Laura Maria Sangalli

This work discusses a theory of functional spaces over metric graphs that permits the definition of penalized likelihood methods for data observed over spatial supports that are graphs. Within the considered mathematical framework, we recover classical results in functional analysis, such as a Poincaré-type inequality. This, in turn, enables us to uplift, to the present setting, the theory of some fundamental penalized likelihood methods. Specifically, we present two important classes of statistical models: nonparametric regression and nonparametric density estimation, here defined for data observed over graphs. We derive theoretical results regarding the well-posedness of the associated estimation problems and the consistency of the estimators. We also demonstrate the performance of the defined estimators with respect to state-of-the-art alternatives.

In this work we propose a novel approach for modeling spatio-temporal data characterized by group structures. In particular, we extend classical mixed effect regression models by introducing a space-time nonparametric component, regularized through a partial differential equation, to embed the physical dynamics of the underlying process, while random effects capture latent variability associated with the group structure present in the data. We propose a two-step procedure to estimate the fixed and random components of the model, relying on a functional version of the Iterative Reweighted Least Squares algorithm. We investigate the asymptotic properties of both fixed and random components, and we assess the performance of the proposed model through a simulation study, comparing it with state-of-the-art alternatives from the literature. The proposed methodology is finally applied to the study of hourly nitrogen dioxide concentration data in Lombardy (Italy), using random effects to account for measurement heterogeneity across monitoring stations equipped with different sensor technologies.