Statistique de la pollution de l'air : méthode mathématiques : applications au cas de la région parisienne
Thèses d'Orsay, no. 538 (1999) , 238 p.

As it occurs in all great cities, Paris has a serious photochemical ozone air pollution problem. Our work is inserted in the project on forecasting ozone episodes in the Paris area realised in coorperation with AIRPARIF, the Paris area air pollution agency.

In the first part, we biefly present the ozone formation phenomena.

In the second part, we analyse the influence of wind on pollution repartition; to achieve this aim, classical data analysis (Principal Component Analysis, Procustean Analysis, Multidimensional Scaling) and linear model were used.

In the field of air pollution control, the rare event is often more significance than the common one. This is evidenced by the content of air quality standards which define acceptable upper limits of air pollution concentrations. The purpose of the third part is to establish whether observed trends in the data of tropospheric ozone are real, meaning that they could be attributed to actual changes in the emissions of toxic gases into atmosphere, or whether they are the result of meteorological changes affecting the conditions under which ozone is generated. To investigate this question, we construct a regression model in which the level of ozone is represented as a function of both meteorological variables and time, in order to determine the significance of the time component when the meteorological variables are taken into account. Also, we propose to use a logistic regression to model the probability of an exceedance of a high threshold level every day, in accounting for the relationship between very high values of ozone and meteorological conditions.Then, we apply the results of the extreme value theory to model the point process consisting of the times and the sizes of high-level exceedances by a non-homogeneous Poisson process. We apply the method to data from the Paris and Los Angeles areas.

In the fourth one, we demonstrate the convergence to a Compound Poisson process of a high-level exceedances point process N n ( B ) = j n B 1 X j , > U n , where X n = φ ( ξ n , Y n ) , φ , a (regular) regression function, u n grows to infinity with n in a suitable way, ξ and Y are mutually independent, ξ is stationary and weakly dependent, and Y is non-stationary, satisfying some ergodicconditions. The basic technique is the study of high-level exceedances of stationary process over suitable collections of random sets.

Mots clés : Principal Component Analysis, Procustean analysis, Multidimensional Scaling, linear model, exceedances, point processes, convergence, logistic regression, diagnostic model testing, generalized Pareto distributions, meteorological conditions, non-homogeneous Poisson process, Bootstrap method, Compound Poisson process, level sets, mean occupation measures, asymptotically ponderable collections of sets.
@phdthesis{BJHTUP11_1999__0538__A1_0,
     author = {Bellanger, Lise},
     title = {Statistique de la pollution de l'air : m\'ethode math\'ematiques : applications au cas de la r\'egion parisienne},
     series = {Th\`eses d'Orsay},
     publisher = {Universit\'e de Paris-Sud U.F.R. Scientifique d'Orsay},
     number = {538},
     year = {1999},
     language = {fr},
     url = {http://archive.numdam.org/item/BJHTUP11_1999__0538__A1_0/}
}
TY  - BOOK
AU  - Bellanger, Lise
TI  - Statistique de la pollution de l'air : méthode mathématiques : applications au cas de la région parisienne
T3  - Thèses d'Orsay
PY  - 1999
IS  - 538
PB  - Université de Paris-Sud U.F.R. Scientifique d'Orsay
UR  - http://archive.numdam.org/item/BJHTUP11_1999__0538__A1_0/
LA  - fr
ID  - BJHTUP11_1999__0538__A1_0
ER  - 
%0 Book
%A Bellanger, Lise
%T Statistique de la pollution de l'air : méthode mathématiques : applications au cas de la région parisienne
%S Thèses d'Orsay
%D 1999
%N 538
%I Université de Paris-Sud U.F.R. Scientifique d'Orsay
%U http://archive.numdam.org/item/BJHTUP11_1999__0538__A1_0/
%G fr
%F BJHTUP11_1999__0538__A1_0
Bellanger, Lise. Statistique de la pollution de l'air : méthode mathématiques : applications au cas de la région parisienne. Thèses d'Orsay, no. 538 (1999), 238 p. http://numdam.org/item/BJHTUP11_1999__0538__A1_0/

[Alpuim, Catkan & Hüsler (1995)] Extremes and clustering of nonstationary max-AR(1) sequences. Stock. Proc. Appl. 56, 171-184. | MR | Zbl | DOI

[Barbour, Holst & Janson (1992)] Poisson approximation. Oxford Studies in Probability 2. Oxford University Press. | Zbl | MR

