Including a Nugget Effect in Lifted Brownian Covariance Models

Kriging is widely used for modeling the response surface in computer simulations in science and engineering. A Brownian integrated (BI) covariance model and its variant, the lifted Brownian (LB) covariance model, were recently proposed as the underlying random field models for kriging, and they were shown to have attractive properties for modeling deterministic computer experiment data. With no nugget effect, kriging models will perfectly interpolate the response data, which is usually desirable for deterministic simulations. However, it is necessary to include a nugget effect when modeling stochastic simulations or when combining simulation with physical experimental data, and inclusion of a nugget effect is often beneficial, even with deterministic simulations, to avoid numerical problems. This is challenging for the LB covariance, which requires translation based on a perfectly observed response value. In this paper, we introduce a novel approach for including a nugget effect in LB covariance models in a manner that preserves their desirable properties. We also derive a number of important theoretical results, including invariance to the choice of translation point and Bayesian connections.