Extreme Value Analysis for Non-Stationary Processes

This project aims to address critical gaps in understanding extreme events, which are rare but high-impact occurrences such as floods, heavy rainfall, earthquakes, and financial crashes. Existing models for studying these events typically assume that the underlying processes are stationary, meaning their statistical properties remain constant over time. However, in reality, many of these processes are non-stationary and vary over time, seasons, or other conditions. For instance, heavy rainfall and floods follow seasonal patterns. Understanding these extreme events in non-stationary contexts is vital, as it enables more accurate predictions, better risk assessments, and the design of effective mitigation strategies. The project’s goal is to expand the existing theory of extreme values to cover more realistic scenarios. Specifically, it aims to develop mathematical foundations and statistical methods to model extreme values when the underlying processes are non-stationary. As traditional approaches assume stationarity, this project aims to debunk the long-held belief that extremes can only be fully rigorously analysed under stationary conditions. Additionally, the project considers observations over arbitrary index sets, enabling more flexibility than traditional methods, which typically rely on regularity in observations. By allowing observations to be irregular, missing, or to exhibit complex spatial-temporal structures, this approach broadens the scope of extreme value analysis to address a wider range of real-world scenarios. The practical implications are substantial, as extreme events pose significant challenges across multiple fields. For example, in climate science, enhanced models could improve disaster prediction and response, supporting efforts to minimise economic and human losses. In finance, understanding how market crashes cluster could inform better regulatory policies and risk management, improving economic stability. In insurance, a better understanding of catastrophic events could lead to more accurate risk models, fairer pricing, and ultimately, more robust industry practices. Overall, by bridging theoretical work with real-world applications, this project has the potential to impact a wide array of sectors and contribute to societal resilience in the face of increasing global challenges. The project is structured into three main work packages to systematically address these goals. The first work package (WP1) will focus on establishing mathematical foundations to understand the asymptotic behaviour of extremes in non-stationary spatio-temporal processes. We we will extend existing theories by creating models that can handle arbitrary index sets, enabling a flexible approach to spatio-temporal data. The second work package (WP2) will develop statistical methods to measure and infer the clustering effects of non-stationary extremes—where an initial extreme event triggers additional extremes within a short period—particularly focusing on cluster indices such as the extremal index. The final work package (WP3) will apply these theoretical findings to real-world data, with an initial focus on analysing the clustering of heavy rainfall in France and the UK. By leveraging publicly available datasets, this case study will illustrate the project’s practical benefits. In addition, (WP3) will study the UK electricity market, where the transition to sustainable energy sources has intensified non-stationary patterns. Forecasting extreme variations in electricity production is critical for grid operators, who must ensure sufficient reserves to prevent blackouts. To maximise the project’s impact, the research team will produce several outputs, including research articles, open-source algorithms, and conference presentations. The team will also engage stakeholders such as the Grantham Institute and the Met Office, to promote real-world applications, particularly in climate forecasting and risk assessment.