Statistical learning for multivariate extreme value theory

多元极值理论为模拟随机向量的尾部行为提供了一个严格的数学框架，这在气候科学、金融和工程等许多应用领域都有重要意义。分析中的一个关键挑战是对极值相关性结构的估计，该结构描述了随机向量各分量之间的尾部相关性。极端相依结构通常由一种称为角度量度的度量来表征，该度量值必须基于少量观察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维度或中等维度的随机向量，因此它们不足以分析极端事件，例如，在大量空间位置上。近年来，来自无监督学习的方法，如聚类和主成分分析(PCA)，已经适应了极端情况，并且更好地配备了处理高维环境的能力。然而，这些方法目前主要用于研究极值依赖结构。该博士项目旨在推进/发展分析高维环境中极端依赖结构的方法，并将这些技术提供的信息用于建模目的。开发的方法也应该适用于生成极端事件的合成集合，这是许多实践者感兴趣的，以帮助未来的规划。将开展的研究可侧重于以下方面：改进现有程序的稳健性；设计新的差异衡量标准，以评估产生极端事件合成集合的方法的性能；开发分析高维环境中极端情况的新方法；对现有方法和新方法的不确定性进行量化。本项目中开发的所有方法都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。本博士项目中考虑的研究课题具有一系列重要的应用，包括在极端气候条件下，例如模拟洪水或野火。改进对极端气候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，这项研究尤其及时。该项目的潜在影响与EPSRC提供经济和社会效益的目标一致。

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.