A Non-Homogeneous Markov Chain Model to Study Ozone Exceedances in Mexico City

Abstract

Additional information is available at the end of the chapter IntroductionIn many cities around the world, air pollution is among the many environmental problems that affect their population. Among the many known facts about the impact of pollution on human health, we have that for ozone concentration levels above 0.11 parts per million (0.11ppm), the susceptible part of the population (e.g., the elderly, ill, and newborn) staying in that environment for a long period of time, may experience serious health deterioration (see, for example, [1][2][3][4][5][6][7][8][9][10]). Therefore, to understand the behaviour of ozone and/or pollutants in general, is a very important issue.It is possible to find in the literature a vast amount of works that try to answer some of the many issues arising in the study of pollutants' behaviour. Depending on the type of questions that one is trying to answer, different methodologies may be used. Among the many works concentrating on the study of ozone behaviour are, [11][12][13] using extreme value theory to study the behaviour of the maximum ozone measurements; [14] using time series analysis; [15] using volatility models to study the variability of the weekly average ozone measurements; [13,16] using homogeneous Poisson processes and [17, 18] using non-homogeneous Poisson models to analyse the probability of having a certain number of ozone exceedances in a time interval of interest; [19] using compound Poisson models to study the occurrence of clusters of ozone exceedances as well as their mean duration time; and [20] using queueing model to study the occurrence of cluster of ozone exceedances as well as their size distribution.In the environmental area, it is also possible to find works using Markov chains models. Some of them are, [21,22] where non-homogeneous Markov models are used to study the occurrence of precipitation. We also have [23] where those types of models are used to study tornado activity. In the case of ozone modelling we have, for instance, the works of [24][25][26] using time homogeneous Markov chains. In those works the interest was in estimating the probability that the ozone measurement would be above (below) a given threshold, conditioned on where it lays in the present and in the past days.© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.In [24], the order of the Markov chain was estimated using auto-correlation function. Its transition matrix was estimated using the maximum likelihood method (see, for instance, [27,28], among others). In [25], the order of the chain was also considered an unknown quantity that needed to be estimated. The Bayesian approach (see, for example, [29]) was used to estimate the order as well as the transition probabilities of the chain. In particular, the maximum à posteriori method wa...

Air Quality Estimation Using Nonhomogeneous Markov Chains: A Case Study Comparing Two Rules Applied to Mexico City Data

“…Z be the process recording the air quality indices in the jth day, 1, 2, , j N = . As in [31] and [32], we assume that…”

“…Therefore, because we have a likelihood function proportional to a product of multinomial distributions and the prior distribution of the vector of parameters also a product of Dirichlet distributions, the marginal conditional posterior distributions of the initial distribution and each row of the transition matrices are also Dirichlet distributions (see, for instance, [32]). The hyperparameters of these posterior distributions are, respectively, ( )…”

“…In the present case, results show that the behaviors of the estimated quantities of interest mimic well those of the observed (see plots given in the appendix) showing the suitability of the model and results presented in this work in the study of air quality indices. We may also compare, as in [22] and [32], the probabilities of having the occurrence of a given air quality index in a given hour of the day when we have information of few hours earlier. For instance, consider region SW and NOM-172 and assume we want to know the probability that at 1 pm the air quality index will be 3, i.e., B under the NOM-172 legislation, and that at 10 am, 11 am, and 12 pm we have the sequence of states 1, 1, and 2 (i.e., G, G, and A air quality).…”