Statistical learning for multivariate extreme value theory

多元极值理论为随机向量的尾部行为建模提供了一个严格的数学框架，这在许多应用领域都很有意义，例如气候科学，金融和工程。分析中的一个关键挑战是极值依赖结构的估计，该结构描述了随机向量的分量之间的尾部依赖。极端依赖结构通常由称为角度度量的度量来表征，该度量必须基于少量观测到的极端事件来估计。传统的角度测量建模方法往往局限于小或中等维度的随机向量，因此不足以分析极端事件，例如，在大量的空间位置，近年来，无监督学习的方法，如聚类和主成分分析（PCA），已经适应了极端情况，更好地处理高维设置。然而，这些方法目前主要被设计用于研究极值依赖结构。这个博士项目旨在推进/开发用于分析高维环境中的极值依赖结构的方法，并使用这些技术提供的信息进行建模。所制定的方法也应该适用于生成综合的极端事件集，这对许多从业人员来说是有意义的，有助于未来的规划。将进行的研究可能侧重于以下方面：提高现有程序的鲁棒性;设计新的差异措施，以评估产生综合极端事件集的方法的性能;开发新的方法来分析高维环境中的极端情况;现有和新方法的不确定性量化。在这个项目中开发的所有方法将使用精心设计的模拟研究进行严格验证，并应用于分析真实世界的环境数据集。在这个博士项目中考虑的研究课题有一系列重要的应用，包括极端气候，例如，模拟洪水或野火。改善气候极端事件的建模和理解对于最大限度地减少与此类事件相关的经济，社会和人力成本非常重要。由于气候变化，极端事件的频率和严重性正在增加，这项研究特别及时。该项目的潜在影响与EPSRC提供经济和社会效益的目标相一致。

Multivariate extreme value theory provides a rigorous mathematical framework for modelling the tail behaviour of random vectors, which is of interest in many application areas, such as climate science, finance, and engineering. One key challenge in the analysis is the estimation of the extremal dependence structure, which describes the tail dependence between the components of the random vector. The extremal dependence structure is usually characterised by a measure, called the angular measure, which must be estimated based on a small number of observed extreme events. Traditional methods for modelling the angular measure tend to be limited to random vectors of small or moderate dimension, and they are thus inadequate to analyse extreme events, for instance, over a large number of spatial locations.In recent years, methods from unsupervised learning, such as clustering and principal component analysis (PCA), have been adapted to extremes and are better-equipped to handle high-dimensional settings. However, these methods are currently mainly designed to investigate the extremal dependence structure. This PhD project aims to advance/develop methodology for analysing the extremal dependence structure in high-dimensional settings and to use the information provided by these techniques for modelling purposes. The developed approaches should also be applicable for generating synthetic sets of extreme events, which are of interest to many practitioners to help with future planning. The research to be conducted may focus on aspects such as: improving robustness of existing procedures; devising novel discrepancy measures to assess performance of approaches generating synthetic sets of extreme events; developing new methodology for analysing extremes in high-dimensional settings; uncertainty quantification for existing and novel methods. All approaches developed in this project will be rigorously validated using well-designed simulation studies and applied to analyse real-world environmental data sets.The research topic considered in this PhD project has a range of important applications, including in climate extremes, e.g., modelling floods or wildfires. Improving the modelling and understanding of climate extremes is important to minimise the economic, social, and human cost associated with such events. This research is particularly timely as the frequency and severity of extreme events is increasing due to climate change. The potential impacts of the project align with EPSRC's objective to deliver economic and social benefits.