Analyzing data in complicated 3D domains: Smoothing, semiparametric regression, and functional principal component analysis

In this work, we introduce a family of methods for the analysis of data observed at locations scattered in three-dimensional (3D) domains, with possibly complicated shapes. The proposed family of methods includes smoothing, regression, and functional principal component analysis for functional signals defined over (possibly nonconvex) 3D domains, appropriately complying with the nontrivial shape of the domain. This constitutes an important advance with respect to the literature, because the available methods to analyze data observed in 3D domains rely on Euclidean distances, which are inappropriate when the shape of the domain influences the phenomenon under study. The common building block of the proposed methods is a nonparametric regression model with differential regularization. We derive the asymptotic properties of the methods and show, through simulation studies, that they are superior to the available alternatives for the analysis of data in 3D domains, even when considering domains with simple shapes. We finally illustrate an application to a neurosciences study, with neuroimaging signals from functional magnetic resonance imaging, measuring neural activity in the gray matter, a nonconvex volume with a highly complicated structure.