Additive Function-on-Function Regression

We study additive function-on-function regression where the mean response at a particular time point depends on the time point itself, as well as the entire covariate trajectory. We develop a computationally efficient estimation methodology based on a novel combination of spline bases with an eigenbasis to represent the trivariate kernel function. We discuss prediction of a new response trajectory, propose an inference procedure that accounts for total variability in the predicted response curves, and construct pointwise prediction intervals. The estimation/inferential procedure accommodates realistic scenarios, such as correlated error structure as well as sparse and/or irregular designs. We investigate our methodology in finite sample size through simulations and two real data applications. Supplementary material for this article is available online.本文研究加性函数对函数回归（additive function-on-function regression）问题：特定时间点处的平均响应不仅取决于该时间点本身，还依赖于完整的协变量轨迹。本文提出一种计算高效的估计方法，该方法通过样条基（spline bases）与特征基（eigenbasis）的创新性结合，用于表征三元核函数。本文讨论了新响应轨迹的预测问题，提出一种可量化预测响应曲线总变异性的推断方法，并构建了逐点预测区间（pointwise prediction intervals）。该估计与推断方法可适配多种实际场景，例如误差相关结构、稀疏观测设计以及不规则观测设计。本文通过仿真实验与两个真实数据集应用，在有限样本规模下验证了所提方法的性能。本文的补充材料可在线获取。