Statistical models for forecasting reliability

Advances in computing, data availability, and understanding of mathematics have resulted in the indispensable role of mathematical models in decision making processes. These models, referred to as simulators, constitute an approximation of some physical phenomena and are often represented using computer code. For decisions made based upon these models to be credible, it is necessary to present them with a corresponding uncertainty estimate. Uncertainty Quantification (UQ) is a multidisciplinary subject which aims to understand the source, effect and magnitude of this uncertainty. To analyse this, we impose a statistical framework upon the problem of interest through the construction of an emulator, a statistical model of the simulator. Gaussian Process (GP) models are nonparametric tools for function estimation which are widely accepted in the literature as essential tools for emulation and will be the focus of this project. GPs are attractive as the mathematics relies mainly on fundamental properties of Gaussian distributions, and they provide an automatic estimation of the uncertainty in their output. However, issues arise during the implementation of GPs due to computational complexity and ill-conditioning of the covariance matrix. This has led to much research involving approximation methods, for example inducing point methods.
GPs are fully determined through their mean and kernel functions, with much research focusing on kernel design. The kernel encodes our prior beliefs about, and ultimately determines, the type of function we can estimate. One potential direction for this project involves the exploitation of prior knowledge of shape constraints for the reduction of uncertainty. Fortunately, the derivative processes of GPs are also GPs, with kernels that can be obtained relatively easily. Thus, it is possible to include prior derivative information within the kernel to reduce uncertainty. There have been results published which express monotonicity and log-concavity constraints through a combination of derivative information with indicator or probit functions. This has been generalised so that GPs can be constrained to satisfy linear operator constraints in the form of partial differential equations (pdes). This may be of specific interest, as the pdes express known
physical laws and may be directly applicable to models used by AWE. An extension of this work would consider nonlinear operator constraints, perhaps via some form of linearisation. An example application is the long-term reliability of waste storage containers. Finite element modelling can be combined with measurement data to assess mechanical properties and their probability of failure under load. Recent research involves the development of an MLMC (Multi-level Monte Carlo) algorithm which improves the efficiency of approximating failure probabilities by altering the fidelity of the simulator. More generally, multi-fidelity emulation is an active area of research.

The COMPASS Centre for Doctoral Training will have the following impact.

Doctoral Students Impact.

I1. Recruit and train over 55 students and provide them with a broad and comprehensive education in contemporary Computational Statistics & Data Science, leading to the award of a PhD. The training environment will be built around a set of multilevel cohorts: a variety of group sizes, within and across year cohort activities, within and across disciplinary boundaries with internal and external partners, where statistics and computation are the common focus, but remaining sensitive to disciplinary needs. Our novel doctoral training environment will powerfully impact on students, opening their eyes to not only a range of modern technical benefits and opportunities, but on the power of team-working with people from a range of backgrounds to solve the most important problems of the day. They will learn to apply their skills to achieve impact by collaborative working with internal and external partners, such as via our Rapid Response Teams, Policy Workshops & Statistical Clinics.

I2. As well as advanced training in computational statistics and data science, our students will be impacted by exposure to, and training in, important cognate topics such as ethics, responsible innovation, equality, diversity and inclusion, policy, effective communication and dissemination, enterprise, impact and consultancy skills. It is vital for our students to understand that their training will enable them to have a powerful impact on the wider world, so, e.g., AI algorithms they develop should not be discriminatory, and statistical methodologies should be reproducible, and statistical results accurately and comprehensibly communicated to the general public and policymakers.

I3. The students will gain experience via collaborations with academic partners within the University in cognate disciplines, and a wide range of external industrial & government partners. The students will be impacted by the structured training programmes of the UK Academy of Postgraduate Training in Statistics, the Bristol Doctoral College, the Jean Golding Institute, the Alan Turing Institute and the Heilbronn Institute for Mathematical Sciences, which will be integrated into our programme.

I4. Having received an excellent training, the students will then impact powerfully on the world in their future fruitful careers, spreading excellence.

Impact on our Partners & ourselves.

I5. Direct impacts will be achieved by students engaging with, and working on projects with, our academic partners, with discipline-specific problems arising in engineering, education, medicine, economics, earth sciences, life sciences and geographical sciences, and our external partners Adarga, the Atomic Weapons Establishment, CheckRisk, EDF, GCHQ, GSK, the Office for National Statistics, Sciex, Shell UK, Trainline and the UK Space Agency. The students will demonstrate a wide range of innovation with these partners, will attract engagement from new partners, and often provide attractive future employment matches for students and partners alike.

Wider Societal Impact

I6. COMPASS will greatly benefit the UK by providing over 55 highly trained PhD graduates in an area that is known to be suffering from extreme, well-known, shortages in the people pipeline nationally. COMPASS CDT graduates will be equipped for jobs in sectors of high economic value and national priority, including data science, analytics, pharmaceuticals, security, energy, communications, government, and indeed all research labs that deal with data. Through their training, they will enable these organisations to make well-informed and statistically principled decisions that will allow them to maximise their international competitiveness and contribution to societal well-being. COMPASS will also impact positively on the wider student community, both now and sustainably into the future.