Graph-constrained Analysis for Multivariate Functional Data.

The manuscript considers multivariate functional data analysis with a known graphical model among the functional variables representing their conditional relationships (e.g., brain region-level fMRI data with a prespecified connectivity graph among brain regions). Functional Gaussian graphical models (GGM) used for analyzing multivariate functional data customarily estimate an unknown graphical model, and cannot preserve knowledge of a given graph. We propose a method for 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 recently for constructing optimal multivariate covariance functions that retain a given graphical model. The theoretical connection helps to design a new algorithm that leverages Dempster's covariance selection for obtaining 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 respecting the graph and retaining computational scalability. The benefits of the proposed algorithms are illustrated using empirical experiments and a neuroimaging application.