Latent variable Gaussian process models: A rank-based analysis and an alternative approach

Gaussian process (GP) models have been extended to emulate expensive computer simulations with both qualitative/categorical and quantitative/continuous variables. Latent variable (LV) GP models, which have been recently developed to map each qualitative variable to some underlying numerical LVs, have strong physics-based justification and have achieved promising performance. Two versions use LVs in Cartesian (LV-Car) space and hyperspherical (LV-sph) space, respectively. Despite their success, the effects of these different LV structures are still poorly understood. This article illuminates this issue with two contributions. First, we develop a theorem on the effect of the ranks of the qualitative factor correlation matrices of mixed-variable GP models, from which we conclude that the LV-sph model restricts the interactions between the input variables and thus restricts the types of response surface data with which the model can be consistent. Second, following a rank-based perspective like in the theorem, we propose a new alternative model named LV-mix that combines the LV-based correlation structures from both LV-Car and LV-sph models to achieve better model flexibility than them. Through extensive case studies, we show that LV-mix achieves higher average accuracy compared with the existing two.