Spatial Extreme Value Research Articles

Detection of non-stationarity in precipitation extremes using a max-stable process model

Non-stationarity in extreme precipitation at sub-daily and daily timescales is assessed using a spatial extreme value model based on max-stable process theory. This approach, which was developed to simulate spatial fields comprising observations from multiple point locations, significantly increases the precision of a statistical inference compared to standard univariate methods. Applying the technique to a field of annual maxima derived from 30 sub-daily gauges in east Australia from 1965 to 2005, we find a statistically significant increase of 18% for 6-min rainfall over this period, with smaller increases for longer duration events. We also find an increase of 5.6% and 22.5% per degree of Australian land surface temperature and global sea surface temperature at 6-min durations, respectively, again with smaller scaling relationships for longer durations. In contrast, limited change could be observed in daily rainfall at most locations, with the exception of a statistically significant decline of 7.4% per degree land surface temperature in southwest Western Australia. These results suggest both the importance of better understanding changes to precipitation at the sub-daily timescale, as well as the need to more precisely simulate temporal variability by accounting for the spatial nature of precipitation in the statistical model.

Testing the independence of maxima: from bivariate vectors to spatial extreme fields

Characterizing the behaviour of multivariate or spatial extreme values is of fundamental interest to understand how extreme events tend to occur. In this paper we propose to test for the asymptotic independence of bivariate maxima vectors. Our test statistic is derived from a madogram, a notion classically used in geostatistics to capture spatial structures. The test can be applied to bivariate vectors, and a generalization to the spatial context is proposed. For bivariate vectors, a comparison to the test by Falk and Michel (Ann Inst Stat Math 58:261–290, 2006) is conducted through a simulation study. In the spatial case, special attention is paid to pairwise dependence. A multiple test procedure is designed to determine at which lag asymptotic independence takes place. This new procedure is based on the bootstrap distribution of the number of times the null hypothesis is rejected. It is then tested on maxima of three classical spatial models and finally applied to two climate datasets.

Continuous Spatial Process Models for Spatial Extreme Values

We propose a hierarchical modeling approach for explaining a collection of point-referenced extreme values. In particular, annual maxima over space and time are assumed to follow generalized extreme value (GEV) distributions, with parameters μ, σ, and ξ specified in the latent stage to reflect underlying spatio-temporal structure. The novelty here is that we relax the conditionally independence assumption in the first stage of the hierarchial model, an assumption which has been adopted in previous work. This assumption implies that realizations of the the surface of spatial maxima will be everywhere discontinuous. For many phenomena including, e.g., temperature and precipitation, this behavior is inappropriate. Instead, we offer a spatial process model for extreme values that provides mean square continuous realizations, where the behavior of the surface is driven by the spatial dependence which is unexplained under the latent spatio-temporal specification for the GEV parameters. In this sense, the first stage smoothing is viewed as fine scale or short range smoothing while the larger scale smoothing will be captured in the second stage of the modeling. In addition, as would be desired, we are able to implement spatial interpolation for extreme values based on this model. A simulation study and a study on actual annual maximum rainfall for a region in South Africa are used to illustrate the performance of the model.

Continuous Spatial Process Models for Spatial Extreme Values

Continuous Spatial Process Models for Spatial Extreme Values

A dependence measure for multivariate and spatial extreme values: Properties and inference

We present properties of a dependence measure that arises in the study of extreme values in multivariate and spatial problems. For multivariate problems the dependence measure characterises dependence at the bivariate level, for all pairs and all higher orders up to and including the dimension of the variable. Necessary and sufficient conditions are given for subsets of dependence measures to be self‐consistent, that is to guarantee the existence of a distribution with such a subset of values for the dependence measure. For pairwise dependence, these conditions are given in terms of positive semidefinite matrices and non‐differentiable, positive definite functions. We construct new nonparametric estimators for the dependence measure which, unlike all naive nonparametric estimators, impose these self‐consistency properties. As the new estimators provide an improvement on the naive methods, both in terms of the inferential and interpretability properties, their use in exploratory extreme value analyses should aid the identification of appropriate dependence models. The methods are illustrated through an analysis of simulated multivariate data, which shows that a lack of self‐consistency is frequently a problem with the existing estimators, and by a spatial analysis of daily rainfall extremes in south‐west England, which finds a smooth decay in extremal dependence with distance.