  Gaussian processes (GPs) are highly flexible function estimators used for
geospatial analysis, nonparametric regression, and machine learning, but they
are computationally infeasible for large datasets. Vecchia approximations of
GPs have been used to enable fast evaluation of the likelihood for parameter
inference. Here, we study Vecchia approximations of spatial predictions at
observed and unobserved locations, including obtaining joint predictive
distributions at large sets of locations. We consider a general Vecchia
framework for GP predictions, which contains some novel and some existing
special cases. We study the accuracy and computational properties of these
approaches theoretically and numerically, proving that our new methods exhibit
linear computational complexity in the total number of spatial locations. We
show that certain choices within the framework can have a strong effect on
uncertainty quantification and computational cost, which leads to specific
recommendations on which methods are most suitable for various settings. We
also apply our methods to a satellite dataset of chlorophyll fluorescence,
showing that the new methods are faster or more accurate than existing methods,
and reduce unrealistic artifacts in prediction maps.
