This paper proposes three new approaches for additive functional regression models with functional responses. The first one is a reformulation of the linear regression model, and the last two are on the yet scarce case of additive nonlinear functional regression models. Both proposals are based on extensions of similar models for scalar responses. One of our nonlinear models is based on constructing a Spectral Additive Model (the word "Spectral" refers to the representation of the covariates in an $\mcal{L}_2$ basis), which is restricted (by construction) to Hilbertian spaces. The other one extends the kernel estimator, and it can be applied to general metric spaces since it is only based on distances. We include our new approaches as well as real datasets in an R package. The performances of the new proposals are compared with previous ones, which we review theoretically and practically in this paper. The simulation results show the advantages of the nonlinear proposals and the small loss of efficiency when the simulation scenario is truly linear. Finally, the supplementary material provides a visualization tool for checking the linearity of the relationship between a single covariate and the response.

翻译：本文提出了三种处理函数型响应下可加函数型回归模型的新方法。第一种方法是对线性回归模型的重新表述，后两种方法针对目前研究尚少的可加非线性函数型回归模型。两种新方法均基于标量响应型相似模型的扩展。其中一种非线性模型通过构建谱可加模型（"谱"一词指协变量在$\mcal{L}_2$基中的表示方法），该模型在构造上局限于希尔伯特空间。另一种模型扩展了核估计器，因其仅基于距离度量，可适用于一般度量空间。我们将所提新方法及其实数据集整合于一个R语言包中。通过理论回顾与实证比较，将新方法的性能与既有方法进行对比。仿真结果表明，非线性方法具有显著优势，且在真实线性场景下效率损失较小。最后，补充材料提供了可视化工具，用于检验单一协变量与响应变量之间的线性关系。