Group Kernels for Gaussian Process Metamodels with Categorical Inputs

Gaussian processes (GP) are widely used as a metamodel for emulating time-consuming computer codes.We focus on problems involving categorical inputs, with a potentially large number L of levels (typically several tens),partitioned in G << L groups of various sizes. Parsimonious covariance functions, or kernels, can then be defined by block covariance matrices T with constant covariances between pairs of blocks and within blocks. However, little is said about the positive definiteness of such matrices, which may limit their practical usage.In this paper, we exploit the hierarchy group/level and provide a parameterization of valid block matrices T, based on a nested Bayesian linear model. The same model can be used when the assumption within blocks is relaxed, giving a flexible parametric family of valid covariance matrices with constant covariances between pairs of blocks. As a by-product, we show that the positive definiteness of T is equivalent to the positive definiteness of a small matrix of size G, obtained by averaging each block.We illustrate with an application in nuclear engineering, where one of the categorical inputs is the atomic number in Mendeleev's periodic table and has more than 90 levels.