[Berman (1992)] Sojourns and Extremes of Stochastic Processes. Wadsworth & Brooks. | MR | Zbl

[Bradley (1986)] Basic properties of strong mixing conditions. Dependence in Probability and Statistics: A Survey of Recent Results (editors: Ebberlein, E. and Taqqu, M). Birhaüser, 165-192. | MR | Zbl | DOI

[Brown & Xia (1995)] On Stein-Chein factors for Poisson approximation. Statist. Probab. Letters 23, 327-332. | MR | Zbl | DOI

[Bryc & Smolenski(1993)] Moment Conditions for almost sure convergence of weakly correlated random variables. Proc. Am. Math. Soc. 119, No. 2, 355-373. | MR | Zbl | DOI

[Cohen (1989)] On the Compound Poisson Limit Theorem for High Level Exceedances. J. Appl. Prob. 26, 458-465. | MR | Zbl | DOI

[Cox & Isham (1992)] Point Processes. Monographs on Apllied Probability and Statistics 12. Chapman & Hall. | MR | Zbl

[Davydov (1973)] Mixing conditions for Markov Chains. Theory Probab. Appl. 18, No. 2, 312-328. | Zbl | DOI

[Doukhan, P. (1995)] Mixing: Properties and Examples. Lectures Notes in Statistics 85, Springer Verlag. | MR | Zbl

[Doukhan & Louichi (1996)] Weak dependence and moment inequalities. Prepublications Mathématiques d'Orsay 97.08.

[Dziubdziela (1988)] A Compound Poisson Limit Theorem for Stationary Mixing Sequences. Rev. Roumaine Math. Pures Appl. 33, 1-2, 39-45. | MR | Zbl

[Dziubdziela (1995)] On the limit distribution of the sums of mixing Bernoulli random variables Statist. Probab. Letters 23, 179-182. | MR | Zbl | DOI

[Falk, Hüsler & Reiss (1994)] Laws of Small Numbers: Extremes and Rare Events. DMV Seminar 23, Birkhäuser-Verlag. | MR | Zbl

[Ferreira (1993)] Joint exceedances of high levels under a local dependence condition. J. Appl. Prob. 30 112-120. | MR | Zbl | DOI

[Guyon (1995)] Random Fields on a Network. Modeling, Statistics and Applications. Probability and its Applications, Springer-Verlag. | MR | Zbl

[Hudson, Tucker & Veeh (1989)] Limit distributions of sums of m-dependent Bernoulli random variables. Probab. Theory Related Fields 82, 9-17. | MR | Zbl | DOI

[Hüsler (1993)] A Note on Exceedances and rare events of non-stationary sequences. J. Appl. Prob. 30, 877-888. | MR | Zbl | DOI

[Hsiau (1997)] Compound Poisson Limit Theorems for Markov Chains. J. Appl. Prob. 34, 24-34. | MR | Zbl | DOI

[Hsing, Hüsler & Leadbetter (1988)] On the Exceedance Point Process for a stationary sequence. Probab. Th. Rel. Fields 78, 97-112. | MR | Zbl | DOI

[Kallenberg (1983)] Random Measures, 3rd. edition. Academic Press. | MR | Zbl

[Leadbetter, Lindgren & Rootzén (1983)] Extremes and Related Properties of Random Sequences and Processes. Spinger Series in Statistics, Springer-Verlag. | MR | Zbl | DOI

[Leadbetter & Rootzén (1988)] Extremal Theory for Stochastic Processes. Ann. Probab. 16, No. 2, 431-478. | MR | Zbl | DOI

[Leadbetter & Nandagopalan (1989)] On Exceedance Point Processes for Stationary Sequences under Mild Oscilation Restrictions. Extreme Value Theory, Proc. Conf Oberwolfach, 1987. Lectures Notes in Statistics 51, 69-80. | MR | Zbl | DOI

[Leadbetter & Hsing (1990)] Limit theorems for strongly mixing stationary random measures. Stoch. Proc. Appl. 36, 231-243. | MR | Zbl | DOI

[Leadbetter (1991)] On a basis for a "Peak over Treshold" modelling. Statist. Probab. Letters 12, 357-362. | MR | Zbl | DOI

