Functional Gaussian graphical regression models for air quality data

Functional data capture a wide range of processes, including growth curves and spectral absorption patterns. In this study, we analyse air pollution data from the In-service Aircraft for a Global Observing System, focusing on the spatial interactions among atmospheric chemicals and their dependence on meteorological conditions. This analysis necessitates functional regression, where both response and covariates are functional objects evolving throughout the troposphere. Quantifying both the functional dependencies between the response and covariates and the interdependencies among multivariate response functions poses significant challenges. To address these challenges, we introduce a functional Gaussian graphical regression model, which extends conditional Gaussian graphical models to partially separable functional data. We propose a doubly penalized estimator for model inference and develop a novel adaptation of Kullback-Leibler cross-validation, specifically tailored for graphical estimators. This criterion, named joint Kullback-Leibler cross-validation, simultaneously accounts for both precision and regression matrices, particularly in scenarios where the population comprises multiple sub-groups. Model performance is evaluated in terms of Kullback-Leibler divergence and graph recovery power.