Advertisement

Bayesian Methods for Non-stationary Extreme Value Analysis

  • Benjamin Renard
  • Xun Sun
  • Michel Lang
Chapter
Part of the Water Science and Technology Library book series (WSTL, volume 65)

Abstract

Non-stationary models for extremes have attracted significant attention in recent years. These models require adapted estimation methods. Bayesian inference offers an attractive framework to estimate non-stationary models and, importantly, to quantify estimation and predictive uncertainties.

This chapter therefore focuses on the application of Bayesian inference to non-stationary extreme models. It is organized as a step-by-step building of non-stationary models of increasing generality. The principles of Bayesian inference are introduced using the simple case of a univariate and stationary distribution. The construction of at-site non-stationary models, using regression functions linking parameter values with time-varying covariates, is then presented. The difficulty of identifying non-stationary components based on the sole use of at-site data is also discussed, and motivates the construction of regional non-stationary models. Such models are based on the concept of “regional parameters”, i.e. parameters being assumed identical for all sites within a homogeneous region. The inference of regional models poses an additional difficulty compared to the at-site approach: the existence of spatial dependences makes the derivation of the inference equations challenging. A practical solution, based on the use of spatial copulas, is briefly presented. Lastly, a generalization of the “regional parameter” paradigm, based on Bayesian hierarchical modeling, is discussed.

Keywords

Posterior Distribution Markov Chain Monte Carlo Prior Distribution Bayesian Inference Generalize Extreme Value 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

Part of this work is funded by the French Research Agency (ANR) through the project EXTRAFLO (https://extraflo.cemagref.fr/). Météo France is gratefully acknowledged for providing the data.

