  Many modern spatial models express the stochastic variation component as a
basis expansion with random coefficients. Low rank models, approximate spectral
decompositions, multiresolution representations, stochastic partial
differential equations and empirical orthogonal functions all fall within this
basic framework. Given a particular basis, stochastic dependence relies on
flexible modeling of the coefficients. Under a Gaussianity assumption, we
propose a graphical model family for the stochastic coefficients by
parameterizing the precision matrix. Sparsity in the precision matrix is
encouraged using a penalized likelihood framework. Computations follow from a
majorization-minimization approach, a byproduct of which is a connection to the
graphical lasso. The result is a flexible nonstationary spatial model that is
adaptable to very large datasets. We apply the model to two large and
heterogeneous spatial datasets in statistical climatology and recover
physically sensible graphical structures. Moreover, the model performs
competitively against the popular LatticeKrig model in predictive
cross-validation, but substantially improves the Akaike information criterion
score.
