Skip to main content

Part of the book series: SpringerBriefs in Mathematics ((BRIEFSMATH))

  • 1224 Accesses

Abstract

In this chapter we consider some counting processes more general than the Poisson process to study the distribution of the time between surpassings of a given environmental standard.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Achcar, J.A., Martínez, E.Z., Rufino-Neto, A., Paulino, C.D., Soares, P.: A statistical model investigating the prevalence of tuberculosis in New York using counting processes with two change-points. Epidemiol. Infect. 136, 1599–1605 (2008a)

    Article  Google Scholar 

  2. Achcar, J.A., Fernández-Bremauntz, A.A., Rodrigues, E.R., Tzintzun, G.: Estimating the number of ozone peaks in Mexico City using a non-homogeneous Poisson model. Environmetrics 19, 469–485 (2008b)

    Article  MathSciNet  Google Scholar 

  3. Achcar, J.A., Rodrigues, E.R., Paulino, C.D., Soares, P.: Non-homogeneous Poisson processes with a change-point: an application to ozone exceedances in Mexico City. Environ. Ecol. Stat. 17, 521–541 (2010a)

    Article  MathSciNet  Google Scholar 

  4. Achcar, J.A., Zozolotto, H.C., Rodrigues, E.R.: Bivariate stochastic volatility models applied to Mexico City ozone pollution data. In: Romano, G.C., Conti, A.G. (eds.) Air Quality in the 21st Century, pp. 241–267. Nova Publisher, New York (2010b)

    Google Scholar 

  5. Achcar, J.A., Rodrigues, E.R., Tzintzun, G.: Using non-homogeneous Poisson models with multiple change-points to estimate the number of ozone exceedances in Mexico City. Environmetrics 22, 1–12 (2011a)

    Article  MathSciNet  Google Scholar 

  6. Achcar, J.A., Ortíz-Rodríguez, G., Rodrigues, E.R.: The behaviour of a Metropolis-Hastings algorith under different prior distributions: an application to ozone measurements in Mexico City. In: Jha, M. (ed.) Latest Trends on Urban Planning and Transportation. Proceedings of the 3rd WSEAS International Conference on Urban Planning and Transportation, pp.160–165. July 22–24, 2010, Corfu, Greece (2011b)

    Google Scholar 

  7. Achcar, J.A., Rodrigues, E.R., Tzintzun, G.: Modelling interoccurrence time between ozone peaks in Mexico City in the presence of multiple change-points. Braz. J. Probab. Stat. 25, 183–204 (2011c)

    Article  MathSciNet  Google Scholar 

  8. Achcar, J.A., Sousa, D.E.F., Rodrigues, E.R., Tzintzun, G.: Comparing the number of ozone exceedances in different seasons of the year in Mexico City. Environ. Model. Assess. 16, 251–264 (2011d)

    Article  Google Scholar 

  9. Achcar, J.A., Rodrigues, E.R., Tarumoto, M.H.: Using counting processes to estimate the number of ozone exceedances: an application to the Mexico City measurements. In: Proceedings of the 58th ISI World Statistics Congress, Dublin, 21–16 August 2011, Dublin, 2011. Available at http://www.isi2011.ie/content/access-congress-proceedings.html/CPS58-04. (2011e)

  10. Achcar, J.A., Zozolotto, H.C., Rodrigues, E.R., Saldiva, P.H.N.: Two Multivariate stochatic volatility models applied to air pollution data from São Paulo, Brazil. Adv. Appl. Stat. 20, 1–23 (2011f)

    MATH  Google Scholar 

  11. Achcar, J.A., Barrios, J.M., Rodrigues, E.R.: Comparing the fitting of some non-homogeneous Poisson models to estimate ozone exceedances in Mexico City. Accepted for publication in the Special Issue Air pollution of the Journal of Environmental Protection (2012a)

    Google Scholar 

  12. Achcar, J.A., Cepeda-Cuervo, E., Rodrigues, E.R.: Weibull and generalized exponential overdispersion models with an application to ozone air pollution. J. Appl. Stat. 39, 1953–1963 (2012b)

    Article  MathSciNet  Google Scholar 

  13. Álvarez, L.J., Rodrigues, E.R.: Trans-dimensional MCMC algorithm to estimate the order of a Markov chain: an application to ozone peaks in Mexico City. Int. J. Pure Appl. Math. 48, 315–331 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Álvarez, L.J., Fernández-Bremauntz, A.A., Rodrigues, E.R., Tzintzun, G.: Maximum a posteriori estimation of the daily ozone peaks in Mexico City. J. Agr. Biol. Environ. Stat. 10, 276–290 (2005)

