Multivariate Gaussian Process Emulators With Nonseparable Covariance Structures

Gaussian process regression models or ‘emulators’ have become popular in the statistical analysis of deterministic computer models (simulators), in particular for computationally expensive models where the emulator is used as a fast surrogate. For models with multivariate output, common practice is to specify a separable covariance structure for the Gaussian process. Though computationally convenient, this can be too restrictive, leading to poor performance of the emulator, particularly when the different simulator outputs represent different physical quantities. Also, treating the simulator outputs as independent can lead to inappropriate representations of joint uncertainty. We develop nonseparable covariance structures for Gaussian process emulators, based on the linear model of coregionalization, and convolution methods. Using two case studies, we compare the performance of these covariances structures both with standard separable covariance structures, and with emulators that assume independence between the outputs. In each case study we find that only emulators with nonseparable covariances structures have sufficient flexibility to give both good predictions and represent joint uncertainty about the simulator outputs appropriately.