Over the last twenty years, significant advancements have been made in extreme value theory (EVT), many of which have been achieved by leveraging concepts from other mathematical areas. One such example in the last five years is graph theory. This has gone both ways: extreme value theory has been used to model characteristics of graphical structure, such as their degree distribution, and graphs have been used for their stochastic representation to aid in dimensionality reduction in EVT models. The work contained in this thesis contributes to both of these areas. For characteristics of graphical structures, modelling the degree distribution (the number of edges connected to a vertex) will be our aim. In the literature, such modelling has several limitations. Some authors approximate the degree distribution, a discrete random variable, with a continuous distribution, while others use a single discrete model for the entire degree distribution, despite the data exhibiting different behaviour in the body and the tail. More concerning is the restriction of most analyses to heavy-tailed distributions. Given these issues, we propose a flexible mixture distribution, where the tail is modelled using an integer generalised Pareto distribution, to model the entire degree distribution. Consequently, the model can capture a variety of behaviour in both the body and the tail, without needing to pre-specify the rate of tail decay. For exploiting the stochastic representation of graphs, this has been achieved under the assumption of full asymptotic dependence (AD). For data exhibiting asymptotic independence (AI), the theory has been developed, but there is no statistical methodology to accompany it. We aim to develop such methodology. We achieve this in a different way than the theory suggests for computational purposes, by proposing an extension to the conditional multivariate extreme value model (CMEVM). Our extension has three ingredients: a new model for the margins of the residual distribution, a novel approach for incorporating graphical structures into the dependence structure of the residual distribution, and a step-wise inference procedure that loses no information compared to a joint estimation procedure to allow for scalable inference to high dimensions. Our results indicate the necessity for a general graphical dependence structure and a flexible dependence model when applied to river discharges in the upper Danube River basin. Treating river discharges as a multivariate problem meant we were unable to obtain predicted river flows at unobserved locations on the river network without post-hoc interpolation, which may not accurately represent river flow, particularly at confluence points. This inspired modelling river discharges as a stochastic process on a non-Euclidean space. However, this has received little attention in the EVT literature, and where it has, the model has assumed full AD. Thus, we propose a further extension of the CMEVM that models the residual distribution using the very recently developed class of Gaussian Whittle-Matern fields for metric graphs. By treating the data in this manner, we can form a geodesic distance metric on the graph and use a correlation function based on river distance to capture the dependence between locations. Consequently, we can obtain fast and fully stochastic simulations for any point on the river network. Although our results are preliminary, they offer valuable insight into potential advancements in stochastic simulations of river flows.
Aiden Farrell (Thu,) studied this question.