A Poisson process reparameterisation for Bayesian inference for extremes

A common approach to modelling extreme values is to consider the excesses above a high threshold as realisations of a non-homogeneous Poisson process. While this method offers the advantage of modelling using threshold-invariant extreme value parameters, the dependence between these parameters makes estimation more dicult. We present a novel approach for Bayesian estimation of the Poisson process model parameters by reparameterising in terms of a tuning parameter m. This paper presents a method for choosing the optimal value of m that near-orthogonalises the parameters, which is achieved by minimising the correlation between the asymptotic posterior distribution of the parameters. This choice of m ensures more rapid convergence and ecient sampling from the joint posterior distribution using Markov Chain Monte Carlo methods. Samples from the parameterisation of interest are then obtained by a simple transform. Results are presented in the cases of identically and non-identically distributed models for extreme rainfall in Cumbria, UK

Modelling the spatial extent and severity of extreme European windstorms

Windstorms are a primary natural hazard affecting Europe that are commonly linked to substantial property and infrastructural damage and are responsible for the largest spatially aggregated financial losses. Such extreme winds are typically generated by extratropical cyclone systems originating in the North Atlantic and passing over Europe. Previous statistical studies tend to model extreme winds at a given set of sites, corresponding to inference in an Eulerian framework. Such inference cannot incorporate knowledge of the life cycle and progression of extratropical cyclones across the region and is forced to make restrictive assumptions about the extremal dependence structure. We take an entirely different approach which overcomes these limitations by working in a Lagrangian framework. Specifically, we model the development of windstorms over time, preserving the physical characteristics linking the windstorm and the cyclone track, the path of local vorticity maxima, and make a key finding that the spatial extent of extratropical windstorms becomes more localized as its magnitude increases irrespective of the location of the storm track. Our model allows simulation of synthetic windstorm events to derive the joint distributional features over any set of sites giving physically consistent extrapolations to rarer events. From such simulations improved estimates of this hazard can be achieved in terms of both intensity and area affected

Statistical models for extreme weather events

Western Europe is typically prone to extreme weather events during the winter months, which typically take the form of windstorms or flooding. The storm Desmond brought strong winds and heavy rain to Ireland, northern England and Scotland in December 2015, resulting in an estimated $500 million worth of damage and extensive flooding, particularly in the region of Cumbria. Accurate modelling of such extreme weather events is necessary to ensure that the societal and infrastructural risk associated with these phenomena is minimised. In statistical modelling, extreme value analysis is typically used to model the rate and size of extreme weather events. Typically, practitioners can use the outputs of such an analysis to design flood defences to a standard such that there is only a small probability that defences are breached in a given year. These models can be applied at individual sites or adapted to address questions related to the spatial extent of an event, which is important for policy makers eager to reduce the economic and societal impacts associated with extreme weather. One aim of this thesis is to improve inference with regarding to existing extreme value methodology. First, we propose a reparameterisation of the likelihood corresponding to the Poisson process model for excesses above a high threshold, which improves mixing in a Bayesian framework and ensures more rapid convergence of the parameter chains in a Markov Chain Monte Carlo routine. The Poisson process model is often preferred for modelling extremes of non-stationary processes as the parameters are invariant to the choice of threshold; our approach may increase the possibility of this model being used more widely. Second, we propose an adjustment to the likelihood when implementing a spatial hierarchical model for extremes, which accounts for the dependence in the data when estimating model uncertainty. In both cases, the improvement in inference should increase confidence among practitioners of the outputs obtained from extreme value models. The main influence of extreme weather events in winter is from the passage of low-pressure extratropical cyclones from the North Atlantic. The second aim of this thesis is to quantify the risk associated with extreme wind speed events, which we call windstorms, arising from an extratropical cyclone system. First, we develop a model capturing the spatial variation of the track associated with the cyclone, from which we can simulate synthetic tracks with the same statistical characteristics of the observed record. Second, we describe an approach for modelling the spatial extent and severity of windstorms relative to the storm track, from which we can provide improved estimates of risk associated with windstorms at individual sites and jointly over a spatial domain. The methods described in this thesis can be used to address multiple questions related to windstorm risk, that is not available using current methodology