@steinchenanitescufiltering and used to good effect in Section \[sec4\] here could perhaps be of value for approximating $\log\det K$, but we do not explore that possibility. See @aune for some recent developments on stochastic approximation of log determinants of positive definite matrices.

Let us consider the four approaches of spectral methods, low-rank approximations, covariance tapering and composite likelihood in turn. Spectral approximations to the likelihood can be fast and accurate for gridded data \[@whittle [@guyon; @dahlhaus]\], although even for gridded data they may require some prefiltering to work well \[@stein1995\]. In addition, the approximations tend to work less well as the number of dimensions increase \[@dahlhaus\] and thus may be problematic for space–time data, especially if the number of spatial dimensions is three. Spectral approximations have been proposed for ungridded data \[@fuentes\], but they do not work as well as they do for gridded data from either a statistical or computational perspective, especially if large subsets of observations do not form a regular grid. Furthermore, in contrast to the approach we propose here, there appears to be no easy way of improving the approximations by doing further calculations, nor is it clear how to assess the loss of efficiency by 