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 non-convex) 3D domains, appropriately complying with the non-trivial shape of the domain. This constitutes an important advance with respect to the literature, since the available methods to analyse data observed in 3D domains rely on Euclidean distances, that 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 grey matter, a non-convex volume with a highly complicated structure. This article is protected by copyright. All rights reserved.