  We study the nonparametric covariance estimation of a stationary Gaussian
field X observed on a regular lattice. In the time series setting, some
procedures like AIC are proved to achieve optimal model selection among
autoregressive models. However, there exists no such equivalent results of
adaptivity in a spatial setting. By considering collections of Gaussian Markov
random fields (GMRF) as approximation sets for the distribution of X, we
introduce a novel model selection procedure for spatial fields. For all
neighborhoods m in a given collection M, this procedure first amounts to
computing a covariance estimator of X within the GMRFs of neighborhood m. Then,
it selects a neighborhood by applying a penalization strategy. The so-defined
method satisfies a nonasymptotic oracle type inequality. If X is a GMRF, the
procedure is also minimax adaptive to the sparsity of its neighborhood. More
generally, the procedure is adaptive to the rate of approximation of the true
distribution by GMRFs with growing neighborhoods.
