Brazilian Journal of Probability and Statistics

Modelling interoccurrence times between ozone peaks in Mexico City in the presence of multiple change points

Jorge Alberto Achcar, Eliane R. Rodrigues, and Guadalupe Tzintzun

Full-text: Open access


In this article we consider the problem of analysing the interoccurrence times between ozone peaks. These interoccurrence times are assumed to have an exponential distribution with some rate λ>0 (which may have different values for different interoccurrence times). We consider four parametric forms for λ. These parametric forms depend on some parameters that will be estimated by using Bayesian inference through Markov Chain Monte Carlo (MCMC) methods. In particular, we use a Gibbs sampling algorithm internally implemented in the software WinBugs. We also present an analysis to detect the possible presence of change points. This is performed using the 95% credible interval of the difference between two consecutive means. Results are applied to the maximum daily ozone measurements provided by the monitoring network of Mexico City. An analysis in terms of the number of possible change points present in the model in terms of different years and seasons of the year is also presented.

Article information

Braz. J. Probab. Stat., Volume 25, Number 2 (2011), 183-204.

First available in Project Euclid: 31 March 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Poisson models multiple change points Bayesian inference ozone peaks interoccurrence times


Achcar, Jorge Alberto; Rodrigues, Eliane R.; Tzintzun, Guadalupe. Modelling interoccurrence times between ozone peaks in Mexico City in the presence of multiple change points. Braz. J. Probab. Stat. 25 (2011), no. 2, 183--204. doi:10.1214/10-BJPS116.