References

  1. AghaKouchak A, Bardossy A, Habib E (2010a) Conditional simulation of remotely sensed rainfall data using a non-Gaussian v-transformed copula. Adv Water Resour 33(6):624–634. doi: 10.1016/j.advwatres.2010.02.010 CrossRefGoogle Scholar
  2. AghaKouchak A, Bardossy A, Habib E (2010b) Copula-based uncertainty modelling: application to multisensor precipitation estimates. Hydrol Process 24(15):2111–2124. doi: 10.1002/hyp. 7632 Google Scholar
  3. AghaKouchak A, Habib E, Bardossy A (2010c) A comparison of three remotely sensed rainfall ensemble generators. Atmos Res 98(2–4):387–399. doi: 10.1016/j.atmosres.2010.07.016 CrossRefGoogle Scholar
  4. AghaKouchak A, Ciach G, Habib E (2010d) Estimation of tail dependence coefficient in rainfall accumulation fields. Adv Water Resour 33(9):1142–1149. doi: 10.1016/j.advwatres.2010.07.003 CrossRefGoogle Scholar
  5. AghaKouchak A, Behrangi A, Sorooshian S, Hsu K, Amitai E (2011) Evaluation of satellite-retrieved extreme precipitation rates across the central United States. J Geophys Res Atmos 116:D02115. doi: 10.1029/2010jd014741 CrossRefGoogle Scholar
  6. Akaike H (1973) Information theory and an extension of the maximum likelihood principle. In: Petrov BN, Csaki F (eds) 2nd International symposium on information theory. Akadémiai Kiadó, BudapestGoogle Scholar
  7. Aryal SK, Bates BC, Campbell EP, Li Y, Palmer MJ, Viney NR (2009) Characterizing and modeling temporal and spatial trends in rainfall extremes. J Hydrometeorol 10(1):241–253CrossRefGoogle Scholar
  8. Bárdossy A, Li J (2008) Geostatistical interpolation using copulas. Water Resour Res 44(7):W07412. doi: 10.1029/2007wr006115 CrossRefGoogle Scholar
  9. Bernier J, Parent E, Boreux J-J (2000) Statistique pour l’environnement: traitement bayésien de l’incertitude. Technique & Documentation, Paris, 364 ppGoogle Scholar
  10. Bos CS (2002) A comparison of marginal likelihood computation methods. In: Hardle W, Ronz B (eds) Compstat2002. Physica-Verlag, HeidelbergGoogle Scholar
  11. Chib S (1995) Marginal likelihood from the Gibbs output. J Am Stat Assoc 90(432):1313–1321CrossRefGoogle Scholar
  12. Chiles J-P, Delfiner P (1999) Geostatistics: modeling spatial uncertainty. Wiley, New YorkCrossRefGoogle Scholar
  13. Clark JS (2005) Why environmental scientists are becoming Bayesians. Ecol Lett 8:2–14CrossRefGoogle Scholar
  14. Coles S (2001) An introduction to statistical modeling of extreme values. Verlag, London, 210 ppGoogle Scholar
  15. Coles SG, Powell EA (1996) Bayesian methods in extreme value modelling: a review and new developments. Int Stat Rev 64(1):119–136CrossRefGoogle Scholar
  16. Coles S, Heffernan JE, Tawn JA (1999) Dependence measures for extreme value analyses. Extremes 2:339–365CrossRefGoogle Scholar
  17. Coles S, Pericchi LR, Sisson S (2003) A fully probabilistic approach to extreme rainfall modelling. J Hydrol 273(1–4):35–50CrossRefGoogle Scholar
  18. Cooley D, Nychka D, Naveau P (2007) Bayesian spatial modeling of extreme precipitation return levels. J Am Stat Assoc 102(479):824–840CrossRefGoogle Scholar
  19. Cox DR, Isham VS, Northrop PJ (2002) Floods: some probabilistic and statistical approaches. Philos Trans R Soc Math Phys Eng Sci 360(1796):1389–1408CrossRefGoogle Scholar
  20. Cunderlik JM, Burn DH (2003) Non-stationary pooled frequency analysis. J Hydrol 276:210–223CrossRefGoogle Scholar
  21. Dalrymple T (1960) Flood frequency analyses. In: US Geological Survey (ed) Water-supply paper 1543-A. U.S. G.P.O, Washington, DCGoogle Scholar
  22. De Haan L, Pereira TT (2006) Spatial extremes: models for the stationary case. Ann Stat 34(1):146–168CrossRefGoogle Scholar
  23. Diggle PJ, Tawn JA, Moyeed RA (1998) Model-based geostatistics. J R Stat Soc Ser C Appl Stat 47:299–326CrossRefGoogle Scholar
  24. Dobson AJ (2001) An introduction to generalised linear models. Chapman & Hall, London, 240 ppCrossRefGoogle Scholar
