Model selection based on validation criteria for Gaussian process regression: An application with highlights on the predictive variance

The manufacturing quality of industrial pieces is linked with the intrinsic concentration field of certain chemical species. Thus, from punctual measurements sparsely distributed on the piece, a robust estimation of the maximum concentration is very valuable. To this aim, Gaussian Process regression models may be used, providing useful confidence interval at each interpolation location. However, the specification of the underlying covariance function from sparse data may be arduous, both in terms of function form and hyperparameters. This specification problem, also referred to as a model selection problem, may be addressed in several ways, from expert knowledge to numerical estimation. Regarding the covariance function form, the predictivity coefficient is commonly used as a primary criteria and focuses on the prediction capabilities of the model predictor. In this paper, other criteria related to the correctness of the model variance, covariance and more generally to the whole conditional distribution are added. This methodology is applied on the Kriging of chemical measurements spatially distributed on circular domains, where several classical covariance functions are fitted and compared. The results highlight the need to include the above mentioned criteria to get a predictive model with robust uncertainty. The Kriging models are then used to quantify the uncertainty on the highest concentration via the use of conditional simulations.