When is multivariate kriging worthwhile? A design-geometry analysis of heterotopic multi-output Gaussian processes

Abstract

Simulation experiments, multi-fidelity computer models and monitoring networks often produce several related outputs observed at different input locations, a sampling pattern known as heterotopic. Whether a joint multivariate kriging metamodel then predicts better than separate univariate metamodels has remained unresolved: careful simulation comparisons on common designs report little or no benefit from multivariate kriging, yet the multi-fidelity and geostatistical literatures are built on the premise that auxiliary outputs help. We show that, for separable multi-output Gaussian processes, the answer is governed by the geometry of the output-specific designs. We introduce model-free diagnostics that can be computed before fitting, namely directed coverage, directed proximity and borrowing potential indices. We derive an exact identity for the oracle prediction gain of joint modelling and bound this gain using local geometry under radial functions. We further prove that the estimability of cross-output dependence is controlled by a kernel-weighted cross-design interaction mass, and extend this result component by component to the linear model of coregionalisation. One consequence is that interleaved and separated designs are not statistically equivalent, even when both have zero overlap. We combine these results into a first-order net benefit criterion for deciding when joint modelling is worthwhile. Controlled synthetic experiments, an M/M/1 queueing illustration and a case study of a multi-pollutant monitoring network turn this criterion into practical guidance.