  We propose a new modeling framework for highly multivariate spatial processes
that synthesizes ideas from recent multiscale and spectral approaches with
graphical models. The basis graphical lasso writes a univariate Gaussian
process as a linear combination of basis functions weighted with entries of a
Gaussian graphical vector whose graph is estimated from optimizing an $\ell_1$
penalized likelihood. This paper extends the setting to a multivariate Gaussian
process where the basis functions are weighted with Gaussian graphical vectors.
We motivate a model where the basis functions represent different levels of
resolution and the graphical vectors for each level are assumed to be
independent. Using an orthogonal basis grants linear complexity and memory
usage in the number of spatial locations, the number of basis functions, and
the number of realizations. An additional fusion penalty encourages a
parsimonious conditional independence structure in the multilevel graphical
model. We illustrate our method on a large climate ensemble from the National
Center for Atmospheric Research's Community Atmosphere Model that involves 40
spatial processes.
