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 experiments), 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 domain. 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 methods. 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. © 2015 The Authors. Published by Elsevier B.V. Peer-review under responsibility of Spatial Statistics 2015: Emerging Patterns committee.