  Undirected graphical models are widely used to model the conditional
independence structure of vector-valued data. However, in many modern
applications, for example those involving EEG and fMRI data, observations are
more appropriately modeled as multivariate random functions rather than
vectors. Functional graphical models have been proposed to model the
conditional independence structure of such functional data. We propose a
neighborhood selection approach to estimate the structure of Gaussian
functional graphical models, where we first estimate the neighborhood of each
node via a function-on-function regression and subsequently recover the entire
graph structure by combining the estimated neighborhoods. Our approach only
requires assumptions on the conditional distributions of random functions, and
we estimate the conditional independence structure directly. We thus circumvent
the need for a well-defined precision operator that may not exist when the
functions are infinite dimensional. Additionally, the neighborhood selection
approach is computationally efficient and can be easily parallelized. The
statistical consistency of the proposed method in the high-dimensional setting
is supported by both theory and experimental results. In addition, we study the
effect of the choice of the function basis used for dimensionality reduction in
an intermediate step. We give a heuristic criterion for choosing a function
basis and motivate two practically useful choices, which we justify by both
theory and experiments.
