An application of Bayesian analysis and Markov chain Monte Carlo methods to the estimation of a regional trend in annual maxima

Abstract

[1] Bayesian analysis is becoming increasingly popular in a number of fields, including hydrology. It appears to be a convenient framework for deriving complex models in agreement with both physical reality and statistical requirements. The aim of this paper is to present an application to the regional frequency analysis of extremes in a nonstationary context. A nonstationary regional model is thus proposed, together with the related hypotheses. The Bayesian inference of this model is then described. Markov chain Monte Carlo (MCMC) methods are needed for this purpose because of the dimensionality of the model and are described in this paper. The usefulness of such a model is then illustrated on a hydrological case study concerning annual maximum discharges of several sites. The advantage of regional analysis compared to at-site estimation is thus highlighted. Moreover, the Bayesian framework allows for a direct and comprehensive inference based on the posterior distribution and is able to take into account modeling uncertainties, which is particularly useful when the stationarity of a series can neither be ensured nor be totally rejected.

Non-stationary return levels of CMIP5 multi-model temperature extremes

“…This Bayesian framework integrates the information brought by a prior distribution p(θ) and the observation vector − → y into the posterior distribution of the GEV parameters. For non-stationary GEV parameter estimation, the Bayes formula can be written as (Winkler 1973;Renard et al 2006;Cheng et al 2014a, b):…”

Analysis of spatial and temporal rainfall trends in Sicily during the 1921–2012 period

“…Indeed, these methodologies can deal with incomplete datasets in trend analysis, taking advantage from the improvement of estimates precision due to the inclusion of informative prior knowledge (Coles and Tawn 1996). In spite of these advantages, the development of Bayesian models used to be limited by numerical difficulties (Renard et al 2006). …”

Use of a Gaussian copula for multivariate extreme value analysis: Some case studies in hydrology

“…Because of the complexity of this model, a Bayesian estimation scheme using MCMC algorithms is applied. Some details of these methods in a hydrological context can be found for instance in the papers by Perreault et al [47,48], Thyer et al [61], Marshall et al [41] and Renard et al [53]. The likelihood of the data is simply computed from the product of the multivariate densities of observations, i.e.…”

“…For the two remaining parameters of model 2, the exponential model described in equation (19) is fitted to the empirical correlations estimated by the method of Phoon, using a classical least-square approach. As a second step, these estimations are used as starting parameters of a Metropolis algorithm (see Renard et al [53] for a detailed description), which was run for 100 000 iterations, and whose convergence was checked using the approach suggested by Gelman et al [20]. Finally, the last 50 000 iterations are used to perform the inference.…”