  Gaussian processes are widely used as priors for unknown functions in
statistics and machine learning. To achieve computationally feasible inference
for large datasets, a popular approach is the Vecchia approximation, which is
an ordered conditional approximation of the data vector that implies a sparse
Cholesky factor of the precision matrix. The ordering and sparsity pattern are
typically determined based on Euclidean distance of the inputs or locations
corresponding to the data points. Here, we propose instead to use a
correlation-based distance metric, which implicitly applies the Vecchia
approximation in a suitable transformed input space. The correlation-based
algorithm can be carried out in quasilinear time in the size of the dataset,
and so it can be applied even for iterative inference on unknown parameters in
the correlation structure. The correlation-based approach has two advantages
for complex settings: It can result in more accurate approximations, and it
offers a simple, automatic strategy that can be applied to any covariance, even
when Euclidean distance is not applicable. We demonstrate these advantages in
several settings, including anisotropic, nonstationary, multivariate, and
spatio-temporal processes. We also illustrate our method on multivariate
spatio-temporal temperature fields produced by a regional climate model.
