathematical Modeling ApproachApplied to an Air Quality Monitoring NetworkC. El Aoun1, N. Boutabba2,3 and H. Eleuch4,∗1 Institut des Hautes Etudes Comerciales de Sfax, B.P 43 - 3061Sfax, Tunisia2 Physics Department, Cegep St-Laurent, Montreal, Qc, H4L 3X7, Canada3 University of Carthage, Tunis 1040, Tunisia4 Department of Physics, McGill University, Montreal, Qc, H3A 2T8, CanadaReceived: 12 Jun. 2014, Revised: 30 Jun. 2014, Accepted: 1 Aug. 2014Published online: 1 Jan. 2015Abstract: In this paper we apply the alpha Model to air quality prediction purposes. This model is used to predict the ground levelozone concentration which is influenced by multiple pollutants such as NO2, CO, Pb and particulate matter. The model was previouslyelaborated and successfully tested on other complex systems. Our main object here is to show that the alpha model is a powerfultechnique for analyzing performance complexities of implementing input-output relations. Air quality is taken as an example.Keywords: Air quality prediction, model, alpha model.1 IntroductionOur modern society is data oriented. The major problemsin our modern life are related to predictions: prediction offinancial stock markets, volcanic eruptions, tsunamis,epidemiological widespread diseases, political stability,economical growth and financial crisis etc. Thesephenomena have extreme complicated dependence on bignumber of observable and non observable parameters.Concrete scientific models based on fundamental lawseven combined with heavy computations fail to describesuch phenomena. Several empirical methods based onhistorical data and time series have been developed forexample neuronal networks [1,2,3] and geneticalgorithms[4]. These techniques are often considered asblack-box models where explicit relations between theinput and the output data can not be seen. Recently, a newmethod, the alpha model has been developed by two of us[5]. The alpha model gives explicit relations between theinputs and the outputs data based on historical or timeseries data. It was applied to forecast of several financialstock markets and diffuse solar irradiance [5,6]. Here weextend the application sets of the alpha model to studyand predict the air pollution. Our main goal is to show theprediction power of this technique.Air pollution occurs when unwanted chemicals orother contaminants are released into the air in largeenough amounts to harm the health of people, plants,animals, and our environment. Air quality degradation isone of the important environmental hazards and a lot ofresearch is therefore conducted in this field. It is the resultof our modern life, in fact one of the main causes of airpollution is manufacturing which produces gases orvapors, dust, smoke or soot. However, Some otherpollutants are emitted from natural processes such asforest fires, decaying vegetation, dust storms, andvolcanic eruptions. The major pollutants are particles ofdiameter 10µm or less (PM10), very small particles ofdiameter 2.5µm or less (PM2.5), nitrogen dioxide (NO2),carbon monoxide (CO) and ozone (O3). Moreover, solarradiation, wind speed, opaque cloud cover and ozoneitself contribute considerably in the ozone formation[7].Several authors have developed models based on neuralnetworks to forecast air quality and air pollutantconcentration [8,9,10]. They are important tools and theyplay a crucial role in protecting air quality. The variousmodels used to predict air quality and to analyze travelroad strategies attempt to minimize possible negativeimpacts of pollution on the environment[11,12,13]. Inthis work, we focus on the analysis of two data sets∗ Corresponding author e-mail:heleuch@physics.mcgill.cac© 2015 NSPNatural Sciences Publishing Cor.28 C. El Aoun et. al. : Alpha Model: A Mathematical Modeling Approach...described below using the alpha model [5,6] to predictthe ozone formation.The analysis part of the modeling did lead to clearresults regarding air quality prediction in the area wherethe data have been collected.2 Method and materials2.1 Data setsTwo data sets were used. The first one is (856 points)describing the daily ozone concentration in Houstonsummer (April-October) along the Gulf coast during theyears 1982-1985 (in ppm). The meteorologic variablesconsidered as predictors are the solar radiation( inW/m2), the wind speed (in m/s), the opaque cloud cover(in percentage) and the ozone of a previous day. Moreinformation about this data can be found in theGeophysical Statistics Project, through NCAR.The second data set are daily max 8-hours ozoneformation, hourly pollutant and meteorological averages(for the first 173 days of 2012) in Los Angeles,California. These data have been taken from the USenvironmental protection agency web site. Pollutants arecarbon monoxide (CO) (in parts per million unit (ppm)),Ozone (in parts per million unit (ppm)), Nitrogen dioxide(in parts per billion