Structured Additive Regression Models for Functional Data
 F. Scheipl

Researchers are increasingly interested in regression models for functional data to relate functional observations to other variables of interest. We will discuss a comprehensive framework for additive (mixed) models for functional responses and/or functional covariates. The guiding principle is to reframe functional regression in terms of corresponding models for scalar data, allowing the adaptation of a large body of existing methods for these novel tasks. The framework encompasses  many existing as well as new models. It includes regression  for ``generalized'' functional data,  mean regression, quantile regression as well as generalized additive models for location, shape and scale (GAMLSS) for functional data. It admits many flexible linear, smooth or interaction terms of scalar and functional covariates as well as (functional) random effects and allows flexible choices of bases - in particular splines and functional principal components - and corresponding penalties for each term. It covers functional data observed on common (dense) or curve-specific (sparse) grids.
Penalized likelihood based and gradient-boosting based inference for these models are implemented in R packages refund and FDboost, respectively. We also discuss identifiability and computational complexity for the functional regression models covered and showcase tidyfun, a new R-package for exploratory functional data analysis.