Functional data has become a commonly encountered data type. In this paper, we contribute to the literature on functional graphical modelling by extending the notion of conditional Gaussian Graphical models and proposing a double-penalized estimator by which to recover the edge-set of the corresponding graph. Penalty parameters play a crucial role in determining the precision matrices for the response variables and the regression matrices. The performance and model selection process in the proposed framework are investigated using information criteria. Moreover, we propose a novel version of the Kullback-Leibler cross-validation designed for conditional joint Gaussian Graphical Models. The evaluation of model performance is done in terms of Kullback-Leibler divergence and graph recovery power.

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