ppb), PM10 and PM2.5 (inµg/m3)and Pb (inµg/m3).2.2 Alpha ModelSeveral Models in finance and economics are inspiredfrom physical concepts such as kinetic exchange modelsof markets [14], quantum finance [15,16], correlationfunction approach [17] and alpha model [6]. Kineticexchange models of markets apply the kinetic theory ofgases to the exchange markets. Quantum finance uses thequantum mechanical formalism to analyze the optionstock markets. We have recently adapted the correlationanalysis [17] (widely used in electronics [18], plasmaphysics[19], astrophysics[20], atomic physics [21,22],statistical mechanics [23], condensed matter physics [24,25,26,27] and telecommunication[28]) to identify thebusiness cycle turning points.The alpha model was previously used for financeapplications in the context of e-commerce negotiation andstock exchange forecast [5]. It was also proposed as amodel for prediction of hourly solar irradiance values [6].In the present paper, we investigate the alpha model forair quality prediction. This model was developed basedon analogy between the physics (Coulomb’s) law andbuyer-seller relation in finance. It describes the attractionbetween buyer and seller analogous to the attractionbetween two opposite charges. The case of seller/seller orbuyer/buyer is considered as two similar charges pushingeach other. The alpha model involves an explicit relationbetween the observable input and output variables withset of parameters adjusted from historical or time seriesdata. As described in our previous work in financeapplications[5], the parameterα represents the ratiobetween the final price and the starting price. It dependson the mean observable variables that influence the finalprice of the auction. The alpha model is a methodaided-decision for selling or buying auctions. In fact, theprediction of the final price is possible if we have haveenough information onα. In this work, we apply thealpha model for the two sets of data described above. Inthis section we are focusing on two goals. The first goal isto identify which predictors in this complex system havemore impact and influence on the ozone concentration.We need then to study the relation between the inputparameters (predictors) and the output variable (ozoneconcentration) at the same day. The second goal isconcentrated on predicting the output variable in dayj +1 from the input parameters at previous dayj. Let usfirst present the alpha model for exploring the dominantpollution factors of the ozone concentration for the twodata sets described in the previous section. The variablepollutants taken into account in the area of Houston (firstdata set) are the opaque cloud cover (opcov), the windspeed (WSPD) and the solar radiation (SR) at the sameday j+1 and the ozone (O3) of the previous dayj. In thiscase the expression of the alpha model can be written as[5,6]αHoustonj+1 =(O3j)2β1[r1+ r01]2 , (1)withr1 =(opcov j+1)β2 (WSPD j+1)β3 (SR j+1)β4. (2)For the second set the input variables of the alphamodel are the pollutants at timej +1 (CO, PM10,NO2,PM2.5, Pb) andO3 at time j while the output is theground level ozone concentration at timej + 1. In thiscase the expression of the alpha model can be written as[5,6]αLos−Angelesj+1 =(O3j)2β5[r2+ r02]2 , (3)withr2 =(CO j+1)β6 (PM10j+1)β7 (NO2j+1)β8×(PM2.5j+1)β9 (Pb j+1)β10. (4)Whereβi andr0 j are the parameters to be calculatedfrom the historical data.r0 j takes into account the residualweak influence of all other variables that are notconsidered here.α j+1 represents the predicted level of thec© 2015 NSPNatural Sciences Publishing Cor.Appl. Math. Inf. Sci.9, No. 1, 27-30 (2015) /www.naturalspublishing.com/Journals.asp 29Table 1: The parameter values of the simulation for Houstonβ1 β2 β3−0.24606 −0.0903078 −0.301688β4 MSE−0.00202147 0.00061273ground ozone concentrationO3j+1 divided by the valueof the ozone concentration a day beforeO3jα j+1 =O3j+1O3j. (5)If α > 1, then the ozone concentration will increase. Forα < 1 the ozone concentration will decrease.For using the alpha model to predict the ozoneconcentration, the two formulas (1) and (2) will beslightly modified. We use in this case the values of thepollutants of the dayj in order to predict the ozoneconcentration in the next dayj + 1, so in this case thealpha model for Houston is then [5,6]αHoustonj+1 =(O3j)2β11[r3+ r03]2 , (6)withr3 = (opcov j)β12 (WSPD j)β13 (SR j)β14, (7)and for Los-Angeles is given by [5,6]αLos−Angelesj+1 =(O3j)2β15[r4+ r04]2 , (8)wherer4 = (CO j)β16 (PM10j)β17 (NO2j)β18×(PM2.5j)β19 (Pb j)β20. (9)3 Results and discussionFor the first simulation, we consider the first set ofvariables described above. 