Soft-constrained interval predictor models and epistemic reliability intervals: A new tool for uncertainty quantification with limited experimental data

Abstract

Interval Predictor Models (IPMs) offer a non-probabilistic, interval-valued, characterization of the uncertainty affecting random data generating processes. IPMs are constructed directly from data, with no assumptions on the distributions of the uncertain factors driving the process, and are therefore exempt from the subjectivity induced by such a practice. The reliability of an IPM defines the probability of correct predictions for future samples and, in practice, its true value is always unknown due to finite samples sizes and limited understanding of the process. This paper proposes an overview of scenario optimization programs for the identification of IPMs. Traditional IPM identification methods are compared with a new scheme which softens the scenario constraints and exploits a trade-off between reliability and accuracy. The new methods allows prescribing predictors that achieve higher accuracy for a quantifiable reduction in the reliability. Scenario optimization theory is the mathematical tool used to prescribe formal epistemic bounds on the predictors reliability. A review of relevant theorems and bounds is proposed in this work. Scenario-based reliability bounds hold distribution-free, non asymptotically, and quantify the uncertainty affecting the model’s ability to correctly predict future data. The applicability of the new approach is tested on three examples: i) on the modelling of a trigonometric function affected by a noise term, ii) on the identification of a black-box system-controller dynamic response model and, iii) on the modelling of the vibration response of a car suspension arm crossed by a crack of unknown length. The points of strength and limitations of the new IPM are discussed based on the accuracy, computational cost, and width of the resulting epistemic bounds.