Statistical models for forecasting reliability

计算、数据可用性和对数学的理解的进步导致了数学模型在决策过程中不可或缺的作用。这些模型，被称为模拟器，构成了一些物理现象的近似，通常使用计算机代码表示。为了使基于这些模型做出的决策可信，有必要对它们进行相应的不确定性估计。不确定性量化（UQ）是一门多学科的学科，旨在了解这种不确定性的来源，影响和大小。为了分析这一点，我们施加一个统计框架上的问题，通过构建一个仿真器，模拟器的统计模型。高斯过程（GP）模型是函数估计的非参数工具，在文献中被广泛接受为仿真的基本工具，将是本项目的重点。GP是有吸引力的，因为数学主要依赖于高斯分布的基本属性，并且它们提供了对其输出中的不确定性的自动估计。然而，由于计算复杂性和协方差矩阵的病态性，GP的实现过程中出现了问题。这导致了许多关于近似方法的研究，例如诱导点方法。GP完全通过其均值和核函数来确定，许多研究集中在核设计上。内核编码了我们先前的信念，并最终决定了我们可以估计的函数类型。这个项目的一个潜在的方向涉及利用先验知识的形状约束减少不确定性。幸运的是，GP的导数过程也是GP，其核可以相对容易地获得。因此，可以在核内包括先验导数信息以减少不确定性。已经发表的结果，表达单调性和对数的约束，通过组合的衍生信息与指标或概率单位功能。这已被推广，使全球定位系统可以被约束，以满足线性算子的约束条件的形式的偏微分方程（PDE）。这可能特别有趣，因为pdes表达了已知的物理定律，并且可能直接适用于AWE使用的模型。这项工作的延伸将考虑非线性算子约束，也许通过某种形式的线性化。废物储存容器的长期可靠性就是一个应用实例。有限元建模可以与测量数据相结合，以评估机械性能及其在负载下的失效概率。最近的研究涉及MLMC（多级蒙特卡罗）算法的发展，它通过改变模拟器的保真度来提高近似故障概率的效率。更一般地，多保真度仿真是一个活跃的研究领域。

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.