Bayesian Estimation for GEV-B-Spline Model


The stationarity hypothesis is essential in hydrological frequency analysis and statistical inference. This assumption is often not fulfilled for large observed datasets, especially in the case of hydro-climatic variables. The Generalized Extreme Value distribution with covariates allows to model data in the presence of non-stationarity and/or dependence on covariates. Linear and non-linear dependence structures have been proposed with the corresponding fitting approach. The objective of the present study is to develop the GEV model with B-Spline in a Bayesian framework. A Markov Chain Monte Carlo (MCMC) algorithm has been developed to estimate quantiles and their posterior distributions. The methods are tested and illustrated using simulated data and applied to meteorological data. Results indicate the better performance of the proposed Bayesian method for rainfall quantile estimation according to BIAS and RMSE criteria especially for high return period events.

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B. Nasri, S. Adlouni and T. Ouarda, "Bayesian Estimation for GEV-B-Spline Model," Open Journal of Statistics, Vol. 3 No. 2, 2013, pp. 118-128. doi: 10.4236/ojs.2013.32013.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] A. F. Jenkinson, “The Frequency Distribution of the Annual Maximum (or Minimum) of Meteorological Elements,” Quarterly Journal of the Royal Meteorological Society, Vol. 81, No. 348, 1955, pp. 158-171. doi:10.1002/qj.49708134804
[2] T. N. Thiele, “Theory of Observations,” C. and E. Layton, London, 1903, pp. 165-308.
[3] R. A. Fisher, “Moments and Product Moments of Sampling Distributions,” Proceedings of the London Mathe matical Society, Vol. 30, No. 1, 1929, pp. 199-238. doi:10.1112/plms/s2-30.1.199
[4] R. L. Smith, “Maximum Likelihood Estimation in a Class of Non-Regular Cases,” Biometrika, Vol. 72, No. 1, 1985, pp. 67-90. doi:10.1093/biomet/72.1.67
[5] J. Hosking, “L-Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics,” Journal of the Royal Statistical Society, Series B, Vol. 52, No. 1, 1990, pp. 105-124.
[6] D. J. Dupuis, “Modeling Waves of Extreme Temperature: The Changing Tails of Four Cities,” Journal of the American Statistical Association, Vol. 107, No. 497, 2012, pp. 24-39. doi:10.1080/01621459.2011.643732
[7] P. C. D. Milly, J. Betancourt, M. Falkenmark, R. M. Hirsch, Z. W. Kundzewicz, D. P. Lettenmaier and R. J. Stouffer, “Stationarity Is Dead: Whither Water Management?” Science, Vol. 319, No. 5863, 2008, pp. 573-574. doi:10.1126/science.1151915
[8] J. R. Olsen, J. R. Stedinger, N. C. Matalas and E. Z. Stakhiv, “Climate Variability and Flood Frequency Estimation for the Upper Mississippi and Lower Missouri Rivers,” Journal of the American Water Resources Association, Vol. 35, No. 6, 1999, pp. 1509-1524. doi:10.1111/j.1752-1688.1999.tb04234.x
[9] S. Coles, “An Introduction to Statistical Modeling of Extreme Values,” Springer, London, 2001.
[10] J. M. Cunderlik, V. Jourdain, T. B. M. J. Ouarda and B. Bobée, “Local Non-Stationary Flood-Duration-Frequency Modeling,” Canadian Water Resources Journal, Vol. 32, No. 1, 2007, pp. 43-58. doi:10.4296/cwrj3201043
[11] S. El Adlouni, T. B. M. J. Ouarda, X. Zhang, R. Roy and B. Bobee, “Generalized Maximum Likelihood Estimators for the Nonstationary Generalized Extreme Value Mod el,” Water Resources Research, Vol. 43, No. 3, 2007, W03410. doi:10.1029/2005WR004545
[12] Y. Hundecha, T. B. M. J. Ouarda and A. Bárdossy, “Re gional Estimation of Parameters of a Rainfall-Runoff Model at Ungauged Watersheds Using the Spatial Structures of the Parameters within a Canonical Physiographic Climatic Space,” Water Resources Research, Vol. 44, No. 1, 2008, W01427. doi:10.1029/2006WR005439