[Leadbetter & Rootzén (1993)] On Central Limit Theory for Families of Strongly Mixing Additive Random Functions. Stochastic Processes: A Festschrift in Honour of Gopinath Kallianpur (editors: S. Cambanis, J.K. Ghosh, R.L. Karandikar, P.K. Sen), Springer-Verlag. | MR | Zbl | DOI

[Leadbetter (1995)] On high level exceedance modeling and tail inference. J. Statist. Plann. Inference 45, 247-260. | MR | Zbl | DOI

[Newman (1980)] Normal Fluctuations and the FKG inequalities. Commun. Math. Phys. 74, 119-128. | MR | Zbl | DOI

[Perera (1994)] Spatial statistics, central limit theorems for mixing random fields and the geometry of d C.R. Acad. Sci. Paris t. 319, Série I, 1083-1088. | MR | Zbl

[Perera (1997)a] Geometry of d and the Central Limit Theorem for weakly dependent random fields. J. Theoret. Probab. Vol. 10, No. 3, 581-603. | MR | Zbl | DOI

[Perera (1997)b] Applications of Central Limit Theorems over asymptotically measurable sets: regression models. C. R. Acad. Sci. Paris t. 324, Série I, p. 1275-1280. | MR | Zbl

[Rényi (1951)] On composed Poisson distribution II. Acta Math. Acad. Sci. Hungar. 2, 83-98. | MR | Zbl | DOI

[Resnick (1987)] Extreme Values, Regular Variation, and Point Processes. Applied Probability Series 4, Springer-Verlag. | MR | Zbl

[Roussas (1994)] Asymptotic Normality of Random Fields of Positively of Negatively Associated Processes. J. Multivariate Anal. 50, 152-173. | MR | Zbl | DOI

[Smith & Shively (1994)] A Point Process Approach to Modelling trends in Tropospheric Ozone Based On Exceedances of a High Treshold. Technical Report 16, National Institute of Statistical Sciences.

[Volkonskii & Rozanov (1959)] Some limit theorems for random functions, I. Theory Probab. Appl. 4, 178-197. | MR | Zbl | DOI

[Volkonskii & Rozanov (1961)] Some limit theorems for random functions, I. Theory Probab. Appl. 6, 186-198. | Zbl | DOI

[Wschebor (1985)] Surfaces aléatoires: mesure géométrique des ensembles de niveau. Lecture Notes in Mathematics 1147, Springer-Verlag. | Zbl | MR

[1] Académie Des Sciences. Ozone et propriétés oxydantes de la troposphère : Essai d'évaluation scientifique. Technique & Documentation-Lavoisier, Paris, Octobre 1993. Rapport n 30.

[2] Airparif. Surveillance de la qualité de l'Air en Ile-de-France : Les résultats 1997. 1996.

[3] Azais J.-M. Analyse de variance non orthogonale, l'exemple de SAS/GLM. Revue de Statistique Appliquée, XLII(2): 27-41, 1994.

[4] Bel L., Bellanger L., Bobbia M., Ciuperca G., Dacunha-Castelle D., Gilibert E., Jackubowicz P., Oppenheim G., Tomassone R. On forecasting ozone episodes in the Paris area. Listy Biometryczne- Biometrical Letters, 35(1), 1998.

[5] Bel L., Bellanger L., Bonneau V., Ciuperca G., Dacunha-Castelle D., Deniau C., Ghattas B., Misiti M., Misiti Y., Oppenheim G., Poggi J.M., Tomassone R. Eléments de comparaison de prévisions statistiques des pics d'ozone. Revue de Statistique Appliquée, 1998. à paraître.

[6] Bel L., Bellanger L., Bonneau V., Ciuperca G., Dacunha-Castelle D., Deniau C., Ghattas B., Misiti M., Misiti Y., Oppenheim G., Poggi J.M., Tomassone R. Prévision des pointes de pollution dans la région parisienne. Technical report, Laboratoire Modélisation Stochastique et Statistique, Université de Paris-Sud, Orsay, 1997.

[7] Bellanger L., Tomassone R. Wind direction and maximum pollutants concentration : a case-study. In Blasco ed., editor, Advances in Environmental and Ecological Modelling. Kluwer, Amsterdam, 1998. à paraître.

[8] Bloomfield P., Royle A., Yang Q. Accounting for meteorological effects in measuring urban ozone levels and trends. Technical Report 1, National Institute of Statistical Sciences, P.O. Box 14162, Research Triangle Park, N.C. 27709, 1993.

