Statistical models for forecasting reliability

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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.