    Article  Google Scholar 

  15. Barreto-Souza, W., de Morais, A.L., Cordeiro, G.M.: The Weibull-geometric distribution. J. Stat. Comput. Simulat. 81, 645–657 (2011)

    Article  MATH  Google Scholar 

  16. Bell, M.L., Peng, R., Dominici, F.: The exposure-response curve for ozone and risk of mortality and the adequacy of current ozone regulations. Environ. Health. Perspect. 114, 532–536 (2005)

    Article  Google Scholar 

  17. Bell, M.L., Goldberg, R., Rogrefe, C., Kinney, P.L., Knowlton, K., Lynn, B., Rosenthal, J., Rosenzwei, C., Patz, J.A.: Climate change, ambient ozone, and health in 50 US cities. Clim. Change 82, 61–76 (2007)

    Article  Google Scholar 

  18. Bernardo, J.M., Smith, A.F.M.: Bayesian Theory. Wiley, London (1995)

    Google Scholar 

  19. Boys, R.J., Henderson, D.A.: On determining the order of Markov dependence of an observed process governed by a hidden Markov model. Special Issue Sci. Program. 10, 241–251 (2002)

    Google Scholar 

  20. Boys, R.J., Henderson, D.A.: Bayesian approach to DNA segmentation. Biometrics 60, 573–588 (2004)

    MathSciNet  MATH  Google Scholar 

  21. Burham, K.P., Anderson, D.A.: Model Selection and Multivariate Inference: A Practical Information – Theoretical Approach, 2nd edn. Springer, New York (2002)

    Google Scholar 

  22. Carlin, B.P., Chib, S.: Bayesian model choice via Markov chain Monte Carlo methods. J. Roy. Stat. Soc. B 57, 473–484 (1995)

    MATH  Google Scholar 

  23. Carlin, B.P., Louis, T.A.: Bayes and Empirical Bayes Methods for Data Analysis, 2nd edn. Chapman and Hall/CRC, Boca Raton (2000)

    Book  MATH  Google Scholar 

  24. Chen, M.-H., Shao, Q.-M., Ibrahim, J.G.: Monte Carlo Methods in Bayesian Computation. Springer, New York (2000)

    Book  MATH  Google Scholar 

  25. Chib, S., Greenberg, E.: Understanding the Metropolis-Hastings algorithm. American Statistician 49, 327–335 (1995)

    Google Scholar 

  26. Cordeiro, G.M., Simas, A.B., Stošić, B.D.: Closed form expressions for the moments of the Beta-Weibull distribution. Ann. Braz. Acad. Sci. 83, 357–373 (2011)

    MATH  Google Scholar 

  27. Cox, D.R., Lewis, P.A.: Statistical Analysis of Series Events. Methuen, London (1966)

    MATH  Google Scholar 

  28. Dacunha-Castelle, D., Dulfo, M.: Probability and Statistics, vol. II. Springer, New York (1986)

    Book  Google Scholar 

  29. Dockery, D.W., Schwartz, J., Spengler, J.D.: Air pollution and daily mortality: association with particulates and acid aerosols. Environ. Res. 59, 362–373 (1992)

    Article  Google Scholar 

  30. Evans, M., Swartz, T.: Approximating Integrals via Monte Carlo and Deterministic Methods. Oxford Statistical Sciences Series 20. Oxford University Press, Oxford (2000)

    Google Scholar 

  31. Fan, T.-H., Tsai, C.-A.: A Bayesian method in determining the order of a finite state Markov chain. Comm. Stat. Theor. Meth. 28, 1711–1730 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  32. Famoye, F., Lee, C., Olumolade, O.: The Beta-Weibull distribution. J. Stat. Theory Appl. 4, 121–136 (2005)

    MathSciNet  Google Scholar 

  33. Feller, W.: An Introduction to Probability Theory and Its Applications, 3rd edn. Wiley, London (1968)

    MATH  Google Scholar 

  34. Gamerman, D.: Markov chain Monte Carlo. Stochastic Simulation for Bayesian Inference. Chapman and Hall, Boca Raton (1997)

    MATH  Google Scholar 

  35. Gelfand, A.E., Smith, A.F.M.: Sampling-based approaches to calculating marginal densities. J. Am. Stat. Assoc. 85, 398–409 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  36. Gelman, A., Rubin, D.B.: Inference from iterative simulation using multiple sequences. Stat. Sci. 7, 457–511 (1992)

