Exploring and exploiting new representations for multivariate extremes

在生活的各个方面，需要按比例减轻罕见但可能是灾难性事件造成的风险。例如，我们通过建造防御设施来保护自己免受洪水的侵袭：在这样做的过程中，我们通过建造足够高的防御设施来平衡风险和成本，使它们在生命周期内被破坏的概率很小，但又不会太高，以至于浪费金钱来消除无限小的风险。通常情况下，我们会试图防止比我们观察到的更大的极端事件，这意味着直接估计现有数据的违约概率是不可能的。极值理论是解决这一问题的一种方法：我们可以从我们已经看到的极值中学习，以估计尚未目睹的事件的概率。造成最大影响的事件通常是多变量或空间的。例如，在大风期间可能会对建筑物造成损坏，但在伴随着高降雨时，这种损坏的代价可能要高得多。同样，金融投资组合中单个要素的巨额损失比全面的多个损失更少灾难性。为了了解这些现象带来的风险，我们需要工具来模拟极端水平下过程之间的依赖性。到目前为止，有各种各样的方法，每一种都基于不同的基本假设，这些方法在多大程度上代表了良好的统计模型，这在很大程度上取决于未知的基本依赖性。在这项工作中，我们将利用新的表示和最近发现的这些方法之间的联系，以统一这些不同的方法，并提供一个单一的，可靠的策略建模多元极端。

In all aspects of life, there is a need to proportionally mitigate against the risk posed by rare but potentially catastrophic events. For example, we protect ourselves from flooding through the construction of defences: in doing so, we balance risk and cost by building these high enough such that the probability of them being breached over their lifetime is small, but not so high that money is wasted on eliminating infinitesimal risks. Usually, we will be trying to protect against extreme events that are larger than we have ever observed, meaning that direct estimation of the probability of breach from existing data is impossible. Extreme value theory is the mathematically-justified approach for tackling this problem: we can learn from extremes we have seen to estimate probabilities of events not yet witnessed.Events that cause the most impact are often multivariate or spatial in nature. For example, damage to a structure may occur during a period of high winds, but that damage could be far more costly when accompanied by high rainfall. Equally, a large loss in a single element of a financial portfolio is less disastrous than multiple losses across the board. In order to understand the risks posed by such phenomena, we need tools for modelling the dependence between processes at extreme levels. To date there are a variety of methods available, each based on different underlying assumptions, and the extent to which these represent good statistical models rests strongly on the unknown underlying dependence. In this work we will exploit novel representations and recently-uncovered links between these methods to unify these disparate methodologies and provide a single, reliable strategy for modelling multivariate extremes.