[9] Bloomfield P., Royle A., Yang Q. Rural ozone and meteorology: Analysis and comparison with urban ozone. Technical Report 5, National Institute of Statistical Sciences, P.O. Box 14162, Research Triangle Park, N.C. 27709, 1993.

[10] Brion D., Gilibert E. Forecasting atmospheric pollution peaks over the Ile-de-France, à paraître, 1996.

[11] Carlier P., Mouvier G. Initiation à la physico-chimie de la basse troposphère. Pollution Atmosphérique, Janvier-Mars: 12-24, 1988.

[12] Cox D. R., Lewis P. A. The Statistical Analysis of Series of Events. John Wiley, New York, 1966. | MR | Zbl

[13] Cox W. M., Shao-Hang C. Meteorologically adjusted ozone trends in urban areas: A probabilistic approach. Atmospheric Environment, 27B(4): 425-434, 1993.

[14] Crow L.H. Reliability analysis for complex repairable systems. In F. Proschan and R.J. Serfling (eds), editors, Reliability and Biometry SIAM, pages 379-410. Philadelphia, PA, 1974.

[15] Davidson A.C. Modelling excesses over high thresholds, with an application. In Statistical extremes and applications (ed. J.Tiago de Oliveira), pages 621-638, 1984 Dordrecht: Reidel.

[16] Davidson A.C., Smith R. L. Models for exceedances over high thresholds (with discussion). J.R. Statist. Soc., 52: 393-442, 1990. | MR | Zbl

[17] Davis J.M., Eder B.K., Bloomfield P. Modeling Ozone in the Chicago Urban Area. In Cox L.H. Nychka D., Piegorsch W.W., editor, Case Studies in Environmental Statistics. Springier Verlag, New York, 1998. | Zbl | DOI

[18] Eder B. E., Davis J. M., Bloomfield P. An automated classification scheme designed to better elucidate the dependence of ozone on meteorology. Journal of Applied Meteorology, 33, 1994.

[19] Gao F., Sacks J., Welch W. J. Predicting the urban ozone levels and trends with semiparametric modelling. Technical Report 14, National Institute of Statistical Sciences, P.O. Box 14162, Research Triangle Park, N.C. 27709, 1994. | MR

[20] Graf-Jaccottet M., Jaunin M.-H. Predictive models for ground ozone and nitrogen dioxyde time series. Environmetrics, 9: 393-406, 1998. | DOI

[21] Hosking J.M.R., Wallis J.R. Parameter and quantite estimation for the generalized Pareto distribution. Technometrics, 29: 339-349, 1987. | MR | Zbl | DOI

[22] Hosmer D. W., Lemeshow S. Applied Logistic Regression. John Wiley & Sons, New York, 1989. | Zbl

[23] Johnson R. A., Wehrly T. Measures and models for angular correlation and angular-linear correlation. Journal of the Royal Statistical Society, (Serie B) 39: 222-229, 1977. | MR | Zbl

[24] Krzanowski, W. J. Principal component analysis in the presence of group structure. Applied Statistics, 33: 164-168, 1984. | DOI

[25] Leadbetter M. R. On a basis for "Peaks over Threshold" modeling. Statistics and Probability Letters, 12: 357-362, 1991. | MR | Zbl | DOI

[26] Leadbetter M. R. On high level exceedance modelling and tail inference. J. Stat. Plan. Inference, 45(1-2): 247-260, 1995. | MR | Zbl | DOI

[27] Leadbetter M.R. On exceedance based environmental criteria. Technical Report 9, National Institute of Statistical Sciences, P.O. Box 14162, Research Triangle Park, N.C. 27709, 1993.

[28] Leadbetter M.R., Lindgren G., Rootzén H. Extremes and Related Properties of Random Sequences and Series. Springer Verlag, New York, 1983. | MR | Zbl

[29] Lee L. Testing adequacy of the Weibull and log linear rate models for a Poisson process. Technometrics, 22(2): 195-199, 1980. | Zbl | DOI

[30] Legay J.M., Tomassone R. La comparaison de régressions orthogonales. Revue de Statistique Appliquée, 1998. à paraître.

[31] Mardia K. V., Kent J. T., Bibby J. M. Multivariate Analysis. Academic Press Inc., London, 1982. | MR | Zbl

[32] Math Works Inc. MATLAB Reference Guide,. Cary, The Math Works Inc., 1995.

[33] Math Works Inc. MATLAB Userff's Guide,. Cary, The Math Works Inc., 1995.

