Recent developments in extreme value statistics have established the so-called geometric approach as a powerful modelling tool for multivariate extremes. We tailor these methods to the case of spatial modelling and examine their efficacy at inferring extremal dependence and performing extrapolation. The geometric approach is based around a limit set described by a gauge function, which is a key target for inference. We consider a variety of spatially-parameterised gauge functions and perform inference on them by building on the framework of Wadsworth and Campbell (2024), where extreme radii are modelled via a truncated gamma distribution. We also consider spatial modelling of the angular distribution, for which we propose two candidate models. Estimation of extreme event probabilities is possible by combining draws from the radial and angular models respectively. We compare our method with two other established frameworks for spatial extreme value analysis and show that our approach generally allows for unbiased, albeit more uncertain, inference compared to the more classical models. We apply the methodology to a space weather dataset of daily geomagnetic field fluctuations.

翻译：极值统计学的最新进展确立了所谓的几何方法作为多元极值建模的强大工具。我们将这些方法应用于空间建模场景，并检验其在推断极值依赖性和进行外推方面的有效性。几何方法基于由规范函数描述的极限集，该函数是推断的关键目标。我们考虑多种空间参数化的规范函数，并基于Wadsworth和 Campbell (2024) 的框架进行推断，其中极端半径通过截断伽马分布建模。我们还考虑了角度分布的空间建模，为此提出了两种候选模型。通过分别结合径向模型和角度模型的抽样结果，可以实现极端事件概率的估计。我们将本方法与空间极值分析中另外两种成熟框架进行比较，结果表明相较于经典模型，我们的方法通常能够实现无偏推断，尽管不确定性更高。我们将该方法应用于每日地磁场波动的空间天气数据集。