  We study the nonparametric covariance estimation of a stationary Gaussian
field X observed on a lattice. To tackle this issue, a neighborhood selection
procedure has been recently introduced. This procedure amounts to selecting a
neighborhood m by a penalization method and estimating the covariance of X in
the space of Gaussian Markov random fields (GMRFs) with neighborhood m. Such a
strategy is shown to satisfy oracle inequalities as well as minimax adaptive
properties. However, it suffers several drawbacks which make the method
difficult to apply in practice. On the one hand, the penalty depends on some
unknown quantities. On the other hand, the procedure is only defined for
toroidal lattices. The present contribution is threefold. A data-driven
algorithm is proposed for tuning the penalty function. Moreover, the procedure
is extended to non-toroidal lattices. Finally, numerical study illustrate the
performances of the method on simulated examples. These simulations suggest
that Gaussian Markov random field selection is often a good alternative to
variogram estimation.
