  We propose an extensive framework for additive regression models for
correlated functional responses, allowing for multiple partially nested or
crossed functional random effects with flexible correlation structures for,
e.g., spatial, temporal, or longitudinal functional data. Additionally, our
framework includes linear and nonlinear effects of functional and scalar
covariates that may vary smoothly over the index of the functional response. It
accommodates densely or sparsely observed functional responses and predictors
which may be observed with additional error and includes both spline-based and
functional principal component-based terms. Estimation and inference in this
framework is based on standard additive mixed models, allowing us to take
advantage of established methods and robust, flexible algorithms. We provide
easy-to-use open source software in the pffr() function for the R-package
refund. Simulations show that the proposed method recovers relevant effects
reliably, handles small sample sizes well and also scales to larger data sets.
Applications with spatially and longitudinally observed functional data
demonstrate the flexibility in modeling and interpretability of results of our
approach.
