G-optimal grid designs for kriging models

Abstract: This work is focused on finding G-optimal designs theoretically for kriging models with two-dimensional inputs and separable exponential covariance structures. For design comparison, the notion of evenness of two-dimensional grid designs is developed. The mathematical relationship between the design and the supremum of the mean squared prediction error ($SMSPE$) function is studied and then optimal designs are explored for both prospective and retrospective design scenarios. In the case of prospective designs, the new design is developed before the experiment is conducted and the regularly spaced grid is shown to be the G-optimal design. The retrospective designs are constructed by adding or deleting points from an already existing design. Deterministic algorithms are developed to find the best possible retrospective designs (which minimizes the $SMSPE$). It is found that a more evenly spread design under the G-optimality criterion leads to the best possible retrospective design. For all the cases of finding the optimal prospective designs and the best possible retrospective designs, both frequentist and Bayesian frameworks have been considered. The proposed methodology for finding retrospective designs is illustrated with a methane flux monitoring design.