Group kernels for Gaussian process metamodels with categorical inputs

Abstract

Gaussian processes (GPs) 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 $G \ll L$ 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. We study the positive definiteness of such matrices to encourage their practical use. The hierarchical group/level structure, equivalent to a nested Bayesian linear model, provides a parameterization of valid block matrices $T$. The same model can then 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. The positive definiteness of ${T}$ is equivalent to the positive definiteness of a smaller matrix of size $G$, obtained by averaging each block. The model is applied to a problem in nuclear waste analysis, where one of the categorical inputs is atomic number, which has more than 90 levels.