[13] S. El Adlouni and T. B. M. J. Ouarda, “Joint Bayesian Model Selection and Parameter Estimation of the Generalized Extreme Value Model with Covariates Using Birth-Death Markov Chain Monte Carlo,” Water Re sources Research, Vol. 45, No. 6, 2009, W06403. doi:10.1029/2007WR006427
[14] A. Cannon, “A Flexible Nonlinear Modelling Framework for Nonstationary Generalized Extreme Value Analysis in Hydroclimatology,” Hydrological Process, Vol. 24, No. 6, 2010, pp. 673-685. doi:10.1002/hyp.7506
[15] J. M. Cunderlik and T. B. M. J. Ouarda, “Regional Flood Duration—Frequency Modeling in a Changing Environment,” Journal of Hydrology, Vol. 318, No. 1-4, 2006, pp. 276-291. doi:10.1016/j.jhydrol.2005.06.020
[16] M. Leclerc and T. B. M. J. Ouarda, “Non-Stationary Regional Flood Frequency Analysis at Ungauged Sites,” Journal of Hydrology, Vol. 343, No. 3-4, 2007, pp. 254-265. doi:10.1016/j.jhydrol.2007.06.021
[17] S. El Adlouni and T. B. M. J. Ouarda, “Comparison of Methods for Estimating the Parameters of the Non-Stationary GEV Model,” Revue des Sciences de l’Eau, Vol. 21, No. 1, 2008, pp. 35-50.
[18] V. Chavez-Demoulin and A. Davison, “Generalized Additive Modeling of Sample Extremes,” Applied Statistics, Vol. 54, No. 1, 2005, pp. 207-222. doi:10.1111/j.1467-9876.2005.00479.x
[19] C. De Boor, “A Practical Guide to Spline,” Springer, London, 2001, 208 pp.
[20] H. G. Müller and J. L. Wang, “Density and Failure Rate Estimation,” In: F. Ruggeri, R. Kenett and F. W. Faltin, Eds., Encyclopedia of Statistics in Quality and Reliability, Wiley, Chichester, 2007, pp. 517-522.
[21] S. A. Padoan and M. P. Wand, “Mixed Model-Based Ad ditive Models for Sample Extremes Statistics and Probability,” Letters, Vol. 78, No. 17, 2008, pp. 2850-2858.
[22] R. A. Fisher and L. H. C. Tippett, “Limiting Forms of the Frequency Distribution of the Largest or Smallest Member of a Sample,” Proceedings of the Cambridge Philosophical Society, Vol. 24, No. 2, 1928, pp. 180-190. doi:10.1017/S0305004100015681
[23] E. S. Martins and J. R. Stedinger, “Generalized Maximum Likelihood GEV Quantile Estimators for Hydrologic Data,” Water Resources Research, Vol. 36, No. 3, 2000, pp. 737-744. doi:10.1029/1999WR900330
[24] S. Neville, M. J. Palmer and M. P. Wand, “Generalised Extreme Value Geoadditive Model Analysis via Mean Field Variational Bayes,” Australian and New Zealand Journal of Statistics, Vol. 53, No. 3, 2011, pp. 305-330. doi:10.1111/j.1467-842X.2011.00637.x
[25] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller and E. Teller, “Equation of State Calculations by Fast Computing Machines,” The Journal of Chemical Physics, Vol. 21, No. 6, 1953, pp. 1087-1092.
[26] 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
[27] W. R. Gilks, S. Richardson and D. J. Spiegelhalter, “Markov Chain Monte Carlo in Practice,” Chapman and Hall, London, 1996.
[28] S. El Adlouni, A. C. Favre and B. Bobée, “Comparison of Methodologies to Assess the Convergence of Markov Chain Monte Carlo Methods,” Computational Statistics and Data Analysis, Vol. 50, No. 10, 2006, pp. 2685-2701. doi:10.1016/j.csda.2005.04.018
[29] D. P. Brown and A. C. Comrie, “Sub-Regional Seasonal Precipitation Linkages to SOI and PDO in the Southwest United States,” Atmospheric Science Letters, Vol. 3, No. 2-4, 2002, pp. 94-102. doi:10.1006/asle.2002.0057
[30] M. Buchinsky, “The Dynamics of Changes in the Female Wage Distribution in the USA: A Quantile Regression Approach,” Journal of Applied Econometrics, Vol. 13, No. 1, 1998, pp. 1-30.
[31] T. H. Jagger and B. E. James, “Climatology Models for Extreme Hurricane Winds near the United States,” Journal of Climate, Vol. 19, No. 13, 2006, pp. 3220-3236. doi:10.1175/JCLI3913.1

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