Using Counting Processes To Estimate the Number of Ozone Exceedances: An Application to Mexico City Ozone Measurements
Jorge A. Achcar¹, Eliane R. Rodrigues², Mario H. Tarumoto³
¹Faculdade de Medicina de Ribeirao Preto, Universidade de Sao Paulo - USP, Ribeirao Preto, Sao Paulo, Brazil; ²Instituto de Matematicas - UNAM, Universidad Nacional Autonoma de Mexico, Mexico City, DF, Mexico; ³Faculdade de Ciencias e Tecnologia, Universidade Estadual Paulista - UNESP, Presidente Prudente, Sao Paulo, Brazil

In this paper we consider the problem of estimating the number of times that the concentration of a pollutant of interest surpasses a threshold in a given time interval. That question is also related to the estimation of the distribution of the time between two consecutive days in which the threshold is surpassed by the pollutant's concentration. One approach to the problem is to consider that the number of times that the threshold is surpassed can be described by a non-homogeneous Poisson process (see for instance Achcar et al, 2008). However, depending on the value of the threshold, the hypothesis of independence between two consecutive time intervals between surpassings may not be true. Hence, the use of Poisson models may not be adequate. In this work we consider two cases. In the first of them, we still assume independence between two time intervals between surpassings, but instead of having an exponential distribution for them, we consider that the time intervals between surpassings have as distribution a Gamma density. In a second case, we drop the independence hypothesis and assume a first order dependence between two consecutive time intervals between surpassings. The distribution of the time intervals in here is also related to a Gamma distribution. In the case of dependent time intervals, we take into account a autoregressive model used by Sim (1990, 1992) in problems related to queueing theory. In both cases, the parameters present in the model are related to parameters of the Gamma distribution of the time interval between surpassings. In order to estimate the parameters of the models, a Bayesian point of view is used. Inference is performed using a sample drawn from the posterior distribution using a combination of Gibbs sampling and Metropolis-Hastings algorithms. Everything is programmed using R. Convergence of the algorithm is monitored using the Gelman-Rubin method as well as an analysis of the traceplot produced. The theoretical part of this work is applied to ozone data obtained from the Mexico City monitoring network.

1. Achcar, J.A., Fernandez-Bremauntz, A.A., Rodrigues, E.R. and Tzintzun, G., 2008. Estimating the number of ozone peaks in Mexico City using a non-homogeneous Poisson model. Environmetrics 19, 469-485.

2. Sim, C. H. 1990. First-order autoregressive models for Gamma and Exponential processes. Journal of Applied Probability 27, 325-332.

3. Sim, C. H. 1992. Point processes with correlated Gamma interarrival times. Statistics and Probability Letters 15, 135-141.

Keywords: Couting processes; Dependent interarrival times; Markov chain Monte Carlo methods; Ozone air pollution

Biography: Eliane Rodrigues is a researcher at Institute of Mathematics of the National University in Mexico. Her field of research is applied probability with focuss on stochastic modelling applied mainly to air pollution problems. She has been working on the subject since 2003.