This paper proposes three new approaches for additive functional regression models with a functional response. The first is based on a reformulation of the linear regression model, while the last two are on the yet scarce case of additive nonlinear functional regression models. All proposals are based on extensions of similar models for scalar responses. One of the proposed nonlinear models is based on constructing a Spectral Additive Model, which is restricted to Hilbertian spaces. The other one extends the kernel estimator for functional response and more than one functional covariate. The latter can be applied to general metric spaces since it is only based on distances. The new approaches as well as real data sets are included in the developer version of R package \texttt{fda.usc} available on GitHub. 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. The supplementary material also provides a visualization tool for checking the linearity of the relationship between a single covariate and the response, as well as more simulation and data analysis results.

翻译：本文提出了三种针对带函数响应的可加函数回归模型的新方法。第一种基于线性回归模型的重新表述，后两种则针对目前尚属空白的可加非线性函数回归模型。所有方法均基于对标量响应类似模型的扩展。其中一种非线性模型通过构建谱可加模型实现，但仅限于希尔伯特空间；另一种方法则将核估计器扩展至函数响应及多个函数协变量的情形，该方法仅基于距离度量，因而可适用于一般度量空间。新方法及其实例数据集已包含在GitHub上发布的R包\texttt{fda.usc}开发版本中。本文从理论与实证两方面回顾了既有方法，并将新提出的方法与它们进行了性能比较。模拟结果显示，非线性方法具有显著优势，且当模拟场景为真实线性时，效率损失极小。补充材料还提供了用于检验单一协变量与响应之间线性关系的可视化工具，以及更多模拟和数据分析结果。