Skip to main content
Log in

Use of non-homogeneous Poisson process (NHPP) in presence of change-points to analyze drought periods: a case study in Brazil

  • Published:
Environmental and Ecological Statistics Aims and scope Submit manuscript

Abstract

Rain precipitation in the last years has been very atypical in different regions of the world, possibly, due to climate changes. We analyze Standard Precipitation Index (SPI) measures (1, 3, 6 and 12-month timescales) for a large city in Brazil: Campinas located in the southeast region of Brazil, São Paulo State, ranging from January 01, 1947 to May 01, 2011. A Bayesian analysis of non-homogeneous Poisson processes in presence or not of change-points is developed using Markov Chain Monte Carlo methods in the data analysis. We consider a special class of models: the power law process. We also discuss some discrimination methods for the choice of the better model to be used for the rain precipitation data.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Achcar JA, Bolfarine H (1989) Constant hazard against a change-point alternative: a Bayesian approach with censored data. Commun Stat Theory Methods 18(10):3801–3819. doi:10.1080/03610928908830124

    Article  Google Scholar 

  • Achcar JA, Fernández-Bremauntz AA, Rodrigues ER, Tzintzun G (2008) Estimating the number of ozone peaks in Mexico City using a non-homogeneous Poisson model. Environmetrics 19(5):469–485. doi:10.1002/env.890

    Article  CAS  Google Scholar 

  • Achcar JA, Loibel S (1998) Constant hazard models with a change-point: a Bayesian analysis using Markov chain Monte Carlo methods. Biom J 40:543–555

    Article  Google Scholar 

  • Achcar JA, Rodrigues ER, Paulino CD, Soares P (2010) Non-homogeneous Poisson models with a change-point: an application to ozone peaks in Mexico City. Environ Ecol Stat 17(4):521–541. doi:10.1007/s10651-009-0114-3

    Article  CAS  Google Scholar 

  • Barlow R, Hunter L (1960) Optimum preventive maintenance policies. Oper Res 8:90–100

    Article  Google Scholar 

  • Carlin BP, Gelfand AE, Smith AF (1992) Hierarchical Bayesian analysis of changepoint problems. Appl Stat 41(2):389–405

    Article  Google Scholar 

  • Chib S, Greenberg E (1995) Undestanding the Metropolis–Hastings algorithm. Am Stat 49(4):327–335

    Google Scholar 

  • Chortaria C, Karavitis CA, Alexandris S (2010) Development of the spi drought index for greece using geo-statistical methods. In: Proceedings of BALWOIS 2010 international conference

  • Cid JER, Achcar JA (1999) Bayesian inference for nonhomogeneous Poisson processes in software reliability models assuming nonmonotonic intensity functions. Comput Stat Data Anal 32(2):147–159

    Article  Google Scholar 

  • Cox DR, Lewis PAW (1966) The statistical analysis of series of events. Methuen & Co., Ltd.; Wiley, London; New York

    Book  Google Scholar 

  • Crow LH (1974) Reliability analysis for complex systems. In: Proschan F, Serfling RJ (eds) Reliability and biometry. SIAM, Philadelphia, pp 379–410

    Google Scholar 

  • Dey DK, Purkayastha S (1997) Bayesian approach to change point problems. Commun Stat Theory Methods 26(8):2035–2047. doi:10.1080/03610929708832030

    Article  Google Scholar 

  • Edwards DC (1997) Characteristics of 20th century drought in the united states at multiple time scales. Tech. rep., DTIC Document

  • Gelfand AE, Smith AFM (1990) Sampling-based approaches to calculating marginal densities. J Am Stat Assoc 85(410):398–409

    Article  Google Scholar 

  • Gelman A, Rubin DB (1992) Inference from iterative simulation using multiple sequences. Stat Sci 7(4):457–472

    Article  Google Scholar 

  • Ghysels E (1996) Stochastic volatility. Statistical methods on finance. Springer, New York

    Google Scholar 

  • Guttman NB (1994) On the sensitivity of sample l moments to sample size. J Clim 7(6):1026–1029

    Article  Google Scholar 

  • Hayes MJ, Svoboda MD, Wilhite DA, Vanyarkho OV (1999) Monitoring the 1996 drought using the standardized precipitation index. Bull Am Meteorol Soc 80(3):429–438

    Article  Google Scholar 

  • Karavitis CA (1998) Drought and urban water supplies: the case of metropolitan athens. Water Policy 1(5):505–524

    Article  Google Scholar 

  • Karavitis CA (1999) Decision support systems for drought management strategies in metropolitan athens. Water Int 24(1):10–21

