Toward a diagnostic toolkit for linear models with Gaussian-process distributed random effects [article]

Maitreyee Bose, James S. Hodges, Sudipto Banerjee
2018 arXiv   pre-print
Gaussian processes (GPs) are widely used as distributions of random effects in linear mixed models, which are fit using the restricted likelihood or the closely-related Bayesian analysis. This article addresses two problems. First, we propose tools for understanding how data determine estimates in these models, using a spectral basis approximation to the GP under which the restricted likelihood is formally identical to the likelihood for a gamma-errors GLM with identity link. Second, to examine
more » ... the data's support for a covariate and to understand how adding that covariate moves variation in the outcome y out of the GP and error parts of the fit, we apply a linear-model diagnostic, the added variable plot (AVP), both to the original observations and to projections of the data onto the spectral basis functions. The spectral- and observation-domain AVPs estimate the same coefficient for a covariate but emphasize low- and high-frequency data features respectively and thus highlight the covariate's effect on the GP and error parts of the fit respectively. The spectral approximation applies to data observed on a regular grid; for data observed at irregular locations, we propose smoothing the data to a grid before applying our methods. The methods are illustrated using the forest-biomass data of Finley et al.~(2008).
arXiv:1805.01010v1 fatcat:75og37e2yfdddgafh4hqxdwwp4