  25. Efron B (2005) Bayesians, frequentists, and scientists. J Am Stat Assoc 100(469):1–5. doi: 10.1198/01621450500003 CrossRefGoogle Scholar
  26. El Adlouni S, Favre AC, Bobee B (2006) Comparison of methodologies to assess the convergence of Markov chain Monte Carlo methods. Comput Stat Data Anal 50(10):2685–2701CrossRefGoogle Scholar
  27. El Adlouni S, Ouarda TBMJ, Zhang X, Roy R, Bobée B (2007) Generalized maximum likelihood estimators for the nonstationary generalized extreme value model. Water Resour Res 43(3):W03410. doi: 10.1029/2005wr004545 CrossRefGoogle Scholar
  28. Garavaglia F, Lang M, Paquet E, Gailhard J, Garcon R, Renard B (2011) Reliability and robustness of a rainfall compound distribution model based on weather pattern sub-sampling. Hydrol Earth Syst Sci 15(2):519–532. doi: 10.5194/hess-15-519-2011 CrossRefGoogle Scholar
  29. Gelfand AE, Sahu SK (1999) Identifiability, improper priors, and Gibbs sampling for generalized linear models. J Am Stat Assoc 94:247–253CrossRefGoogle Scholar
  30. Gelman A (2008) Objections to Bayesian statistics. Bayesian Anal 3(3):445–450Google Scholar
  31. Gelman A, Carlin JB, Stern HS, Rubin DB (2004) Bayesian data analysis, 2nd edn. Chapman & Hall, London, 696 ppGoogle Scholar
  32. Genest C, Favre AC, Béliveau J, Jacques C (2007) Metaelliptical copulas and their use in frequency analysis of multivariate hydrological data. Water Resour Res 43:W09401. doi: 10.1029/2006WR005275
  33. Geweke J (1992) Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In: Bernardo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics 4. Oxford University Press, Oxford, pp 169–193Google Scholar
  34. Gunasekara TAG, Cunnane C (1992) Split sampling technique for selecting a flood frequency-analysis procedure. J Hydrol 130(1–4):189–200CrossRefGoogle Scholar
  35. Haario H, Saksman E, Tamminen J (2001) An adaptive metropolis algorithm. Bernoulli 7(2):223–242CrossRefGoogle Scholar
  36. Haario H, Saksman E, Tamminen J (2005) Componentwise adaptation for high dimensional MCMC. Comput Stat 20(2):265–273CrossRefGoogle Scholar
  37. Hanel M, Buishand TA, Ferro CAT (2009) A nonstationary index flood model for precipitation extremes in transient regional climate model simulations. J Geophys Res Atmos 114:D15107. doi: 10.1029/2009jd011712 CrossRefGoogle Scholar
  38. Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57:97–109CrossRefGoogle Scholar
  39. Heffernan JE, Tawn JA (2004) A conditional approach for multivariate extreme values. J R Stat Soc 66:497–546CrossRefGoogle Scholar
  40. Interagency Advisory Committee on Water Data (1982) Guidelines for determining flood-flow frequency: Bulletin 17B of the Hydrology Subcommittee. U.S. Geological Survey, RestonGoogle Scholar
  41. Jeffreys H (1946) An invariant form for the prior probability in estimation problems. Proc R Soc Lond A Math Phys Sci 186(1007):453–461CrossRefGoogle Scholar
  42. Kass RE, Raftery AE (1995) Bayes factors. J Am Stat Assoc 90(430):773–795CrossRefGoogle Scholar
  43. Katz RW, Parlange MB, Naveau P (2002) Statistics of extremes in hydrology. Adv Water Resour 25(8–12):1287–1304CrossRefGoogle Scholar
  44. Keef C, Svensson C, Tawn JA (2009) Spatial dependence in extreme river flows and precipitation for Great Britain. J Hydrol 378(3–4):240–252CrossRefGoogle Scholar
  45. Khaliq MN, Ouarda TBMJ, Ondo JC, Gachon P, Bobee B (2006) Frequency analysis of a sequence of dependent and/or non-stationary hydro-meteorological observations: a review. Water Resour Res 329(3–4):534–552Google Scholar
  46. Kuczera G (1999) Comprehensive at-site flood frequency analysis using Monte Carlo Bayesian inference. Water Resour Res 35(5):1551–1557CrossRefGoogle Scholar
  47. Laio F, Tamea S (2007) Verification tools for probabilistic forecasts of continuous hydrological variables. Hydrol Earth Syst Sci 11(4):1267–1277CrossRefGoogle Scholar
