Estimating the number of ozone peaks in Mexico City using a non‐homogeneous Poisson model

In this paper, we consider the problem of estimating the number of times an air quality standard is exceeded in a given period of time. A non-homogeneous Poisson model is proposed to analyse this issue. The rate at which the Poisson events occur is given by a rate function λ(t), t ≥ 0. This rate function also depends on some parameters that need to be estimated. Two forms of λ(t), t ≥ 0 are considered. One of them is of the Weibull form and the other is of the exponentiated-Weibull form. The parameters estimation is made using a Bayesian formulation based on the Gibbs sampling algorithm. The assignation of the prior distributions for the parameters is made in two stages. In the first stage, non-informative prior distributions are considered. Using the information provided by the first stage, more informative prior distributions are used in the second one. The theoretical development is applied to data provided by the monitoring network of Mexico City. The rate function that best fit the data varies according to the region of the city and/or threshold that is considered. In some cases the best fit is the Weibull form and in other cases the best option is the exponentiated-Weibull. Copyright © 2007 John Wiley & Sons, Ltd.

[1]  Gabriel Huerta,et al.  Time-varying models for extreme values , 2007, Environmental and Ecological Statistics.

[2]  Gary A. Churchill,et al.  Bayesian Restoration of a Hidden Markov Chain with Applications to DNA Sequencing , 1999, J. Comput. Biol..

[3]  B. Carlin,et al.  Bayesian Model Choice Via Markov Chain Monte Carlo Methods , 1995 .

[4]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[5]  E. M. Roberts,et al.  Review of Statistics of Extreme Values with Applications to Air Quality Data Part I. Review , 1979 .

[6]  SEARCHING FOR LOW DIMENSIONALITY IN AIR POLLUTION TIME SERIES , 1997 .

[7]  J. Achcar.  Ch. 29. Bayesian analysis for software reliability data , 2001 .

[8]  Eliane R. Rodrigues,et al.  Maximum a posteriori estimation of the daily ozone peaks in Mexico City , 2005 .

[9]  H. Bolfarine,et al.  A Bayesian analysis of the exponentiated-Weibull distribution , 1999 .

[10]  Richard J. Boys,et al.  On determining the order of Markov dependence of an observed process governed by a hidden Markov model , 2002, Sci. Program..

[11]  P. Green.  Reversible jump Markov chain Monte Carlo computation and Bayesian model determination , 1995 .

[12]  Bradley P. Carlin,et al.  Bayesian measures of model complexity and fit , 2002 .

[13]  Simon P. Wilson.  Hierarchical modelling of orthopaedic hip replacement damage accumulation and reliability , 2005 .

[14]  H. Javitz.  Statistical Interdependencies in the Ozone National Ambient Air Quality Standard , 1980 .

[15]  Malcolm R Leadbetter,et al.  On a basis for 'Peaks over Threshold' modeling , 1991 .

[16]  R Guardani,et al.  Study of atmospheric ozone formation by means of a neural network-based model. , 1999, Journal of the Air & Waste Management Association.

[17]  P. Kinney,et al.  Long-term residence in areas of high ozone: associations with respiratory health in a nationwide sample of nonsmoking young adults [dsee comments] , 1999, Environmental health perspectives.

[18]  Eliane R. Rodrigues,et al.  Comparing the performance of a reversible jump Markov chain Monte Carlo algorithm for DNA sequences alignment , 2006 .

[19]  Simon P. Wilson,et al.  Predicting future discoveries of European marine species by using a non‐homogeneous renewal process , 2005 .

[20]  Walter W. Piegorsch,et al.  Statistical advances in environmental science , 1998 .

[21]  Gordon Johnston,et al.  Statistical Models and Methods for Lifetime Data , 2003, Technometrics.

[22]  C. Nascimento,et al.  Ground-Level Ozone Mapping in Large Urban Areas Using Multivariate Statistical Analysis: Application to the São Paulo Metropolitan Area , 2003, Journal of the Air & Waste Management Association.

[23]  Dipak K. Dey,et al.  A Bayesian Approach Using Nonhomogeneous Poisson Process for Software Reliability Models , 2007 .

[24]  Joel L. Horowitz,et al.  Extreme Values from a Nonstationary Stochastic Process: An Application to Air Quality Analysis , 1980 .

[25]  I. Zurbenko,et al.  Moderating the Influence of Meteorological Conditions on Ambient Ozone Concentrations. , 1996, Journal of the Air & Waste Management Association.

[26]  A characterization of the weekday-weekend behavior of ambient ozone concentrations in California , 2004 .

[27]  Sylvia Richardson,et al.  Markov Chain Monte Carlo in Practice , 1997 .

[28]  D. Geman.  Random fields and inverse problems in imaging , 1990 .

[29]  E. M. Roberts,et al.  Review of Statistics of Extreme Values with Applications to Air Quality Data: Part II. Applications , 1979 .

[30]  M. O'Neill,et al.  Ozone, area social conditions, and mortality in Mexico City. , 2004, Environmental research.

[31]  K. Berhane,et al.  The effect of air pollution on lung development from 10 to 18 years of age. , 2004, The New England journal of medicine.

[32]  A. Raftery.  Extreme Value Analysis of Environmental Time Series: An Application to Trend Detection in Ground-Level Ozone: Comment: Are Ozone Exceedance Rates Decreasing? , 1989 .

[33]  David R. Cox,et al.  The statistical analysis of series of events , 1966 .

[34]  C. Shy,et al.  Ozone exposure and daily mortality in Mexico City: a time-series analysis. , 1996, Research report.

[35]  C. Rosenzweig,et al.  Climate change, ambient ozone, and health in 50 US cities , 2007 .

[36]  A. Frigessi,et al.  Fast Approximate Maximum a Posteriori Restoration of Multicolour Images , 1995 .

[37]  Martijn Gough.  Climate change , 2009, Canadian Medical Association Journal.

[38]  Richard L. Smith.  Extreme Value Analysis of Environmental Time Series: An Application to Trend Detection in Ground-Level Ozone , 1989 .

[39]  Jen Nan Pan,et al.  Monitoring long‐memory air quality data using ARFIMA model , 2008 .

[40]  Deo Kumar Srivastava,et al.  The exponentiated Weibull family: a reanalysis of the bus-motor-failure data , 1995 .

[41]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[42]  Roger D. Peng,et al.  The Exposure–Response Curve for Ozone and Risk of Mortality and the Adequacy of Current Ozone Regulations , 2006, Environmental health perspectives.

[43]  F. Dominici,et al.  Ozone and short-term mortality in 95 US urban communities, 1987-2000. , 2004, JAMA.

[44]  Jorge Alberto Achcar,et al.  Bayesian inference for nonhomogeneous Poisson processes in software reliability models assuming nonmonotonic intensity functions , 1999 .