  Functional data describe a wide range of processes encountered in practice,
such as growth curves and spectral absorption. Functional regression considers
a version of regression, where both the response and covariates are functional
data. Evaluating both the functional relatedness between the response and
covariates and the relatedness of a multivariate response function can be
challenging. In this paper, we propose a solution for both these issues, by
means of a functional Gaussian graphical regression model. It extends the
notion of conditional Gaussian graphical models to partially separable
functions. For inference, we propose a double-penalized estimator.
Additionally, we present a novel adaptation of Kullback-Leibler
cross-validation tailored for graph estimators which accounts for precision and
regression matrices when the population presents one or more sub-groups, named
joint Kullback-Leibler cross-validation. Evaluation of model performance is
done in terms of Kullback-Leibler divergence and graph recovery power. We
illustrate the method on a air pollution dataset.
