The covariance structure of multivariate functional data can be highly
complex, especially if the multivariate dimension is large, making extension of
statistical methods for standard multivariate data to the functional data
setting quite 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 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 Gaussian graphical
model that can be identified with a sequence of finite-dimensional graphical
models, each of 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.Comment: 39 pages, 5 figure
