Methodology for High-Dimensional Multivariate Extremes

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Most people accept that there are risks in our day-to-day lives. However, we also expect that these risks are managed so that the probability of catastrophe is acceptably low, without infringing on our ability to get on with daily life. For example, we could perhaps eliminate flooding by building very high flood defences on all riverbanks, but we choose not to because this would be disproportionate to the risk: diverting money from other necessary services, and creating other inconveniences.In order to manage the risk proportionately, we need to be well informed about the probability of such rare but disastrous events. The question is how can we do this when we may never have witnessed an event of the size we with to protect against? Extreme value theory is the probabilistic theory of rare events, and provides a rational framework for drawing inference about the likelihood of future extremes given data on past extremes. Models for extremes of a single variable (e.g. river flow at a particular gauging station) are relatively well developed. However, most catastrophic events occur when extremes of different variables combine, or aggregate over space. In order to fully understand the risks we therefore need multivariate and spatial models, and in order that these models produce reliable estimates, they should be motivated by extreme value theory. The multivariate and spatial extreme value models that are commonly used today suffer from restrictive assumptions and / or can only be applied to very low dimensions (e.g. to the joint extremes of two variables). The goal of this project is to build new models for multivariate and spatial extremes that are appropriate under more general assumptions, and to extremes of a greater number of variables, so that they are more applicable to the problems of interest. This will enable better estimation of the probability of extreme events, and thus improve our management of these risks.