Bayesian smoothing with Gaussian processes using Fourier basis functions in the spectralGP package

The spectral representation of stationary Gaussian processes via the Fourier basis provides a computationally efficient specification of spatial surfaces and nonparametric regression functions for use in various statistical models. I describe the representation in detail and introduce the spectralGP package in R for computations. Because of the large number of basis coefficients, some form of shrinkage is necessary; I focus on a natural Bayesian approach via a particular parameterized prior structure that approximates stationary Gaussian processes on a regular grid. I review several models from the literature for data that do not lie on a grid, suggest a simple model modification, and provide example code demonstrating MCMC sampling using the spectralGP package. I describe reasons that mixing can be slow in certain situations and provide some suggestions for MCMC techniques to improve mixing, also with example code, and some general recommendations grounded in experience.