    Article  Google Scholar 

  37. Geman, D.: Random fields and inverse problems in imaging. Lect. Note Math. 1427, 113–193 (1990)

    Article  MathSciNet  Google Scholar 

  38. Gilks, W.R., Richardson, S., Spiegelhalter, D.J.: Markov Chain Monte Carlo in Practice. Chapman and Hall, Boca Raton (1996)

    MATH  Google Scholar 

  39. Goel, A.L., Okumoto, K.: An analysis of recurrent software failures on a real-time control system. In: Proceedings of ACM Conference, pp. 496–500. Washington, D.C., USA (1978)

    Google Scholar 

  40. Green, P.J.: Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. J. Roy. Stat. Soc. B 57, 711–732 (1995)

    Google Scholar 

  41. Grimmett, G.R., Stirzaker, D.R.: Probability and Random Processes. Clarendon, Oxford (1982)

    MATH  Google Scholar 

  42. Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970)

    Article  MATH  Google Scholar 

  43. Itô, K., Thurston, G.D.: Daily PM10/mortality association: an investigation of at-risk subpopulations. J. Expo. Anal. Environ. Epidemiol. 6, 79–95 (1996)

    Google Scholar 

  44. Jara-Ettinger, J.A.: Estimando el número de excedencias de ozono en la Ciudad de México usando procesos de Poisson no-homogéneos y el muestreador de Gibbs. Undergraduate final year report. Facultad de Ciencias Físico-Matemáticas. Universidad Michoacana de San Nicolás Hidalgo, Mexico (2011) (In Spanish.)

    Google Scholar 

  45. Javits, J.S.: Statistical interdependencies in the ozone national ambiente air quality standard. J. Air. Poll. Contr. Assoc. 30, 58–59 (1980)

    Article  Google Scholar 

  46. Jelinski, Z., Moranda, P.B.: Software reliability research. In: Freiberger, W. (ed.) Statistical Computer Performance Evaluation, pp. 465–497. Academic, New York (1972)

    Google Scholar 

  47. Karlin, S., Taylor, H.M.: A First Course in Stochastic Processes, 2nd edn. Academic, New York (1975)

    MATH  Google Scholar 

  48. Larsen, L.C., Bradley, R.A., Honcoop, G.L.: A new method of characterizing the variability of air quality-related indicators. In: Air and Waste Management Association’s InternationalSpecialty Conference of Tropospheric Ozone and the Environment. Los Angeles, CA (1990)

    Google Scholar 

  49. Lawless, J.F.: Statistical Models and Methods for Lifetime Data. Wiley, Hoboken, NJ (1982)

    MATH  Google Scholar 

  50. Lee, P.M.: Bayesian Statistics: An Introduction, 2nd edn. Arnold, London (1997)

    MATH  Google Scholar 

  51. Likens, G. (Lead Author): Environmental Protection Agency (Content source), Davis, W., Zaikowski, L., Nodvin, S.C. (Topic Editors). Acid rain. In: Cutler J (ed) Encyclopedia of Earth. Cleveland. Washington, D.C. Environmental Information Coalition, National Council for Science and the Environment. First published in the Encyclopedia of Earth October 9, 2006;. Last revised January 2, 2010. http://www.eoearth.org/article/Acid\_rain. Retrieved January 22, 2010.

  52. Lunn, D.J., Thomas, A., Best, N., Spiegelhalter, D.: WinBugs - a Bayesian modelling framework: concepts, structure, and extensibility. Stat. Comput. 10, 325–337 (2000)

    Article  Google Scholar 

  53. Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., Teller, E.: Equations of state calculations by fast computing machine. J. Chem. Phys. 21, 1087–1091 (1953)

    Article  Google Scholar 

  54. Mohnen, V.A.: The challenge of acid rain. Sci. Am. 259, 30–38 (1988)

    Article  Google Scholar 

  55. Moranda, P.B.: Prediction of software reliability and its applications. In: Proceedings of the Annual Reliability and maintainability Symposium, pp. 327–332. Washington, D.C., USA (1975)

    Google Scholar 

  56. Morgenstern, D.: Einfache Beispiele Zweidimensionaler Verteilungen. Mitteilingsblatt fur Mathematische 8, 234–253 (1956)

    MathSciNet  MATH  Google Scholar 

  57. Muldholkar, G.S., Srivastava, D.K., Friemer, M.: The exponentiated-Weibull family: a reanalysis of the bus-motor failure data. Technometrics 37, 436–445 (1995)

    Article  Google Scholar 

  58. Musa, J.D., Okumoto, K.: A logarithmic Poisson execution time model for software reliability measurement. In: Proceedings of Seventh International Conference on Software Engineering, pp. 230–238. Orlando, USA (1984)

    Google Scholar 

  59. NOM: Modificación a la Norma Oficial Mexicana NOM-020-SSA1–1993. Diario Oficial de la Federación. 30 October 2002. Mexico (2002) (In Spanish.)

