Cross-Validation Estimations of Hyper-Parameters of Gaussian Processes with Inequality Constraints

In many situations physical systems may be known to satisfy inequality constraints with respect to some or all input parameters. When building a surrogate model of this system (like in the framework of computer experiments7), one should integrate such expert knowledge inside the emulator structure. We proposed a new methodology to incorporate both equality conditions and inequality constraints into a Gaussian process emulator such that all conditional simulations satisfy the inequality constraints in the whole domain6. An estimator called mode (maximum a posteriori) is calculated and satisfies the inequality constraints.Herein we focus on the estimation of covariance hyper-parameters and cross validation methods1. We prove that these methods are suited to inequality constraints. Applied to real data in two dimensions, the numerical results show that the Leave-One-Out mean square error criterion using the mode is more efficient than the usual (unconstrained) Kriging mean.