Functional Gaussian graphical models (GGM) used for analyzing multivariate
functional data customarily estimate an unknown graphical model representing
the conditional relationships between the functional variables. However, in
many applications of multivariate functional data, the graph is known and
existing functional GGM methods cannot preserve a given graphical constraint.
In this manuscript, we demonstrate how to conduct multivariate functional
analysis that exactly conforms to a given inter-variable graph. We first show
the equivalence between partially separable functional GGM and graphical
Gaussian processes (GP), proposed originally for constructing optimal
covariance functions for multivariate spatial data that retain the conditional
independence relations in a given graphical model. The theoretical connection
help design a new algorithm that leverages Dempster's covariance selection to
calculate the maximum likelihood estimate of the covariance function for
multivariate functional data under graphical constraints. We also show that the
finite term truncation of functional GGM basis expansion used in practice is
equivalent to a low-rank graphical GP, which is known to oversmooth marginal
distributions. To remedy this, we extend our algorithm to better preserve
marginal distributions while still respecting the graph and retaining
computational scalability. The insights obtained from the new results presented
in this manuscript will help practitioners better understand the relationship
between these graphical models and in deciding on the appropriate method for
their specific multivariate data analysis task. The benefits of the proposed
algorithms are illustrated using empirical experiments and an application to
functional modeling of neuroimaging data using the connectivity graph among
regions of the brain.Comment: 23 pages, 6 figure