  48. Lima CHR, Lall U (2009) Hierarchical Bayesian modeling of multisite daily rainfall occurrence: rainy season onset, peak, and end. Water Resour Res 45:W07422. doi: 10.1029/2008WR007485
  49. Lima CHR, Lall U (2010) Spatial scaling in a changing climate: a hierarchical Bayesian model for non-stationary multi-site annual maximum and monthly streamflow. J Hydrol 383(3–4):307–318CrossRefGoogle Scholar
  50. Lunn DJ, Thomas A, Best N, Spiegelhalter D (2000) WinBUGS – a Bayesian modelling framework: concepts, structure, and extensibility. Stat Comput 10(4):325–337CrossRefGoogle Scholar
  51. Maraun D, Rust HW, Osborn TJ (2010) Synoptic airflow and UK daily precipitation extremes development and validation of a vector generalised linear model. Extremes 13(2):133–153. doi: 10.1007/s10687-010-0102-x CrossRefGoogle Scholar
  52. Marshall L, Nott D, Sharma A (2004) A comparative study of Markov chain Monte Carlo methods for conceptual rainfall-runoff modeling. Water Resour Res 40:W02501. doi: 10.1029/2003WR002378
  53. Martin AD, Quinn KM, Park JH (2011) MCMCpack: Markov chain Monte Carlo in R. J Stat Softw 42(9)Google Scholar
  54. Martins ES, Stedinger JR (2000) Generalized maximum-likelihood generalized extreme-value quantile estimators for hydrologic data. Water Resour Res 36(3):737–744CrossRefGoogle Scholar
  55. Merz R, Bloschl G (2008a) Flood frequency hydrology: 1. Temporal, spatial, and causal expansion of information. Water Resour Res 44:W08432. doi: 10.1029/2007WR006744
  56. Merz R, Bloschl G (2008b) Flood frequency hydrology: 2. Combining data evidence. Water Resour Res 44:W08433. doi: 10.1029/2007WR006745
  57. Metropolis N, Ulam S (1949) The Monte Carlo method. J Am Stat Assoc 44:335–341CrossRefGoogle Scholar
  58. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21:1087–1092CrossRefGoogle Scholar
  59. Meylan P, Favre A-C, Musy A (2008) Hydrologie fréquentielle: Une science prédictive. Presses polytechniques et universitaires romandes, Lausanne, 173 ppGoogle Scholar
  60. Micevski T (2007) Nonhomogeneity in eastern Australian flood frequency data: identification and regionalisation. PhD thesis, University of Newcastle, Newcastle, Australia, 129 ppGoogle Scholar
  61. Micevski T, Kuczera G, Franks SW (2006) A Bayesian hierarchical regional flood model. Paper presented at 30th hydrology and water resources symposium, Engineers Australia, Launceston, Tas, Australia, 4–7 DecemberGoogle Scholar
  62. Mikosch T (2005) How to model multivariate extremes if one must? Stat Neerl 59(3):324–338CrossRefGoogle Scholar
  63. Naveau P, Cooley D, Poncet P (2005) Spatial extremes analysis in climate studies. Paper presented at extreme value analysis, Gothenburg, SwedenGoogle Scholar
  64. Padoan SA, Ribatet M, Sisson SA (2010) Likelihood-based inference for max-stable processes. J Am Stat Assoc 105(489):263–277CrossRefGoogle Scholar
  65. Parent E, Bernier J (2003) Encoding prior experts judgments to improve risk analysis of extreme hydrological events via POT modeling. J Hydrol 283(1–4):1–18CrossRefGoogle Scholar
  66. Perreault L (2000) Analyse bayésienne rétrospective d’une rupture dans les séquences de variables aléatoires hydrologiques. PhD thesis, ENGREF/INRS-Eau, 200 ppGoogle Scholar
  67. Perreault L, Bernier J, Bobee B, Parent E (2000a) Bayesian change-point analysis in hydrometeorological time series. Part 2. Comparison of change-point models and forecasting. J Hydrol 235(3–4):242–263CrossRefGoogle Scholar
  68. Perreault L, Bernier J, Bobee B, Parent E (2000b) Bayesian change-point analysis in hydrometeorological time series. Part 1. The normal model revisited. J Hydrol 235(3–4):221–241CrossRefGoogle Scholar
  69. Perreault L, Parent E, Bernier J, Bobee B, Slivitzky M (2000c) Retrospective multivariate Bayesian change-point analysis: a simultaneous single change in the mean of several hydrological sequences. Stoch Environ Res Risk Assess 14(4–5):243–261CrossRefGoogle Scholar
