On modeling the maximum duration of dry spells: a simulation study under a Bayesian approach

  • Davi Butturi-GomesEmail author
  • Luiz Alberto Beijo
  • Fabricio Goecking Avelar
Original Paper


Dry spell and drought are hydrological phenomena with serious socioeconomic effects and, despite recent efforts, substantial scientific and statistical comprehension are still lacking—especially when considering their extreme events. Such events are usually modeled using the generalized extreme value (GEV) distribution, whose prediction performance, at least under a Bayesian approach, remain poorly understood when fitted to a discrete series (the simplest way to record dry spell occurrence and duration). Thus, in this study, we aim at evaluating point and interval prediction performances of the GEV distribution when fitted to dry spell data, using computer simulations of different realistic scenarios (variations in the number of days per dry spells, number of dry spells per year, sample sizes, and available prior information). While sample size increase produced generally expected results over point performance (i.e., stronger bias in small samples), counterintuitive patterns arose when we evaluated the accuracy of prediction credible intervals. We also found a positive correlation between prediction bias and the GEV shape parameter estimate, a fact we believe to be related to the discrete nature of the data. Furthermore, we noticed the best interval performances occurred in increasing levels of information rendered by prior distributions. Finally, we consider all these results to be general enough to apply to different extreme discrete phenomena, since we found no effect of neither the duration nor the frequency of dry spells. Although typical issues in discrete data (e.g., overdispersion) and time series data (e.g., trend) should be considered in future investigations, one must be aware that whenever attempting to fit dry spell duration series to the GEV distribution in the absence of substantial prior information will frequently lead to underestimated predictions—the worst kind for dry spell strategic management—which may further compromise scientists, practitioners, and their community responsibilities.



The authors thank Mr. Fábio F. Marchetti for his kind and insightful considerations.

Funding information

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

Supplementary material

704_2018_2684_MOESM1_ESM.pdf (7.2 mb)
ESM 1 (PDF 7337 kb)


