Additive models for functional responses beyond L2-spaces: probability densities and shapes of plane curves

Over the last decades various statistical methods for scalar and multivariate data have been generalized to functional data. This includes (generalized) additive regression models providing a versatile toolbox for practical data problems. However, approaches to functional response regression so far are mostly designed for one-dimensional curves (often considered elements of L²-spaces) and, in particular, typically modeling (effectively) point-wise mean curves. We explore additive models for functional geometries beyond such common scenarios, coming with additional challenges in model formulation, fitting and interpretation. More specifically, such regression approaches are presented for a) probability densities in Bayes Hilbert spaces and b) shapes of plane curves considered elements of Riemannian manifolds due to their inherent invariance under, e.g., rotation.