A general framework for functional regression modelling

Abstract

Researchers are increasingly interested in regression models for functional data. This article discusses a comprehensive framework for additive (mixed) models for functional responses and/or functional covariates based on the guiding principle of reframing 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-particularly 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. A running example on a longitudinal multiple sclerosis imaging study serves to illustrate the flexibility and utility of the proposed model class. Reproducible code for this case study is made available online.

On the estimation of variance parameters in non-standard generalised linear mixed models: application to penalised smoothing

“…This situation was discussed in the paper by Rodríguez-Álvarez et al (2015b) where the SAP From left to right: results using the SOP method and the functional regression approach by Greven and Scheipl (2017). Solid red line: MS patients.…”

“…We analyse the DTI dataset, that can be found in the R-package refund (Goldsmith et al, 2016). A detailed description of the study and data can be found in Goldsmith et al (2011), Goldsmith et al (2012) and Greven and Scheipl (2017). In brief, the study aimed at comparing the white matter tracts in patients affected by multiple sclerosis (MS) and healthy individuals.…”

Prediction of Weights during Growth Stages of Onion Using Agricultural Data Analysis Method

“…We considered a functional sequence regression model with p functional covariates to determine how p environmental factors influence onion weights n over time [12][13][14]. This model is defined as follows:…”

Comments on: Inference and computation with Generalized Additive Models and their extensions

“…As an example of the extensions possible with the discussed model class, we briefly discuss a general framework of functional regression models that we proposed in Scheipl et al (2015Scheipl et al ( , 2016, Brockhaus et al (2015Brockhaus et al ( , 2016 and summarized in Greven and Scheipl (2017), with the accompanying R packages refund (Goldsmith et al 2018) and FDboost (Brockhaus et al 2020) internally using the mgcv and mboost (Hothorn et al 2010) packages for the model fitting, respectively.…”