  70. Plummer M, Best N, Cowles K, Vines K (2006) CODA: convergence diagnosis and output analysis for MCMC. R News 6(1):7–11Google Scholar
  71. Pujol N, Neppel L, Sabatier R (2007) Regional tests for trend detection in maximum precipitation series in the French Mediterranean region. Hydrol Sci J J Sci Hydrol 52(5):956–973Google Scholar
  72. Raftery AE (1996) Approximate Bayes factors and accounting for model uncertainty in generalized linear models. Biometrika 83(2):251–266CrossRefGoogle Scholar
  73. Reis DS, Stedinger JR, Martins ES (2005) Bayesian generalized least squares regression with application to log Pearson type 3 regional skew estimation. Water Resour Res 41(10)Google Scholar
  74. Renard B, Lang M (2007) Use of a Gaussian copula for multivariate extreme value analysis: some case studies in hydrology. Adv Water Resour 30(4):897–912CrossRefGoogle Scholar
  75. Renard B, Garreta V, Lang M (2006) An application of Bayesian analysis and MCMC methods to the estimation of a regional trend in annual maxima. Water Resour Res 42(12)Google Scholar
  76. Renard B, Kavetski D, Thyer M, Kuczera G, Franks SW (2010) Understanding predictive uncertainty in hydrologic modeling: the challenge of identifying input and structural errors. Water Resour Res 46:W05521. doi: 10.1029/2009WR008328
  77. Ribatet M, Sauquet E, Gresillon JM, Ouarda TBMJ (2006) A regional Bayesian POT model for flood frequency analysis. Stoch Environ Res Risk Assess 21(4):327–339CrossRefGoogle Scholar
  78. Robert CP (2001) The Bayesian choice: from decision-theoretic motivations to computational implementation. Springer, New YorkGoogle Scholar
  79. Robert CP, Casella G (2004) Monte Carlo statistical methods. Springer, New York, 650 ppGoogle Scholar
  80. Salvadori G, De Michele C, Kottegoda NT, Rosso R (2007) Extremes in nature: an approach using copulas. Springer, Dordrecht, 292 ppGoogle Scholar
  81. Schalther M, Tawn JA (2003) A dependence measure for multivariate and spatial extreme values: properties and inference. Biometrika 90(1):139–156CrossRefGoogle Scholar
  82. Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6(2):461–464CrossRefGoogle Scholar
  83. Spiegelhalter DJ, Best NG, Carlin BR, van der Linde A (2002) Bayesian measures of model complexity and fit. J R Stat Soc Ser B Stat Methodol 64:583–616CrossRefGoogle Scholar
  84. Stedinger JR (1983) Design-events with specified flood risk. Water Resour Res 19(2):511–522CrossRefGoogle Scholar
  85. Stedinger JR, Tasker GD (1985) Regional hydrologic analysis: 1. Ordinary, weighted and generalized least squares compared. Water Resour Res 21(9):1421–1432, [Correction, Water Resour Res 1422(1425): 1844, 1986.]CrossRefGoogle Scholar
  86. Strupczewski WG, Kaczmarek Z (2001) Non-stationary approach to at-site flood frequency modelling II. Weighted least squares estimation. J Hydrol 248(1–4):143–151CrossRefGoogle Scholar
  87. Strupczewski WG, Singh VP, Feluch W (2001) Non-stationary approach to at-site flood frequency modelling I. Maximum likelihood estimation. J Hydrol 248(1–4):123–142CrossRefGoogle Scholar
  88. Thyer M, Kuczera G (2000) Modeling long-term persistence in hydroclimatic time series using a hidden state Markov model. Water Resour Res 36(11):3301–3310CrossRefGoogle Scholar
  89. Thyer M, Kuczera G (2003a) A hidden Markov model for modelling long-term persistence in multi-site rainfall time series. 2. Real data analysis. J Hydrol 275(1–2):27–48CrossRefGoogle Scholar
  90. Thyer M, Kuczera G (2003b) A hidden Markov model for modelling long-term persistence in multi-site rainfall time series 1. Model calibration using a Bayesian approach. J Hydrol 275(1–2):12–26CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Irstea, UR HHLY, Hydrology-HydraulicsLyonFrance

Personalised recommendations