On Circulant Embedding for Gaussian Random Fields in R

The high-dimensionality typically associated with discretized approximations to Gaussian random fields is a considerable hinderance to computationally efficient methods for their simulation. Many direct approaches require spectral decompositions of the associated covariance matrix and so are unable to complete the solving process in a timely fashion, if at all. However under certain conditions, we may construct block-circulant versions of the covariance matrix at hand thereby allowing access to fast-Fourier methods to perform the required operations with impressive speed. We demonstrate how circulant embedding and subsequent simulation can be performed directly in the R language. The approach is currently implemented in C for the R package RandomFields, and used in the recently released package lgcp. Motivated by applications dealing with spatial point processes we restrict attention to stationary Gaussian fields on R2, where sparsity of the covariance matrix cannot necessarily be assumed.