Probabilistic Analysis with Sparse Data

Abstract

In this paper, the problem of reliability analysis under both aleatory uncertainty (natural variability) and epistemic uncertainty (arising when the only knowledge about the random variables is sparse-point data) is addressed. First considered is epistemic uncertainty arising from a lack of knowledge of the distribution type of the random variables. To address this uncertainty in distribution type, the use of a flexible family of distributions is proposed. The Johnson family of distributions has the ability to reproduce the shape of many named continuous probability distributions and therefore alleviate the difficulty of determining an appropriate named distribution type for the random variable. Next considered is uncertainty in the distribution parameters themselves, and methods to determine probability distributions for the distribution parameters are proposed. As a result, the uncertainty in reliability estimates for limit-state functions having random variables with imprecise probability distributions as their arguments is computed. An optimization-based approach for computing the bounds on the reliability of a design that allows for the decoupling of epistemic and aleatory uncertainty analysis is proposed, enabling computationally affordable approaches to reliability analysis under aleatory and epistemic uncertainty. The proposed methods are illustrated for two problems of uncertainty quantification in the performance of hypersonic aerospace vehicles.