Enhancing Reliability Analysis with Limited Observations: A Statistical Framework for System Safety Margins
Résumé
Reliability analysis of complex systems is essential for cost-effective evaluations. This study addresses scenarios where optimal system operation is analyzed using deterministic black-box models. The input vector, x, characterizes both the system and environmental conditions, while y represents the variable of interest associated to a "failure domain''. Given the uncertainties in x, this deterministic model requires a probabilistic analysis for robust safety demonstrations. Such an analysis typically involves two key steps: first, estimating the system's probability of failure (denoted by p) which is the statistical quantity of interest, and then, evaluating it against safety standards or expert knowledge. While considerable effort has been invested in proposing efficient methods for estimating p, little attention has been paid to the decision phase, which should take into account the uncertainties (e.g., in the p estimate). This work focuses on the definition and use of safety margins in system reliability analysis with a final decision making purpose, especially when the knowledge of the input vectors x is limited to a finite set of n observations. An important distinction is made between cases where n is large or small relative to 1/p. The main contributions of the paper focus on the scenario where n is relatively small, and propose two different approaches, each of which allowing the definition of reasonable safety margins. The first method focuses on the estimation of the probability distribution of the code inputs x, while the second method, closely related to extreme value theory, focuses on the influence of the direct estimation of the tail of the model output probability distribution. The developed framework is then numerically validated on two test cases.
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