  The covariance structure of multivariate functional data can be highly
complex, especially if the multivariate dimension is large, making extensions
of statistical methods for standard multivariate data to the functional data
setting challenging. For example, Gaussian graphical models have recently been
extended to the setting of multivariate functional data by applying
multivariate methods to the coefficients of truncated basis expansions.
However, a key difficulty compared to multivariate data is that the covariance
operator is compact, and thus not invertible. The methodology in this paper
addresses the general problem of covariance modeling for multivariate
functional data, and functional Gaussian graphical models in particular. As a
first step, a new notion of separability for the covariance operator of
multivariate functional data is proposed, termed partial separability, leading
to a novel Karhunen-Lo\`eve-type expansion for such data. Next, the partial
separability structure is shown to be particularly useful in order to provide a
well-defined functional Gaussian graphical model that can be identified with a
sequence of finite-dimensional graphical models, each of identical fixed
dimension. This motivates a simple and efficient estimation procedure through
application of the joint graphical lasso. Empirical performance of the method
for graphical model estimation is assessed through simulation and analysis of
functional brain connectivity during a motor task. %Empirical performance of
the method for graphical model estimation is assessed through simulation and
analysis of functional brain connectivity during a motor task.
