On the maximum likelihood training of gradient-enhanced spatial Gaussian processes

Spatial Gaussian processes, alias spatial linear models or Kriging estimators, are a
powerful and well-established tool for the design and analysis of computer experiments in a multitude
of engineering applications. A key challenge in constructing spatial Gaussian processes is the training
of the predictor by numerically optimizing its associated maximum likelihood function depending on
so-called hyper-parameters. This is well understood for standard Kriging predictors, i.e., without
considering derivative information. For gradient-enhanced Kriging predictors it is an open question
of whether to incorporate the cross-correlations between the function values and their partial deriva-
tives in the maximum likelihood estimation. In this paper it is proved that in consistency with the
model assumptions, both the autocorrelations and the aforementioned cross-correlations must be
considered when optimizing the gradient-enhanced predictor’s likelihood function. The proof works
by computational rather than probabilistic arguments and exposes as a secondary effect the connec-
tion between the direct and the indirect approach to gradient-enhanced Kriging, both of which are
widely used in applications. The theoretical findings are illustrated on an academic example as well
as on an aerodynamic engineering application.