Forward and Inverse Approaches to Model Calibration for Uncertain Data

This article proposes a framework for calibrating parametric models according to data subject to uncertainty. Data uncertainty might be caused by a poor metrology system, measurement noise, model-form uncertainty or by the inability to directly measure the inputs and/or outputs of interest. The formulations developed, called Forward Maximum Likelihood (FML) and Inverse Maximum Likelihood (IML), are applicable to datasets with and without uncertainty. The FML approach performs the calibration in the space of the model's output thereby requiring repeated model simulations. Conversely, the IML approach leverages an ensemble of solutions to an inverse problem in order to perform the calibration in the space of the model's parameters. The potential loss of performance incurred by the IML approach is often justified by a sizable reduction in computational cost. In addition, we use chance-constrained optimization to eliminate the effects of outliers on the calibrated model. This practice yields a model that increases the likelihood of most of the data in exchange for a reduction in the likelihood of a few of the worst-performing data points. Metrics for evaluating the benefits and risks of outlier elimination are also presented.