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 coverage (confidence) intervals and ellipsoids based on unbiased estimators for means and covariances of the outputs, even where f is sharply non-linear, and even when the level of probability required makes the use of Monte Carlo techniques computationally problematic.