Abstract
In this chapter some Markov chain models are presented together with some of their applications to air pollution problems. One of the questions answered is related to the probability of having a pollutant’s concentration between two thresholds of interest. A particular case of this problem is related to the probability of surpassing or not surpassing environmental standards.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
Barreto-Souza, W., de Morais, A.L., Cordeiro, G.M.: The Weibull-geometric distribution. J. Stat. Comput. Simulat. 81, 645–657 (2011)
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)
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)
Bernardo, J.M., Smith, A.F.M.: Bayesian Theory. Wiley, London (1995)
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)
Boys, R.J., Henderson, D.A.: Bayesian approach to DNA segmentation. Biometrics 60, 573–588 (2004)
Burham, K.P., Anderson, D.A.: Model Selection and Multivariate Inference: A Practical Information – Theoretical Approach, 2nd edn. Springer, New York (2002)
Carlin, B.P., Chib, S.: Bayesian model choice via Markov chain Monte Carlo methods. J. Roy. Stat. Soc. B 57, 473–484 (1995)
Carlin, B.P., Louis, T.A.: Bayes and Empirical Bayes Methods for Data Analysis, 2nd edn. Chapman and Hall/CRC, Boca Raton (2000)
Chen, M.-H., Shao, Q.-M., Ibrahim, J.G.: Monte Carlo Methods in Bayesian Computation. Springer, New York (2000)
Chib, S., Greenberg, E.: Understanding the Metropolis-Hastings algorithm. American Statistician 49, 327–335 (1995)
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)
Cox, D.R., Lewis, P.A.: Statistical Analysis of Series Events. Methuen, London (1966)
Dacunha-Castelle, D., Dulfo, M.: Probability and Statistics, vol. II. Springer, New York (1986)
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)
Evans, M., Swartz, T.: Approximating Integrals via Monte Carlo and Deterministic Methods. Oxford Statistical Sciences Series 20. Oxford University Press, Oxford (2000)
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)
Famoye, F., Lee, C., Olumolade, O.: The Beta-Weibull distribution. J. Stat. Theory Appl. 4, 121–136 (2005)
Feller, W.: An Introduction to Probability Theory and Its Applications, 3rd edn. Wiley, London (1968)
Gamerman, D.: Markov chain Monte Carlo. Stochastic Simulation for Bayesian Inference. Chapman and Hall, Boca Raton (1997)
Gelfand, A.E., Smith, A.F.M.: Sampling-based approaches to calculating marginal densities. J. Am. Stat. Assoc. 85, 398–409 (1990)
Gelman, A., Rubin, D.B.: Inference from iterative simulation using multiple sequences. Stat. Sci. 7, 457–511 (1992)
Geman, D.: Random fields and inverse problems in imaging. Lect. Note Math. 1427, 113–193 (1990)
Gilks, W.R., Richardson, S., Spiegelhalter, D.J.: Markov Chain Monte Carlo in Practice. Chapman and Hall, Boca Raton (1996)
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)
Green, P.J.: Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. J. Roy. Stat. Soc. B 57, 711–732 (1995)
Grimmett, G.R., Stirzaker, D.R.: Probability and Random Processes. Clarendon, Oxford (1982)
Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970)
Itô, K., Thurston, G.D.: Daily PM10/mortality association: an investigation of at-risk subpopulations. J. Expo. Anal. Environ. Epidemiol. 6, 79–95 (1996)
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.)
Javits, J.S.: Statistical interdependencies in the ozone national ambiente air quality standard. J. Air. Poll. Contr. Assoc. 30, 58–59 (1980)
Jelinski, Z., Moranda, P.B.: Software reliability research. In: Freiberger, W. (ed.) Statistical Computer Performance Evaluation, pp. 465–497. Academic, New York (1972)
Karlin, S., Taylor, H.M.: A First Course in Stochastic Processes, 2nd edn. Academic, New York (1975)
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)
Lawless, J.F.: Statistical Models and Methods for Lifetime Data. Wiley, Hoboken, NJ (1982)
Lee, P.M.: Bayesian Statistics: An Introduction, 2nd edn. Arnold, London (1997)
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.
Lunn, D.J., Thomas, A., Best, N., Spiegelhalter, D.: WinBugs - a Bayesian modelling framework: concepts, structure, and extensibility. Stat. Comput. 10, 325–337 (2000)
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)
Mohnen, V.A.: The challenge of acid rain. Sci. Am. 259, 30–38 (1988)
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)
Morgenstern, D.: Einfache Beispiele Zweidimensionaler Verteilungen. Mitteilingsblatt fur Mathematische 8, 234–253 (1956)
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)
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)
NOM: Modificación a la Norma Oficial Mexicana NOM-020-SSA1–1993. Diario Oficial de la Federación. 30 October 2002. Mexico (2002) (In Spanish.)
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.)
Ott, W.R.: Environmental Statistics and Data Analysis. Lewis Publishers, CRC Press, Boca Raton, FL (1995)
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.)
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)
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).
Rice, J.A.: Mathematical Statistics and Data Analysis. Brook and Cole, Pacific Grove, CA (1988)
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)
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)
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)
Robert, C.P., Casella, G.: Monte Carlo Statistical Methods. Springer, New York (1999)
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)
Ross, S.M.: Stochastic Processes, 2nd edn. Wiley, London (1996)
Sim, C.H.: First-order autoregressive models for Gamma and Exponential processes. J. Appl. Probab. 27, 325–332 (1990)
Sim, C.H.: Point processes with correlated Gamma inter arrival times. Stat. Probab. Lett. 15, 135–141 (1992)
Spiegelhalter, D.J., Thomas, A., Best, N.G., Gilks, W.R.: Winbugs: Bayesian Inference Using Gibbs Sampling. MRC Biostatistics Unit, Cambridge (1999)
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)
Verzani, J.: Using R for Introductory Statistics. Chapman and Hall, Boca Raton (2005)
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)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media New York
About this chapter
Cite this chapter
Rodrigues, E.R., Achcar, J.A. (2012). Markov Chain Models. 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_2
Download citation
DOI: https://doi.org/10.1007/978-1-4614-4645-3_2
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-4644-6
Online ISBN: 978-1-4614-4645-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)