756 points are used to estimatethe fitting parameters of the alpha model while the otherset of 100 points is used to validate the model and topredict the ozone formation. The result of this simulationis plotted in Fig.1 where the daily ozone concentration isplotted during the last 100 days of summer 1985(July24-October31) in Houston. The values of the fittingparameters of the alpha model are given in table 1. WhereMSE represents the mean squared error. These parametersshow that the ozone concentration in Houston during1986 was less sensitive to opaque cloud and very sensitiveto the wind speed and highly sensitive to the previous dayozone concentration. We can deduce that the ozoneconcentration in Houston is approximatively proportionalTable 2: The parameter values of the simulation for Los Anglosβ5 β6 β7 β8−0.0335725 −0.194444 −0.0623862 0.0433624β9 β10 MSE−0.0177553 0.14281 (8.17)10−5to wind speed in power of 0.6 (WSPD0.6) and to thesquare root of the ozone concentration of the previousday. The second simulation is calculated for theconcentration of ozone in Los Angeles during the first173 days of 2012. 142 points are used to estimate thefitting parameters of the alpha-model. 31 daily measuresof the ozone concentration (May 22- June 21, 2012) areused to validate the alpha-model prediction. Thecalculated values of the fitting parameters for thissimulation using the second set of the observablevariables are given in table 2.As β9 has the smallest value andβ6 has the highestvalue, the ozone concentration in Los Angeles is lesssensitive to thePM2.5 andNO2 (nitrogen dioxide) andhighly sensitive toCO andPb as well as the previous dayozone concentration. As a good approximation, the ozoneconcentration is proportional to the previous day ozoneconcentration, toCO1/3 and toPb−1/4.Fig. 1: Ozone daily prediction during 100 days of 1985 (July24-October31) in Houston.Let us now analyze the predictability of the ozoneconcentration using the alpha model. Fig.1 and Fig.2show good fits between the predicted values withalpha-model and the real values. So, the alpha-model hassmall error for predicting the ozone concentration. Forthese simulations we obtain the following values of theβparameters (see table 3 and 4).4 ConclusionIn this paper, we have shown that the alpha-model can beapplied to air quality prediction which enlarges the fieldsc© 2015 NSPNatural Sciences Publishing Cor.30 C. El Aoun et. al. : Alpha Model: A Mathematical Modeling Approach...Fig. 2: Daily ozone prediction during one month (May 22- June21, 2012) in Los Angeles.Table 3: The parameter values of the ozone concentrationforecast for Houstonβ11 β12 β13−0.166322 −0.05572 −0.198179β14 MSE0.01568 0.00101191Table 4: The parameter values of the ozone concentrationforecast for Los Angelesβ15 β16 β17 β18−0.0175861 −0.045441 −0.029776 −0.0615832β19 β20 MSE0.0428351 −0.167781 (9.34281)10−5of applications for this model. The advantage of thismodel compared to the input-output methods such asneuronal networks or genetic algorithms is the explicitanalytical relation between the input and the outputvariables that can be established from historical data. Thisallows a better understanding of the dynamical behaviorof the studied systems. It also identifies the most influentand the less influent factors on the dynamical system.Furthermore it opens the door for more general andexplicit relations between the input and the outputvariables in particular systems. For example in this studywe have deduced that, on one hand, that in Los Angelesthe CO and Pb pollution factors are the most influentparameters, as well as the ozone concentration of theprevious day, on the air quality. On the other hand, theozone concentration in Houston is mostly determined bythe wind speed and the ozone concentration of theprevious day. Quantitatively speaking, the ozoneconcentration in Houston is approximatively proportionalto the wind speed power 0.6 and to the square root of theozone concentration of the previous day.AcknowledgementsNB and HE would like to thank Prof. G. Brassard andProf. E. Aimer (University of Montreal) for their fruitfulcomments.References[1] G. E. Hinton, R. R. Salakhutdinov Science5786, 313(2006).[2] D. Hansel and H. Sompolinsky Phys. Rev. Lett.68, 718(1992).[3] A.T.C. Goh, Artif. Intell. Eng.9, 3 (1995).[4] Mitchell Melanie, An Introduction to Genetic Algorithms,A Bradford Book, MIT Press, Cambridge, 1999.[5] Chokri El Aoun, Hichem Eleuch, Esma Aı̈meur, Hella BenAyed, and Farouk Kamoun, Journal of Decision Systems16,393 (2007).[6] A. Mellit, H. Eleuch, M. Benghanem, C. Elaoun, A. MassiPavan Energy Conversion and Management51, 771 (2010).[7] M. Christian, Atmospheric Environment35, 1 (2001).[8] K. Atakan , A. B. Oktay, Expert Systems with Applications37, 7986 (2010).[9] W. G. 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Alpha Model: A Mathematical Modeling Approach Applied to an Air Quality Monitoring Network

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Alpha Model: A Mathematical Modeling Approach Applied to an Air Quality Monitoring Network

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