Statistical learning for multivariate extreme value theory

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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.