Toward A Diagnostic Toolkit for Linear Models with Gaussian-Process Distributed Random Effects

Abstract

Summary 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 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).

Approximate Reference Prior for Gaussian Random Fields

“…The starting step to obtain a convenient approximation to the integrated likelihood in (2.1) (or (2.2)) is the spectral representation of stationary random fields. Such approximation has been described and used for different purposes by Royle and Wikle (2005), Paciorek (2007) and Bose et al (2018). Unlike these works, this device is employed here to approximate the random field over a set of locations that may or may not be the sampling design.…”

“…In the expression above and in what follows, ω denotes 'angular frequency', as commonly used in statistics, rather than 'frequency', as used by Bose et al (2018).…”

“…Note that Paciorek (2007) and Bose et al (2018) assumed f ∆ ϑ (ω) = f ϑ (ω), effectively ignoring the aliasing effect. The spectral representation of stationary random fields in Z 2 states that for any u i,j = ∆(i, j) ⊤ ∈ U M it holds that where i = √ −1 and U ∆ (•) is a complex zero-mean random orthogonal measure in the plane (Yaglom, 1987).…”

“…In this work, we use the spectral approximation to stationary random fields to derive a new class of easy-to-compute default priors that approximate reference priors. Spectral approximations have been used for likelihood approximation, Bayesian inference, and model diagnostics by Royle and Wikle (2005), Paciorek (2007) and Bose, Hodges, and Banerjee (2018), among others. We use them here for default prior elicitation, but unlike previous works, we do not assume the sampling design is regular.…”

Bayesian Functional Mixed-Effects Model with Gaussian Process Responses for Wavelet Spectra of Spatiotemporal Colonic Manometry Signals

“…There are many previous publications on functional data analysis, with one of the first text books on the topic published in 1997 [16]. Functions in mixed-effects models are often represented by sums of basis functions, such as splines [17], [18], [19], sines and cosines [20], [21], or discrete wavelet bases [22], [23], [24], [25], [26]. Non-parametric forms such as Gaussian processes (GP) are more expressive but computationally expensive.…”