Understanding Gaussian Process Fits and Some Model Building Tools Using an Approximate Form of the Restricted Likelihood

Abstract

Gaussian processes (GPs) are widely used in statistical modeling, often as random effects in a linear mixed model, with their unknowns estimated by maximizing the restricted likelihood or doing a Bayesian analysis, which are closely related. However, it is unclear how a GP's variance and range and the error variance are fit to features in the data. To get a better understanding of that, we applied the spectral approximation to the intercept-only GP. The restricted likelihood from this approximate model has a simple interpretable form, which is identical to the likelihood arising from a gamma-errors generalized linear model with the identity link. If there are covariates in the model, we regress them out and approximate the residuals using an intercept-only GP. Incorporating ideas from linear models, we propose a few tools for systematic model building in linear mixed models where the random effect is a Gaussian process. We present analyses of simulated data and forest inventory data using the spectral basis representation together with added variable plots as diagnostic tools for identifying missing covariates and assessing general goodness of fit.