    Google Scholar 

  60. Ortíz-Rodríguez, G.: Un algoritmo Metropolis-Hastings y un modelo de Poisson no homogéneo para estudiar el número de rebases de la norma ambiental para ozone en la Ciudad de México. Master’s Dissertation. Facultad de Ciencias. Universidad Nacional Autónoma de México. Mexico (2012) (In Spanish.)

    Google Scholar 

  61. Ott, W.R.: Environmental Statistics and Data Analysis. Lewis Publishers, CRC Press, Boca Raton, FL (1995)

    MATH  Google Scholar 

  62. Pérez-Muñoz, R.H.: Los procesos de Bernoulli y Poisson en el estudio de casos de incumpliminento de la norma para ozono. Undergraduate final year report. Facultad de Ciencias. Universidad Nacional Autónoma de México. Mexico (2006) (In Spanish.)

    Google Scholar 

  63. Raftery, A.E.: Hypothesis testing and model selection. In: Gilks, W., Richardson, S., Speigelhalter, D.J. (eds.) Markov Chain Monte Carlo in Practice, pp. 163–187. Chapman and Hall, Boca Raton, FL (1996)

    Google Scholar 

  64. Ramírez-Cid, J.E., Achcar, J.A.: Bayesian inference for nonhomogeneous Poisson processes in software reliability models assuming non monotonic intensity functions. Comput. Stat. Data Anal. 32, 147–159 (1999).

    Article  Google Scholar 

  65. Rice, J.A.: Mathematical Statistics and Data Analysis. Brook and Cole, Pacific Grove, CA (1988)

    MATH  Google Scholar 

  66. Ritz, B., Yu, F.: The effects of ambient carbon monoxide on low birth weight among children born in Southern California between 1989 and 1993. Environ. Health Perspect. 107, 17–25 (1999)

    Article  Google Scholar 

  67. Ritz, B., Yu, F., Chapa, G., Fruin, S.: Effects of air pollution on preterm birth among children born in Southern California between 1989 and 1993. Epidemiology 11, 502–511 (2000)

    Article  Google Scholar 

  68. Ritz, B., Wilhelm, M., Hoggart, K.J., Ghosh, J.K.C.: Ambient air pollution and preterm birth in the environment and pregnancy outcomes study at the University of California, Los Angeles. Am. J. Epidemiol. 1666, 1045–1052 (2007)

    Article  Google Scholar 

  69. Robert, C.P., Casella, G.: Monte Carlo Statistical Methods. Springer, New York (1999)

    MATH  Google Scholar 

  70. Rodrigues, E.R., Achcar, J.A., Jara-Ettinger, J.: A Gibbs sampling algorithm to estimate the occurrence of ozone exceedances in Mexico City. In: Popovic, D. (ed.) Air Quality – Models and Applications, pp. 131–150. InTech Open Access Publisher, Croatia (2011)

    Google Scholar 

  71. Ross, S.M.: Stochastic Processes, 2nd edn. Wiley, London (1996)

    MATH  Google Scholar 

  72. Sim, C.H.: First-order autoregressive models for Gamma and Exponential processes. J. Appl. Probab. 27, 325–332 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  73. Sim, C.H.: Point processes with correlated Gamma inter arrival times. Stat. Probab. Lett. 15, 135–141 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  74. Spiegelhalter, D.J., Thomas, A., Best, N.G., Gilks, W.R.: Winbugs: Bayesian Inference Using Gibbs Sampling. MRC Biostatistics Unit, Cambridge (1999)

    Google Scholar 

  75. Spiegelhalter, D.J., Best, N.G., Carlin, B.P., Van der Linde, A.: Bayesian measures of model complexity and fit (with discussion and rejoinder). J. Roy. Stat. Soc. B 64, 583–639 (2002)

    Article  MATH  Google Scholar 

  76. Verzani, J.: Using R for Introductory Statistics. Chapman and Hall, Boca Raton (2005)

    MATH  Google Scholar 

  77. Yang, T.E., Kuo, L.: Bayesian binary segmentation procedure for a Poisson process with multiple change-points. J. Comput. Graph. Stat. 10, 772–785 (2001)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media New York

About this chapter

Cite this chapter

Rodrigues, E.R., Achcar, J.A. (2012). Some Counting Processes and Ozone Air Pollution. In: Applications of Discrete-time Markov Chains and Poisson Processes to Air Pollution Modeling and Studies. SpringerBriefs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4645-3_5

Download citation

Publish with us

Policies and ethics