Functional data describe a wide range of processes, such as growth curves and spectral absorption. In this study, we analyze air pollution data from the In-service Aircraft for a Global Observing System, focusing on the spatial interactions among chemicals in the atmosphere and their dependence on meteorological conditions. This requires functional regression, where both response and covariates are functional objects evolving over the troposphere. Evaluating both the functional relatedness between the response and covariates and the relatedness of a multivariate response function can be challenging. We propose a solution to these challenges by introducing a functional Gaussian graphical regression model, extending conditional Gaussian graphical models to partially separable functions. To estimate the model, we propose a doubly-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.

翻译：功能数据描述了广泛的物理过程，如生长曲线和光谱吸收。本研究分析了来自全球观测系统在役飞机的空气污染数据，重点关注大气中化学物质间的空间相互作用及其对气象条件的依赖性。这需要采用功能回归方法，其中响应变量和协变量均为在对流层演化的功能对象。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的相关性可能具有挑战性。我们通过引入功能性高斯图回归模型来解决这些挑战，该模型将条件高斯图模型扩展至部分可分离函数。为估计该模型，我们提出了一种双重惩罚估计器。此外，我们提出了一种针对图估计器量身定制的新型Kullback-Leibler交叉验证方法，当总体存在一个或多个子组时，该方法能同时考虑精度矩阵和回归矩阵，我们将其命名为联合Kullback-Leibler交叉验证。模型性能评估通过Kullback-Leibler散度和图恢复能力两个维度进行。