Journal
SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION
Volume 8, Issue 2, Pages 775-806Publisher
SIAM PUBLICATIONS
DOI: 10.1137/18M1209386
Keywords
Gaussian process regression; categorical data; hierarchical model; kriging; qualitative data
Funding
- OQUAIDO Chair
- Isaac Newton Institute for Mathematical Sciences, Cambridge (EPSRC) [EP/K032208/1]
- EPSRC [EP/K032208/1] Funding Source: UKRI
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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 << 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.
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