Bridging asymptotic independence and dependence in spatial extremes using Gaussian scale mixtures

Abstract

Gaussian scale mixtures are constructed as Gaussian processes with a random variance. They have non-Gaussian marginals and can exhibit asymptotic dependence unlike Gaussian processes, which are asymptotically independent except in the case of perfect dependence. In this paper, we study in detail the extremal dependence properties of Gaussian scale mixtures and we unify and extend general results on their joint tail decay rates in both asymptotic dependence and independence cases. Motivated by the analysis of spatial extremes, we propose several flexible yet parsimonious parametric copula models that smoothly interpolate from asymptotic dependence to independence and include the Gaussian dependence as a special case. We show how these new models can be fitted to high threshold exceedances using a censored likelihood approach, and we demonstrate that they provide valuable information about tail characteristics. Our parametric approach outperforms the widely used nonparametric χ and χ statistics often used to guide model choice at an exploratory stage by borrowing strength across locations for better estimation of the asymptotic dependence class. We demonstrate the capacity of our methodology by adequately capturing the extremal properties of wind speed data collected in the Pacific Northwest, US.

Modeling Spatial Processes with Unknown Extremal Dependence Class

“…Construction (4) has superficial similarities with the Gaussian scale mixture models studied by Huser et al (2017), who multiply a Gaussian random field by a random effect that determines the extremal dependence properties. However, in (4) the latent process W (s) does not have Gaussian margins, resulting in a very different construction in practice, and need not have a Gaussian copula structure, which yields a much wider class of models.…”

“…The case δ = 1/2 may either be established independently, or as a limit, from which we get this is the survival function of a log-Gamma random variable with rate and shape parameters both equal to two. Notice that margins are here available in closed form, unlike the Gaussian scale mixture model of Huser et al (2017), or the bivariate model of Wadsworth et al (2017).…”

“…However our construction here is simpler and substantially more amenable to higher-dimensional inference. Other related work that allows for both asymptotic dependence structures in a spatial setting is the Gaussian scale mixture models proposed in the recent work of Huser et al (2017), but their models either make the transition at a boundary point of the parameter space, or are inflexible in their representation of asymptotic independence structures.…”

Extremal dependence of random scale constructions

“…Hashorva (2010) details calculation of η X assuming R to be in the Gumbel MDA, providing an alternative perspective on the derivation. The spatial model of Huser et al (2017) is also covered by this case.…”

“…On top of the support W, to obtain distributions within a particular family, R or (W 1 , W 2 ) may be specified to have a certain distribution. Where W is two-dimensional, it may sometimes be reduced to the one-dimensional case by redefining R, such as in the Gaussian scale mixtures of Huser et al (2017); other times, such as for the scale mixtures of log-Gaussian variables in Krupskii et al (2016), or the model presented in Huser and Wadsworth (2018), this cannot be done. Where W is two-dimensional, the possible constructions stemming from (1) form an especially large class, since (W 1 , W 2 ) can itself have any copula.…”