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varycoef: An R Package to Model Spatially Varying Coefficients using Gaussian Processes

About

The R package varycoef is the software implementation of Gaussian process-based spatially varying coefficient models by Dambon et al. (2021a). It extends linear regression models such that the coefficients are depending on some coordinates in a d dimensional space, i.e., the coefficient b_j for a covariate j is depending on coordinates s and therefore of the form b_j(s). These coefficients are modeled using Gaussian processes. In most applications, the coordinates s tend to be observation locations like longitude and latitude (see Dambon et al. (2022) as an example). However, the concept can be extended in the number of dimensions or, say, using observation time points to model time-varying coefficients.

The method relies on maximum likelihood estimation. It has been optimized to work with large data sets by applying covariance tapering by Furrer et al. (2006) if necessary and allows for a moderate number of spatially varying coefficients. The R package contains methods to estimate Gaussian process-based (spatially) varying coefficient models, (spatially) predict coefficients as well as the response, and variable selection methods. Latter are based on Dambon et al. (2021b).

Getting Started

To install it, run

devtools::install_github("jakobdambon/varycoef")

for the latest version on this repository or download it from CRAN.

Model Assumptions

Note: The exact definition of the model is given in Dambon et al. (2021a).

Linear Model

Let y be the response vector, let X be the covariate matrix, and let xi be the error term sampled from a zero-mean normal distribution with variance s2. Then the linear model is given by

y = Xb + xi

with coefficient vector b. The coefficients are also called fixed effects.

Spatially Varying Coefficients

We now allow the coefficients to vary from their respective mean. Let e(s) contain the location-dependent differences. In a first step, with some slight abuse of notation, the linear model from above is extended by:

y = X(b + e(s)) + xi

Note that not all coefficients necessarily must be varying. Therefore, we introduce a second covariate matrix W to specify which coefficients are spatially varying. In the case where all coefficients are varying, we have W = X. Again with some abuse of notation, we have the SVC model:

y = Xb + We(s) + xi

Gaussian Process-based Coefficients

We assume that the spatially varying coefficients are defined by Gaussian processes. That is, the deviations per covariate e_k(s) are defined as a zero-mean Gaussian process defined by some covariance function c(d, par) that models the spatial dependence between observations using the pairwise distances. Each coefficient is parameterized by a tuple of parameters par that consists of a range and variance. One main assumption of the model is that the individual coefficients per covariate, i.e., the individual Gaussian processes, are mutually independent and independent of the error.

Connection to Mixed Effect Models

For a finite number of observations n, the model can be expressed as a so-called mixed effect model. That is, using the covariance functions of the Gaussian processes and the distance matrix of the observations, we can express each e_k(s) as a multivariate, zero-mean normal distribution. Another common name for these effects are random effects. Together with the fixed effects from an ordinary linear model, we receive a so-called mixed effect model. The assumption of mutual independence between the Gaussian allows an easy construction of the joint covariance matrix S_y of the response y.

Examples

The R package contains a vignette, which is also linked here:

• 01_Introduction

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