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We construct validation designs $Z_m$ aimed at estimating the integratedsquared prediction error of a given design $X_n$. Our approach is based on theminimization of a maximum mean discrepancy for a particular kernel, conditionalon $X_n$, so that sequences of nested validation designs can be constructedincrementally by kernel herding. Numerical experiments show that key featuresfor a good validation design are its space-filling properties, in order to fillthe holes left by $X_n$ and properly explore the whole design space, and thesuitable weighting of its points, since evaluations far from $X_n$ tend tooverestimate the global error. A dedicated weighting method, based on aparticular kernel, is proposed. Numerical simulations with random functionsshow the superiority the method over more traditional validation based onrandom designs, low-discrepancy sequences, or leave-one-out cross validation.

Validation design I: construction of validation designs via kernel herding