[34] Milionis A. E., Davies T. D. Regression and stochastic models for air pollution-I. Review, comments and suggestions. Atmospheric Environment, 28(17): 2801-2810, 1994. | DOI

[35] Pickands J. The two-dimensional Poisson process and extremal processes. J. Appl. Probab., 8: 745-756, 1971. | MR | Zbl | DOI

[36] Pickands J. Statistical inference using extreme order statistics. Ann. Statist., 3: 119-131, 1975. | MR | Zbl

[37] Pugh D., Vassie J.M. Applications of the joint probability method for extreme sea level computations. Proc. Instn Civ. Engrs, Part 2, 69: 959-975, 1980.

[38] Rootzén H., Leadbetter M. R., De Haan L. On the distribution of tail array sums for strongly mixing stationary sequences. 1994 en preparation. | Zbl

[39] Ross G.J.S. Nonlinear Estimation. Springer-Verlag, London, 1990. | Zbl

[40] Saporta G. Probabilités Analyse des données et statistique. Editions Technip, Paris, 1990. | Zbl

[41] Sas Institute Inc. SAS/IML software: Usage and reference, Version 6, First Edition. Cary, NC: SAS Institute Inc., 1989.

[42] Sas Institute Inc. SAS/STAT User's Guide : Version 6, Fourth Edition, Vol. 1 and 2. Cary, NC: SAS Institute Inc., 1994.

[43] Shively T.S. An analysis of the long-term trend in ozone data from two Houston Texas monitoring sites. Atmospheric Environment, 24B: 293-301, 1990.

[44] Shively T.S. An analysis of the trend in ground-level ozone using nonhomogeneous Poisson processes. Atmospheric Environment, 25B: 387-396, 1991.

[45] Smith R.L. Threshold methods for sample extremes. In Statistical extremes and applications (ed. J.Tiago de Oliveira), pages 621-638, 1984. Dordrecht: Reidel. | MR | Zbl | DOI

[46] Smith R.L. Maximum likelihood estimation in a class of nonregular cases. Biometrika, 72(1): 67-90, 1985. | MR | Zbl | DOI

[47] Smith R.L. Extreme value theory based on the r largest annual events. J. Hydrology, 86: 27-43, 1986. | DOI

[48] Smith R.L. Estimating tails of probability distributions. The annals of Statistics, 15(3): 1174-1207, 1987. | MR | Zbl | DOI

[49] Smith R.L. Extreme value analysis of environmental time series : An application to trend detection in ground-level ozone (with discussion). Statistical Science, 4: 367-393, 1989. | MR | Zbl

[50] Smith R.L., Shively T.S. Point process approach to modeling trends in tropospheric ozone based on exceedances of a high threshold. Atmospheric Environment, 29(23): 3489-3499, 1995. | DOI

[51] Somerville M.C., Mukerjee S., Fox D. Estimating the wind direction of maximum air pollutant concentration. Environmetrics, 7: 231-243, 1996. | DOI

[52] Tomassone R., Dervin C. et Masson J.P. Biométrie: Modélisation de phénomènes biologiques. Masson, Paris, 1995. | Zbl

[53] Toupance G. L'ozone dans la basse troposphère, théorie et pratique. Pollution Atmosphérique, Janvier-Mars: 32-42, 1988.

[54] Toupance G., Perros P., Soedomo M. The rapid rotation of the wind direction and sharp ozone peak features. Physicochemical Behavior of Atmospheric Pollutants, 1986.

[55] Vaquera-Huerta H., Villasenor J.A., Hughes J. Statistical analysis of trends in urban ozone. Statistics for the Environment 3: Pollution Assessment and Control (edited by Barnett V. and Turkman K.F.), pages 175-183, 1997. John Wiley & Sons Ltd.

[56] Vautard R., Beekman M., Honoré C., Deleuze I. La pollution photochimique en région parisienne simulée par le modèle chimere et l'influence du transport régional d'ozone. Technical report, 1998.

[57] Weissman I. Estimation of parameters and large quantiles based on the k largest observations. J. Amer. Statist. Assoc., 73: 812-815, 1978. | MR | Zbl

[58] Zeldin M.D., Cassmassi J.C. Development of improved methods for predicting air quality levels in the south coast air bassin. Final Report to California Air Ressources Board, Technology Service Corp., Santa Monica, CA, Contract AG-192-30, 1979.