Non-Homogeneous Poisson Processes Applied to Count Data: A Bayesian Approach Considering Different Prior Distributions

Affiliation(s)

Department of Statistics, Federal University of Santa Maria, Santa Maria, Brazil.

Department of Statistics, University of Campinas, Campinas, Brazil.

Department of Social Medicine, Medical School, University of S?o Paulo, Ribeir?o Preto, Brazil.

Department of Statistics, Federal University of Santa Maria, Santa Maria, Brazil.

Department of Statistics, University of Campinas, Campinas, Brazil.

Department of Social Medicine, Medical School, University of S?o Paulo, Ribeir?o Preto, Brazil.

ABSTRACT

This article discusses the Bayesian approach for count data using non-homogeneous Poisson processes, considering different prior distributions for the model parameters. A Bayesian approach using Markov Chain Monte Carlo (MCMC) simulation methods for this model was first introduced by [1], taking into account software reliability data and considering non-informative prior distributions for the parameters of the model. With the non-informative prior distributions presented by these authors, computational difficulties may occur when using MCMC methods. This article considers different prior distributions for the parameters of the proposed model, and studies the effect of such prior distributions on the convergence and accuracy of the results. In order to illustrate the proposed methodology, two examples are considered: the first one has simulated data, and the second has a set of data for pollution issues at a region in Mexico City.

This article discusses the Bayesian approach for count data using non-homogeneous Poisson processes, considering different prior distributions for the model parameters. A Bayesian approach using Markov Chain Monte Carlo (MCMC) simulation methods for this model was first introduced by [1], taking into account software reliability data and considering non-informative prior distributions for the parameters of the model. With the non-informative prior distributions presented by these authors, computational difficulties may occur when using MCMC methods. This article considers different prior distributions for the parameters of the proposed model, and studies the effect of such prior distributions on the convergence and accuracy of the results. In order to illustrate the proposed methodology, two examples are considered: the first one has simulated data, and the second has a set of data for pollution issues at a region in Mexico City.

Cite this paper

L. Vicini, L. Hotta and J. Achcar, "Non-Homogeneous Poisson Processes Applied to Count Data: A Bayesian Approach Considering Different Prior Distributions,"*Journal of Environmental Protection*, Vol. 3 No. 10, 2012, pp. 1336-1345. doi: 10.4236/jep.2012.310152.

L. Vicini, L. Hotta and J. Achcar, "Non-Homogeneous Poisson Processes Applied to Count Data: A Bayesian Approach Considering Different Prior Distributions,"

References

[1] J. A. Achcar, K. D. Dey and M. Niverthi, “A Bayesian Approach Using Nonhomogeneous Poisson Process for Software Reliability Models,” In: A. S. Basu, S. K. Basu and S. Mukhopadhyay, Eds., Frontiers in Reliability, World Scientic Publishing Co., Singapore, 1998, pp. 1- 18. doi:10.1142/9789812816580_0001

[2] A. L. Goel and K. Okumoto, “An Analysis of Recurrent Software Errors in a Real-Time Control System,” Proceedings of the 1978 Annual Conference, ACM’78, Washington DC, 4-6 December 1978, pp. 496-501. doi:10.1145/800127.804160

[3] A. L. Goel, “A Guidebook for Software Reliability Assessment,” Technical Report, Syracuse University, Sy- racse, 1983.

[4] G. S. Muldholkar, D. K. Srivastava and M. Friemer, “The Exponentiated-Weibull Family: A Reanalysis of the Bus- Motor Failure Data,” Technometrics, Vol. 37, No. 4, 1995, pp. 436-445. doi:10.1080/00401706.1995.10484376

[5] V. G. Cancho, H. Bolfarine and J. A. Achcar, “A Bayesian Analysis of the Exponentiated-Weibull Distribution,” Journal of Applied Statistical Science, Vol. 4, 1999, pp. 227-242.

[6] J. E. R. Cid and J. A. Achcar, “Bayesian Inference for Non- homogeneous Poisson Processes in Software Reliability Models Assuming Nonmonotonic Intensity Functions,” Computational Statistics and Data Analysis, Vol. 32, No. 2, 1999, pp. 147-159.doi:10.1016/S0167-9473(99)00028-6

[7] A. Gelman and D. R. Rubin, “A Single Series from the Gibbs Sampler Provides a False Sense of Security,” In: J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, Eds., Bayesian Statistics 4, Oxford University Press, Oxford, 1992, pp. 625-631.

[8] S. P. Brook and A. Gelman, “General Methods for Monitoring Convergence of Iterative Simulations,” Journal of Computational and Graphical Statistics, Vol. 7, No. 4, 1997, pp. 434-455. doi:10.2307/1390675

[9] A. Gelman “Inference and Monitoring Convergence,” In: W. R. Gilks, S. Richardson and D. J. Spiegelhalter, Eds., Markov Chain Monte Carlo in Practice, Chapman and Hall, London, 1996, pp. 131-143.