    Article  Google Scholar 

  • Karavitis CA, Alexandris S, Tsesmelis DE, Athanasopoulos G (2011) Application of the standardized precipitation index (spi) in Greece. Water 3(3):787–805

    Article  Google Scholar 

  • Kim S, Shephard N, Chib S (1998) Stochastic volatility: likelihood inference and comparison with arch models. Rev Econ Stud 65(3):361–393

    Article  Google Scholar 

  • Kuo L, Yang TY (1996) Bayesian computation for nonhomogeneous Poisson processes in software reliability. J Am Stat Assoc 91(434):763–773

    Article  Google Scholar 

  • Lana X, Serra C, Burgueño A (2001) Patterns of monthly rainfall shortage and excess in terms of the standardized precipitation index for catalonia (ne spain). Int J Climatol 21(13):1669–1691

    Article  Google Scholar 

  • Livada I, Assimakopoulos V (2007) Spatial and temporal analysis of drought in Greece using the standardized precipitation index (spi). Theor Appl Climatol 89(3–4):143–153

    Article  Google Scholar 

  • Matthews DE, Farewell VT (1982) On testing for a constant hazard against a change-point alternative. Biometrics 38(2):463–468

    Article  CAS  PubMed  Google Scholar 

  • McKee TB, Doesken NJ, Kleist J (1995) Drought monitoring with multiple time scales. In: Ninth conference on applied climatology. American Meteorological Society, Boston

  • McKee TB, Doesken NJ, Kleist J et al (1993) The relationship of drought frequency and duration to time scales. In: Proceedings of the 8th conference on applied climatology, vol 17. American Meteorological Society, Boston, MA, USA, pp 179–183

  • Meyer R, Yu J (2000) Bugs for a Bayesian analysis of stochastic volatility models. Econ J 3(2):198–215

    Google Scholar 

  • Musa JD, Iannino A, Okumoto K (1987) Software reliability: measurement, prediction, application. McGraw-Hill Inc, New York

    Google Scholar 

  • Musa JD, Okumoto K (1984) A logarithmic poisson execution time model for software reliability measurement. In: Proceedings of the 7th international conference on software engineering, IEEE Press, pp 230–238

  • Palmer WC (1965) Meteorological drought, weather bureau research paper no. 45. US Department of Commerce, Washington

    Google Scholar 

  • Pievatolo A, Ruggeri F (2004) Bayesian reliability analysis of complex repairable systems. Appl Stoch Model Bus Ind 20(3):253–264. doi:10.1002/asmb.522

    Article  Google Scholar 

  • R Core Team(2015) R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/

  • Raftery A, Akman V (1986) Bayesian analysis of a Poisson process with a change-point. Biometrika 71(1):85–89

    Article  Google Scholar 

  • Ramsay JO, Silverman BW (2006) Functional data analysis. Wiley, New York

    Book  Google Scholar 

  • Ruggeri F, Sivaganesan S (2005) On modeling change points in non-homogeneous Poisson processes. Stat Inference Stoch Process 8(3):311–329

    Article  Google Scholar 

  • Smith AFM, Roberts GO (1993) Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. J R Stat Soc Ser B Methodol 55(1): 3–23. http://links.jstor.org/sici?sici=0035-9246(1993)55:1<3:BCVTGS>2.0.CO;2-#&origin=MSN

  • Sönmez FK, Koemuescue AU, Erkan A, Turgu E (2005) An analysis of spatial and temporal dimension of drought vulnerability in Turkey using the standardized precipitation index. Nat Hazards 35(2):243–264

    Article  Google Scholar 

  • Spiegelhalter DJ, Best NG, Carlin BP, Van Der Linde A (2002) Bayesian measures of model complexity and fit. J R Stat Soc Ser B Stat Methodol 64(4):583–639

    Article  Google Scholar 

  • Su YS, Yajima M (2015) R2jags: Using R to Run ’JAGS’ . R package version 0.5-6. http://CRAN.R-project.org/package=R2jags

Download references

Acknowledgments

The authors are very grateful to the Editor and a referee for their helpful and useful comments that improved the manuscript. The first author was partially supported by CNPq-Brazil.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Emílio Augusto Coelho-Barros.

Additional information

Handling Editor Bryan F. J. Manly.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Achcar, J.A., Coelho-Barros, E.A. & de Souza, R.M. Use of non-homogeneous Poisson process (NHPP) in presence of change-points to analyze drought periods: a case study in Brazil. Environ Ecol Stat 23, 405–419 (2016). https://doi.org/10.1007/s10651-016-0345-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10651-016-0345-z

Keywords

Navigation