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 derivatives 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 connection 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. [PUBLICATION ABSTRACT]