On the local metric property for Hüsler–Reiss graphical models

Many multivariate data sets exhibit a form of positive dependence, which can either appear globally between all variables or only locally within particular subgroups. A popular notion of positive dependence that allows for localized positivity is positive association. In this work we introduce a notion of positive dependence based on positive association for the parametric family of Hüsler–Reiss distributions, which can be considered as an analogue of multivariate Gaussians in multivariate extremes. Due to its parametric description we call this new notion of extremal positive dependence the metric property. As the parameter of a Hüsler–Reiss distribution is a Euclidean distance matrix, we observe that the metric property relates to research in electrical network theory and Euclidean geometry. We show that the metric property can be localized for Hüsler–Reiss graphical models and study surrogate likelihood inference for this setting. This gives rise to a two-step estimation procedure for locally metrical Hüsler–Reiss graphical models. The second step allows for a simple dual problem, which is implemented via a gradient descent algorithm. Finally, we demonstrate our results on simulated and real data.