Random Variable Estimation and Model Calibration in the Presence of Epistemic and Aleatory Uncertainties

This article presents strategies for evaluating the mean, variance, and failure probability of a response variable given measurements subject to both epistemic and aleatory uncertainties. We focus on a case in which standard sensor calibration techniques cannot be used to eliminate measurement error since the uncertainties affecting the metrology system depend upon the measurement itself (e.g., the sensor bias is not constant and the measurement noise is colored). To this end, we first characterize all possible realizations of the true response that might have led to each of such measurements. This process yields a surrogate of the data for the unobservable true response taking the form of a random variable. Each of these variables, called a Random Datum Model (RDM), is manufactured according to a measurement and to the underlying structure of the uncertainty. Several random variable estimation and model calibration techniques are used within the RDM framework to approximate and bound the three metrics of interest. In contrast to all approximations, the bounding techniques account for the irreducible prediction error caused by the uncertainty. The convergence of the predictions as a function of the number of observations available is evaluated numerically for several datasets. The model of the metrology system and the main goals of this article were taken from the Sandia uncertainty quantification challenge [1]. The framework proposed not only applies to the metrology system posed therein but to systems having uncertainties that depend arbitrarily on the measurement.