Cross-validation estimations of hyper-parameters of Gaussian processes with inequality constraints

Abstract : 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 7), 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 6. 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 1. 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.
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Hassan Maatouk, Olivier Roustant, Yann Richet. Cross-validation estimations of hyper-parameters of Gaussian processes with inequality constraints. Spatial Statistics 2015: Emerging Patterns committee, Jun 2015, Avignon, France. pp.38-44, ⟨10.1016/j.proenv.2015.07.105⟩. ⟨hal-01539509⟩

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