[10] D. R. Cox and P. A. Lewis, “Statistical Analysis of Series of Events,” Methuen, London, 1966. doi:10.1063/1.1699114

[11] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, “Equations of State Calculations by Fast Computing Machines,” Journal of Chemical Physics, Vol. 21, No. 6, 1953, pp. 1087-1092.doi:10.1063/1.1699114

[12] W. K. Hastings, “Monte Carlo Sampling Methods Using Markov Chains and Their Applications,” Biometrika, Vol. 57, No. 1, 1970, pp. 97-109. doi:10.1093/biomet/57.1.97

[13] G. Casella, and R. L. Berger, “Statistical Inference,” 2nd Edition, Duxbury Press, Pacific Grove, 2001.

[14] J. A. Achcar, E. R. Rodrigues, C. D. Paulino and P. Soares, “Non-homogeneous Poisson Models With a Change-Point: An Application to Ozone Peaks in Mexico City,” Environmental and Ecological Statistics, Vol. 17, No. 4, 2010, pp. 521-541. doi:10.1007/s10651-009-0114-3

[1] J. A. Achcar, K. D. Dey and M. Niverthi, “A Bayesian Approach Using Nonhomogeneous Poisson Process for Software Reliability Models,” In: A. S. Basu, S. K. Basu and S. Mukhopadhyay, Eds., Frontiers in Reliability, World Scientic Publishing Co., Singapore, 1998, pp. 1- 18. doi:10.1142/9789812816580_0001

[2] A. L. Goel and K. Okumoto, “An Analysis of Recurrent Software Errors in a Real-Time Control System,” Proceedings of the 1978 Annual Conference, ACM’78, Washington DC, 4-6 December 1978, pp. 496-501. doi:10.1145/800127.804160

[3] A. L. Goel, “A Guidebook for Software Reliability Assessment,” Technical Report, Syracuse University, Sy- racse, 1983.

[4] G. S. Muldholkar, D. K. Srivastava and M. Friemer, “The Exponentiated-Weibull Family: A Reanalysis of the Bus- Motor Failure Data,” Technometrics, Vol. 37, No. 4, 1995, pp. 436-445. doi:10.1080/00401706.1995.10484376

[5] V. G. Cancho, H. Bolfarine and J. A. Achcar, “A Bayesian Analysis of the Exponentiated-Weibull Distribution,” Journal of Applied Statistical Science, Vol. 4, 1999, pp. 227-242.

[6] J. E. R. Cid and J. A. Achcar, “Bayesian Inference for Non- homogeneous Poisson Processes in Software Reliability Models Assuming Nonmonotonic Intensity Functions,” Computational Statistics and Data Analysis, Vol. 32, No. 2, 1999, pp. 147-159.doi:10.1016/S0167-9473(99)00028-6

[7] A. Gelman and D. R. Rubin, “A Single Series from the Gibbs Sampler Provides a False Sense of Security,” In: J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, Eds., Bayesian Statistics 4, Oxford University Press, Oxford, 1992, pp. 625-631.

[8] S. P. Brook and A. Gelman, “General Methods for Monitoring Convergence of Iterative Simulations,” Journal of Computational and Graphical Statistics, Vol. 7, No. 4, 1997, pp. 434-455. doi:10.2307/1390675

[9] A. Gelman “Inference and Monitoring Convergence,” In: W. R. Gilks, S. Richardson and D. J. Spiegelhalter, Eds., Markov Chain Monte Carlo in Practice, Chapman and Hall, London, 1996, pp. 131-143.

[10] D. R. Cox and P. A. Lewis, “Statistical Analysis of Series of Events,” Methuen, London, 1966. doi:10.1063/1.1699114

[11] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, “Equations of State Calculations by Fast Computing Machines,” Journal of Chemical Physics, Vol. 21, No. 6, 1953, pp. 1087-1092.doi:10.1063/1.1699114

[12] W. K. Hastings, “Monte Carlo Sampling Methods Using Markov Chains and Their Applications,” Biometrika, Vol. 57, No. 1, 1970, pp. 97-109. doi:10.1093/biomet/57.1.97

[13] G. Casella, and R. L. Berger, “Statistical Inference,” 2nd Edition, Duxbury Press, Pacific Grove, 2001.

[14] J. A. Achcar, E. R. Rodrigues, C. D. Paulino and P. Soares, “Non-homogeneous Poisson Models With a Change-Point: An Application to Ozone Peaks in Mexico City,” Environmental and Ecological Statistics, Vol. 17, No. 4, 2010, pp. 521-541. doi:10.1007/s10651-009-0114-3