  1. Agarwal A, Babel MS, Maskey S (2014) Analysis of future precipitation in the Koshi river basin, Nepal. J Hydrol 513:422–434. CrossRefGoogle Scholar
  2. Assis JMO (2012) Análise de tendências de mudanças climáticas no semiárido de Pernambuco 166Google Scholar
  3. Beijo LA, Vivanco MJF, Muniz JA (2009) Bayesian analysis for estimating the return period of maximum precipitation at Jaboticabal São Paulo state, Brazil. Ciência e Agrotecnologiancia e Agrotecnologia 33:261–270CrossRefGoogle Scholar
  4. Bloomfield JP, Allen DJ, Griffiths KJ (2009) Examining geological controls on baseflow index (BFI) using regression analysis: an illustration from the Thames Basin, UK. J Hydrol 373:164–176CrossRefGoogle Scholar
  5. Bond NR, Lake PS, Arthington AH (2008) The impacts of drought on freshwater ecosystems: an Australian perspective. Hydrobiologia 600:1–14. CrossRefGoogle Scholar
  6. Bouagila B, Sushama L (2013) On the current and future dry spell characteristics over Africa. Atmosphere (Basel) 4:272–298CrossRefGoogle Scholar
  7. Cancelliere A, Salas JD (2010) Drought probabilities and return period for annual streamflows series. J Hydrol 391:77–89CrossRefGoogle Scholar
  8. Coles S (2001) An introduction to statistical modeling of extreme values. Springer-Verlag, LondonCrossRefGoogle Scholar
  9. Coles S, Pericchi L (2003) Anticipating catastrophes through extreme value modelling. J R Stat Soc: Ser C: Appl Stat 52:405–416CrossRefGoogle Scholar
  10. Coles S, Powell EA (1996) Bayesian methods in extreme value modelling: a review and new developments. Int Stat Rev 64:119–136CrossRefGoogle Scholar
  11. Coles S, Tawn J (2005) Bayesian modelling of extreme surges on the UK east coast. Philos Trans R Soc A 363:1387–1406CrossRefGoogle Scholar
  12. Collischonn W, Tucci CEM, Clarke RT (2001) Further evidence of changes in the hydrological regime of the River Paraguay: part of a wider phenomenon of climate change? J Hydrol 245:218–238. CrossRefGoogle Scholar
  13. Dubrovsky M, Svoboda MD, Trnka M, Hayes MJ, Wilhite DA, Zalud Z, Hlavinka P (2009) Application of relative drought indices in assessing climate-change impacts on drought conditions in Czechia. Theor Appl Climatol 96:155–171. CrossRefGoogle Scholar
  14. Farokhnia A, Morid S, Byun HR (2011) Application of global SST and SLP data for drought forecasting on Tehran plain using data mining and ANFIS techniques. Theor Appl Climatol 104:71–81. CrossRefGoogle Scholar
  15. Fisher RA, Tippett LHC (1928) Limiting forms of the frequency distribution of the largest or smallest member of a sample. Math Proc Camb Philos Soc 24:180–190CrossRefGoogle Scholar
  16. Genest C, Favre AC (2007) Everything you always wanted to know about copula modeling but were afraid to ask. J Hydrol Eng 12:347–368CrossRefGoogle Scholar
  17. Geweke J (1992) Evaluating the accuracy of sampling-based approaches to calculating posterior moments. In: Bernardo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics 4. Clarendon Press, Oxford, p 859Google Scholar
  18. Ghile YB, Schulze RE (2009) Use of an ensemble re-ordering method for disaggregation of seasonal categorical rainfall forecasts into conditioned ensembles of daily rainfall for hydrological forecasting. J Hydrol 371:85–97. CrossRefGoogle Scholar
  19. Gnedenko B (1943) Sur la distribution limite du terme maximum d’une serie aleatoire. Ann Math 44:423–453CrossRefGoogle Scholar
  20. Govindaraju RS, Rao AR (2000) Artificial neural networks in hydrology. Kluwer Academic Publishers, AmsterdamCrossRefGoogle Scholar
  21. Heidelberger P, Welch PD (1983) Simulation run length control in the presence of an initial transient. Oper Res 31:1109–1144. CrossRefGoogle Scholar
  22. IBGE (2010) Brazilian socio-economic database. In: Munic. Soc. Indic. Accessed 10 Sep 2017
  23. IBGE (2013) Brazilian socio-economic database. In: Vectorial SHP files Polit. Div. Brazil (in Port. Accessed 17 Aug 2018
  24. Jury MR (2018) Uganda rainfall variability and prediction. Theor Appl Climatol 132:905–919. CrossRefGoogle Scholar
  25. Kebede A, Diekkrüger B, Edossa DC (2017) Dry spell, onset and cessation of the wet season rainfall in the Upper Baro-Akobo Basin, Ethiopia. Theor Appl Climatol 129:849–858. CrossRefGoogle Scholar
  26. Kumagai T, Saitoh TM, Sato Y et al (2004) Transpiration, canopy conductance and the decoupling coefficient of a lowland mixed dipterocarp forest in Sarawak, Borneo: dry spell effects. J Hydrol 287:237–251. CrossRefGoogle Scholar
  27. Lana X, Burgueño A (1998) Probabilities of repeated long dry episodes based on the Poisson distribution. Theor Appl Climatol 60:111–120CrossRefGoogle Scholar
  28. Lana X, Burgueño A, Martínez MD, Serra C (2006) Statistical distributions and sampling strategies for the analysis of extreme dry spells in Catalonia (NE Spain). J Hydrol 324:94–114. CrossRefGoogle Scholar
