Statistical learning for multivariate extreme value theory

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.

Combining specialised modelling techniques with complex data analysis in order to deliver prediction with quantified uncertainties lies at the heart of many of the major challenges facing UK industry and society over the next decades. Indeed, the recent Government Office for Science report "Computational Modelling, Technological Futures, 2018" specifies putting the UK at the forefront of the data revolution as one of their Grand Challenges.

The beneficiaries of our research portfolio will include a wide range of UK industrial sectors such as the pharmaceutical industry, risk consultancy, telecommunications and advanced materials, as well as government bodies, including the NHS, the Met Office and the Environment Agency.

Examples of current impactful projects pursued by students and in collaboration with stake-holders include:

- Using machine learning techniques to develop automated assessment of psoriatic arthritis from hand X-Rays, freeing up consultants' time (with the NHS).

- Uncertainty quantification for the Neutron Transport Equation improving nuclear reactor safety (co-funded by Wood).

- Optimising the resilience and self-configuration of communication networks with the help of random graph colouring problems (co-funded by BT).

- Risk quantification of failure cascades on oil platforms by using Bayesian networks to improve safety assessment for certification (co-funded by DNV-GL).

- Krylov regularisation in a Bayesian framework for low-resolution Nuclear Magnetic Resonance to assess properties of porous media for real-time exploration (co-funded by Schlumberger).

- Machine learning methods to untangle oceanographic sound data for a variety of goals in including the protection of wildlife in shipping lanes (with the Department of Physics).

Future committed partners for SAMBa 2.0 are: BT, Syngenta, Schlumberger, DNV GL, Wood, ONS, AstraZeneca, Roche, Diamond Light Source, GKN, NHS, NPL, Environment Agency, Novartis, Cytel, Mango, Moogsoft, Willis Towers Watson.

SAMBa's core mission is to train the next generation of academic and industrial researchers with the breadth and depth of skills necessary to address these challenges. SAMBa's most sustained impact will be through the contributions these researchers make over the longer term of their careers. To set the students up with the skills needed to maximise this impact, SAMBa has developed a bespoke training experience in collaboration with industry, at the heart of its activities. Integrative Think Tanks (ITTs) are week-long workshops in which industrial partners present high-level research challenges to students and academics. All participants work collaboratively to formulate mathematical
models and questions that address the challenges. These outputs are meaningful both to the non-academic partner, and as a mechanism for identifying mathematical topics which are suitable for PhD research. Through the co-ownership of collaboratively developed projects, SAMBa has the capacity to lead industry in capitalising on recent advances in mathematics. ITTs occur twice a year and excel in the process of problem distillation and formulation, resulting in an exemplary environment for developing impactful projects.

SAMBa's impact on the student experience will be profound, with training in a broad range of mathematical areas, in team working, in academic-industrial collaborations, and in developing skills in communicating with specialist and generalist audiences about their research. Experience with current SAMBa students has proven that these skills are highly prized: "The SAMBa approach was a great template for setting up a productive, creative and collaborative atmosphere. The commitment of the students in getting involved with unfamiliar areas of research and applying their experience towards producing solutions was very impressive." - Dr Mike Marsh, Space weather researcher, Met Office.