Gaussian processes (GP) are widely used to approximate time-consuming computer codes. A reason for their success is their ability in providing probabilistic models which interpolate the data in various frameworks. This is mostly done by choosing an appropriate covariance function, or kernel.
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. We investigate parsimonious kernels defined by block covariance matrices T with constant covariances between pairs of blocks and within blocks. First, we prove that the validity of such kernels is equivalent to the positive definiteness of a smaller matrix of size G, obtained by averaging each block. Second, the hierarchical group/level structure, equivalent to a nested Bayesian linear model, provides a parameterization of valid block matrices T; In particular, the whole range of admissible negative correlations is covered. We also exhibit a wider class of kernels, obtained by relaxing the within-block structure, for which the same analysis can be done.
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. We end by giving some connections with other hierarchical / multi-resolution GPs and next steps for future research.
Keywords: Gaussian process, Categorical data, Hierarchical model, Multi-resolution, Computer experiments.
Associated research project: OQUAIDO Chair in applied mathematics, oquaido.emse.fr