Abstract
Hazard assessment at a regional scale may be performed thanks to a spatial model for maxima including a characterization of the spatial dependence structure. Such a spatial model can be obtained by combining the generalized extreme-value (GEV) distribution for the univariate marginal distributions with extreme-value copulas to describe their dependence structure, as justified by the theory of multivariate extreme values. However, most high-dimensional copulas are too simplistic for a spatial application. Recently, a class of flexible extreme-value copulas called extra-parametrized Gumbel or XGumbel was proposed.The XGumbel copula combines two Gumbel copulas with weight parameters, termed extraparameters, taking values in the unit hyper-cube. In a multisite study, the copula dimension being the number of sites, the XGumbel copula quickly becomes over-parametrized. In addition, interpolation to ungauged locations is not easily achieved. We develop an extension of the XGumbel copula to the spatial framework. Our case study consists of annual maxima of daily precipitation totals at 177 gauged stations over a 57 year period in the French Mediterranean. The pattern of decrease with the distance of the strength of extremal dependence, as described by the upper tail dependence coefficient, indicates asymptotic dependence as it stabilizes at a non-zero value. In addition, non-stationarity is detected by looking at maps of the upper tail dependence coefficients with respect to a reference station. We propose a spatial model for maxima that combines a spatial regression for GEV marginals built with a vector generalized linear model and the spatialized XGumbel copula defined thanks to a spatial mapping for the extra-parameters. The mapping is designed shaped as a disk according to bivariate properties of the XGumbel copula. An Approximate Bayesian Computation(ABC) scheme that seeks to reproduce upper tail dependence coefficients for distance classes is used to infer the parameters. Evaluation of the proposed spatial model for maxima and comparison with a Brown-Resnick process, a spatial process adapted for pointwise maxima, are carried out on our case study. The spatial regression for GEV marginals is assessed in terms of return levels at six stations kept aside for validation purposes. The posterior distribution of the ABC scheme for the spatialized XGumbel yields, except in one instance, interpretable parameters. The comparisons between the spatialized XGumbel copula and the Brown-Resnick process show the following. The pattern of decrease of the upper tail dependence coefficients with the distance is well reproduced in both cases. However, the spatialized XGumbel copula has a clear non-stationary pattern in terms of the upper tail dependence coefficient inherited from the shape of the extra-parameter mapping while the Brown-Resnick process has none, by construction. Differences due to the non-stationarity of the dependence structure or the lack thereof are also notable on data-scale simulations and on 3-dimensional joint tail dependence coefficient estimates, a quantity involving trivariate distributions that could be of interest for a regional hazard analysis.