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 the notion of extremal positive association for multivariate extremes from threshold exceedances. Via a sufficient condition for extremal association, we show that extremal association generalizes extremal tree models. For H\"usler--Reiss distributions the sufficient condition permits a parametric description that we call the metric property. As the parameter of a H\"usler--Reiss distribution is a Euclidean distance matrix, the metric property relates to research in electrical network theory and Euclidean geometry. We show that the metric property can be localized with respect to a graph and study surrogate likelihood inference. This gives rise to a two-step estimation procedure for locally metrical H\"usler--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.

翻译：许多多元数据集展现出某种形式的正相依性，这种相依性可能全局性地存在于所有变量之间，也可能仅局部地存在于特定子群内部。允许局部正相依性的一个流行概念是正关联性。本文针对超阈值极值问题，引入了极端正关联性的概念。通过给出极端关联性的充分条件，我们证明极端关联性是对极端树模型的推广。对于Hüsler-Reiss分布，该充分条件允许一种参数化描述，我们称之为度量性质。由于Hüsler-Reiss分布的参数是欧氏距离矩阵，该度量性质与电网络理论和欧氏几何中的研究相关。我们证明了度量性质可以相对于图结构进行局部化处理，并研究了代理似然推断。由此产生了一个针对局部度量Hüsler-Reiss图模型的两步估计方法。第二步可简化为一个简单的对偶问题，并通过梯度下降算法实现。最后，我们在模拟数据和真实数据上展示了研究结果。