Automatic Propagation of Uncertainties

Motivated by problems in metrology, we consider a numerical evaluation program y = f(x) as a model for a measurement process. We use a probability density function to represent the uncertainties in the inputs x and examine some of the consequences of using Automatic Differentiation to propagate these uncertainties to the outputs y. We show how to use a combination of Taylor series propagation and interval partitioning to obtain confidence intervals and ellipsoids based on unbiased estimators for means and covariances of the outputs, even in the case where f is sharply non-linear, and even when the level of confidence required makes the use of Monte Carlo techniques computationally problematic.