Export citation


  • Achcar, J. A., Fernández-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.
  • Achcar, J. A., Rodrigues, E. R. and Tzintzun, G. (2009a). Using non-homogeneous Poisson models with multiple change-points to estimate the number of ozone exceedances in Mexico City. Environmetrics. Published on line in September 2009, DOI:10.1002/env.1029.
  • Achcar, J. A., Rodrigues, E. R., Paulino, C. A. and Soares, P. (2009b). Non-homogeneous Poisson process with a change-point: An application to ozone exceedances in Mexico City. Environmental and Ecological Statistics. Published on line in May 2009, DOI:10.1007/s10651-009-0114-3.
  • Achcar, J. A., Ortíz-Rodríguez, G. and Rodrigues, E. R. (2009c). Estimating the number of ozone peaks in Mexico City using a non-homogeneous Poisson model and a Metropolis–Hastings algorithm. International Journal of Pure and Applied Mathematics 53, 1–20.
  • Álvarez, L. J., Fernández-Bremauntz, A. A., Rodrigues, E. R. and Tzintzun, G. (2005). Maximum a posteriori estimation of the daily ozone peaks in Mexico City. Journal of Agricultural, Biological, and Environmental Statistics 10, 276–290.
  • ARB (Air Resource Board) (2005). Review of the California ambient air quality standard for ozone. Staff report. California Environmental Protection Agency, USA.
  • Austin, J. and Tran, H. (1999). A characterization of the weekday–weekend behavior of ambient ozone concentrations in California. In Air Pollution VI 645–661. Ashurst Lodge, UK: WIT Press.
  • Bell, M. L., McDermontt, A., Zeger, S. L., Samet, J. M. and Dominici, F. (2004). Ozone and short-term mortality in 95 US urban communities 1987–2000. Journal of the American Medical Society 292, 2372–2378.
  • Bell, M. L., Peng, R. and Dominici, F. (2005). The exposure–response curve for ozone and risk of mortality and the adequacy of current ozone regulations. Environmental Health Perspectives 114, 532–536.
  • Bernardo, J.-M. and Smith, A. F. M. (1994). Bayesian Theory. New York: Wiley.
  • Flaum, J. B., Rao, S. T. and Zurbenko, I. G. (1996). Moderating influence of meteorological conditions on ambient ozone concentrations. J. Air and Waste Management Assoc. 46, 33–46.
  • Gauderman, W. J., Avol, E., Gililand, F., Vora, H., Thomas, D., Berhane, K., McConnel, R., Kuenzli, N., Lurmman, F., Rappaport, E., Margolis, H., Bates, D. and Peter, J. (2004). The effects of air pollution on lung development from 10 to 18 years of age. The New England Journal of Medicine 351, 1057–1067.
  • Gelfand, A. E. and Smith, A. F. M. (1990). Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association 85, 398–409.
  • Guardani, R., Nascimento, C. A. O., Guardani, M. L. G., Martins, M. H. R. B. and Romano, J. (1999). Study of atmospheric ozone formation by means of a neural network based model. J. Air and Waste Manag Assoc. 49, 316–323.
  • Guardani, R., Aguiar, J. L., Nascimento, C. A. O., Lacava, C. I. V. and Yanagi, Y. (2003). Ground-level ozone mapping in large urban areas using multivariate analysis: Application to the São Paulo Metropolitan Area. J. Air and Waste Management Assoc. 53, 553–559.
  • Horowitz, J. (1980). Extreme values from a nonstationary stochastic process: An application to air quality analysis. Technometrics 22, 469–482.
  • Huerta, G. and Sansó, B. (2005). Time-varying models for extreme values. Technical Report 2005–4. Department of Applied Mathematics and Statistics, University of California, Santa Cruz.
  • Itô, K., de León, S. and Lippman, M. (2005). Associations between ozone and daily mortality: A review and additional analysis. Epidemiology 16, 446–457.
  • Javits, J. S. (1980). Statistical interdependencies in the ozone national ambient air quality standard. J. Air Poll. Control Assoc. 30, 58–59.
  • Jelinski, Z. and Moranda, P. B. (1972). Software reliability research. In Statistical Computer Performance Evaluation (Freiberger, W., ed.) 465–497. New York: Academic Press.
  • Lanfredi, M. and Macchiato, M. (1997). Searching for low dimensionality in air pollution time series. Europhysics Lett. 40, 589–594.
  • Larsen, L. C., Bradley, R. A. and Honcoop, G. L. (1990). A new method of characterizing the variability of air quality-related indicators. In Air and Waste Management Association’s International Specialty Conference of Tropospheric Ozone and the Environment. Los Angeles.
  • Leadbetter, M. R. (1991). On a basis for “peak over threshold” modeling. Statistics and Probability Letters 12, 357–362.
  • Loomis, D. P., Borja-Arbuto, V. H., Bangdiwala, S. I. and Shy, C. M. (1996). Ozone exposure and daily mortality in Mexico City: A time series analysis. Health Effects Institute Research Report 75, 1–46.
  • Moranda, P. B. (1975). Prediction of software reliability and its applications. In Proceedings of the Annual Reliability and maintainability Symposium 327–332. Washington, DC.
  • NOM (2002). Modificación a la Norma Oficial Mexicana NOM-020-SSA1–1993. Diario Oficial de la Federación. 30 de Octubre de 2002.
  • O’Neill, M. R., Loomis, D. and Borja-Aburto, V. H. (2004). Ozone, area social conditions and mortality in Mexico City. Environmental Research 94, 234–242.
  • Pan, J.-N. and Chen, S.-T. (2008). Monitoring long-memory air quality data using ARFIMA model. Environmetrics 19, 209–219.
  • Piegorsch, W. W., Smith, E. P., Edwards, D. and Smith, L. (1998). Statistical advances in environmental sciences. Statistical Sciences 13, 186–208.
  • Raftery, A. E. (1989). Are ozone exceedance rate decreasing?, Comment on the paper “Extreme value analysis of environmental time series: An application to trend detection in ground-level ozone” by R. L. Smith. Statistical Sciences 4, 378–381.
  • Roberts, E. M. (1979a). Review of statistics extreme values with applications to air quality data. Part I. Review. Journal of the Air Pollution Control Association 29, 632–637.
  • Roberts, E. M. (1979b). Review of statistics extreme values with applications to air quality data. Part II. Applications. Journal of the Air Pollution Control Association 29, 733–740.
  • Seinfeld, J. H. (2004). Air pollution: A half century of progress. American Institute of Chemical Engineers Journal 50, 1098–1108.
  • Smith, R. L. (1989). Extreme value analysis of environmental time series: An application to trend detection in ground-level ozone. Statistical Sciences 4, 367–377.
  • Smith, A. F. M. and Roberts, G. O. (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. Journal of the Royal Statistical Society Series B 55, 3–23.
  • Spiegelhalter, D. J., Thomas, A. and Best, N. G. (1999). WinBugs: Bayesian inference using Gibbs sampling. Cambridge, UK: MRC Biostatistics Unit.
  • Zolghadri, A. and Henry, D. (2004). Minmax statistical models for air pollution time series. Application to ozone time series data measured in Bordeaux. Environmental Monitoring and Assessment 98, 275–294.