Bayesian non-asymptotic extreme value models for environmental data

Abstract

Motivated by the analysis of extreme rainfall data, we introduce a general Bayesian hierarchical model for estimating the probability distribution of extreme values of intermittent random sequences, a common problem in geophysical and environmental science settings. The approach presented here relaxes the asymptotic assumption typical of the traditional extreme value (EV) theory, and accounts for the possible underlying variability in the distribution of event magnitudes and occurrences, which are described through a latent temporal process. Focusing on daily rainfall extremes, the structure of the proposed model lends itself to incorporating prior geo-physical understanding of the rainfall process. By means of an extensive simulation study, we show that this methodology can significantly reduce estimation uncertainty with respect to Bayesian formulations of traditional asymptotic EV methods, particularly in the case of relatively small samples. The benefits of the approach are further illustrated with an application to a large data set of 479 long daily rainfall historical records from across the continental United States. By comparing measures of in-sample and out-of-sample predictive accuracy, we find that the model structure developed here, combined with the use of all available

A hierarchical Bayesian non-asymptotic extreme value model for spatial data

“…For nj(s) we assume a binomial distribution, with a success probability λ(s) and number of trials Nt equal to the block size (e.g., Nt = 366 in our application to annual maximum daily rainfall). The decision to employ a simple parametric model is supported by the consideration that the distribution of nj mainly affects the distribution of extreme events only through its average value (Zorzetto et al, 2020). We assume that the rainfall occurrence is also affected by geographical characteristics of the sites, as previously done for the location parameters of the Gumbel distributions.…”

“…For the latent Gumbel scale parameters σγ and σ δ , quantifying the between-block variability of the Weibull parameters, we opt for independent inverse gamma distributions, with expectations equal to 25% and 5% of the respective location parameters (µ δ and µγ). This choice reflects the expectation that the scale parameter varies across years more than the shape parameter (Zorzetto et al, 2020).…”

“…In order to evaluate the performance of the proposed spatial hierarchical model we compare it with standard alternative methods. In particular, the methods used to benchmark sHMEV are Bayesian implementations of the classical generalized extreme value distribution (GEV) and the Bayesian hierarchical model (HMEV) described in Zorzetto et al (2020) . For both competing models we take informative priors.…”

“…For both competing models we take informative priors. More precisely, for the HMEV model we follow the specification of Zorzetto et al (2020), while for the GEV model the prior distribution for the shape parameter is centered around the value 0.114 and has a standard deviation of 0.125; these values have been determined from investigations of rainfall records at the global scale (Papalexiou and Koutsoyiannis, 2013).…”

“…Based on these considerations Zorzetto et al (2020) introduced a Bayesian hierarchical model for extreme values building upon the so called metastatistical extreme value (MEV) approach (Marani and Ignaccolo, 2015;Marra et al, 2018). Bayesian hierarchical models for extremes value modelling represent an active area of research (Cooley et al, 2007;Davison et al, 2012;Sang and Gelfand, 2010;Ghosh and Mallick, 2011;Bracken et al, 2018)).…”