  29. Lana X, Martínez MD, Burgueño A, Serra C (2008) Return period maps of dry spells for Catalonia (northeastern Spain) based on the Weibull distribution. Hydrol Sci J 53:48–64. CrossRefGoogle Scholar
  30. Ma Q, Zhang J, Sun C, Zhang F, Wu R, Wu L (2017) Drought characteristics and prediction during pasture growing season in Xilingol grassland, northern China. Theor Appl Climatol 133:1–14. CrossRefGoogle Scholar
  31. McKee T, Doesken NJ, Kleist J (1993) The relationship of drought frequency and duration to time scales. 179–184Google Scholar
  32. Meshram SG, Gautam R, Kahya E (2018) Drought analysis in the Tons River Basin, India during 1969-2008. Theor Appl Climatol 132:939–951. CrossRefGoogle Scholar
  33. Mishra AK, Singh VP (2010) A review of drough concepts. J Hydrol 391:202–216CrossRefGoogle Scholar
  34. Mishra AK, Singh VP (2011) Drought modeling—a review. J Hydrol 403:157–175CrossRefGoogle Scholar
  35. Nadarajah S, Mitov K (2002) Asymptotics of maxima of discrete random variables. Extremes 5:287–294CrossRefGoogle Scholar
  36. Paixão Junior BR, Estrela F, Cruz G, Lima M (2010) CodeGeo. In: Brazil shapefiles download (in Port. Accessed 17 Aug 2018
  37. Pasarić Z, Cindrić K (2018) Generalised Pareto distribution: impact of rounding on parameter estimation. Theor Appl Climatol.
  38. Paudyal P, Bhuju DR, Aryal M (2015) Climate change dry spell impact on agriculture in Salyantar, Dhading, Central Nepal. Nepal J Sci Technol 16:59–68CrossRefGoogle Scholar
  39. Pérez-Sánchez J, Senent-Aparicio J (2017) Analysis of meteorological droughts and dry spells in semiarid regions: a comparative analysis of probability distribution functions in the Segura Basin (SE Spain). Theor Appl Climatol 133:1–14. CrossRefGoogle Scholar
  40. Plummer M, Best N, Cowles K, Vines K (2006) CODA: convergence diagnosis and output analysis for MCMC. R News 6:7–11Google Scholar
  41. R Core Team (2017) R: a language and environment for statistical computingGoogle Scholar
  42. Raftery AE, Lewis SM (1992) One long run with diagnostics: implementation strategies for Markov chain Monte Carlo. Stat Sci 7:493–497. CrossRefGoogle Scholar
  43. Ratan R, Venugopal V (2013) Wet and dry spell characteristics of global tropical rainfall. Water Resour Res 49:3830–3841. CrossRefGoogle Scholar
  44. Roncoli C, Ingram K, Kirshen P (2001) The costs and risks of coping with drought: livelihood impacts and farmers’ responses in Burkina Faso. Clim Res 19:119–132. CrossRefGoogle Scholar
  45. Seleshi Y, Camberlin P (2006) Recent changes in dry spell and extreme rainfall events in Ethiopia. Theor Appl Climatol 83:181–191. CrossRefGoogle Scholar
  46. Sharma TC (1996) Simulation of the Kenyan longest dry and wet spells and the largest rain-sums using a Markov model. J Hydrol 178:55–67. CrossRefGoogle Scholar
  47. Singh D, Tsiang M, Rajaratnam B, Diffenbaugh NS (2014) Observed changes in extreme wet and dry spells during the South Asian summer monsoon season. Nat Clim Chang 4:456–461CrossRefGoogle Scholar
  48. Smith RL (2003) Statistics of extremes, with applications in environment, insurance and finance. In: Finkenstadt B, Rootzen H (eds) Extreme values in finance, telecommunications, and the environment, 1st edn. Chapman and Hall/CRC, pp 1–78Google Scholar
  49. Stephenson AG (2002) evd: extreme value distributions. R News 2:31–32Google Scholar
  50. Sushama L, Khaliq N, Laprise R (2010) Dry spell characteristics over Canada in a changing climate as simulated by the Canadian RCM. Glob Planet Chang 74:1–14CrossRefGoogle Scholar
  51. Thomas T, Nayak PC, Ghosh NC (2014) Irrigation planning for sustainable rain-fed agriculture in the drought-prone Bundelkhand region of Madhya Pradesh, India. J Water Clim Chang 5:408–426. CrossRefGoogle Scholar
  52. Vicente-Serrano SM, Beguería-Portugués S (2003) Estimating extreme dry-spell risk in the middle Ebro valley (northeastern Spain): a comparative analysis of partial duration series with a general Pareto distribution and annual maxima series with a Gumbel distribution. Int J Climatol 23:1103–1118. CrossRefGoogle Scholar
  53. Wilhite DA, Svoboda MD, Hayes MJ (2007) Understanding the complex impacts of drought: a key to enhancing drought mitigation and preparedness. Water Resour Manag 21:763–774. CrossRefGoogle Scholar
  54. Zin WZW, Jemain AA (2010) Statistical distributions of extreme dry spell in Peninsular Malaysia. Theor Appl Climatol 102:253–264. CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Matemática e Estatística, Campus Santo AntônioUniversidade Federal de São João Del-Rei – DEMAT/UFSJSao Joao Del-ReiBrazil
  2. 2.PPG em Estatística Aplicada e Biometria, Instituto de Ciências ExatasUniversidade Federal de Alfenas – ICEx/UNIFAL-MGAlfenasBrazil

Personalised recommendations