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.

翻译：功能数据描述了实践中遇到的各种过程，例如生长曲线和光谱吸收。功能回归考虑了一种回归的变体，其中响应变量和协变量均为功能数据。评估响应变量与协变量之间的功能相关性以及多元响应函数本身的相关性可能存在挑战。在本文中，我们通过一种功能性高斯图回归模型为这两个问题提出了解决方案。该模型将条件高斯图模型的概念推广至部分可分离函数。在推断方面，我们提出了一种双惩罚估计器。此外，我们针对图估计器提出了一种新颖的Kullback-Leibler交叉验证适应性方法，该方法在总体包含一个或多个子组时同时考虑了精度矩阵和回归矩阵，称为联合Kullback-Leibler交叉验证。模型性能评估基于Kullback-Leibler散度和图恢复能力。我们通过一个空气污染数据集展示了该方法的应用。