We propose a computationally efficient inferential procedure for longitudinal function-on-function regression. The method follows a marginal three-step approach: (1) fit massive pointwise longitudinal scalar-on-function regression models, (2) smooth the resulting estimates along the bivariate functional domain, and (3) compute confidence bands using either an analytic approach for Gaussian data or a cluster bootstrap for Gaussian or non-Gaussian data. Simulation studies demonstrate that the proposed method achieves accurate estimation and valid inference, while substantially reducing computational burden compared to existing approaches. Methods are motivated by a physical activity intervention trial in older adults where high-dimensional wearable data were collected longitudinally across multiple visits. Our applications reveal significant increases in physical activity in the morning using interpersonal intervention strategies, but not intrapersonal strategies. The proposed methods are implemented in an R package.

翻译：我们提出了一种计算高效的纵向函数对函数回归推断方法。该方法采用边际三步法：(1) 拟合大规模逐点纵向标量对函数回归模型，(2) 对所得估计量沿双变量函数域进行平滑处理，(3) 通过解析方法（适用于高斯数据）或聚类自助法（适用于高斯或非高斯数据）计算置信带。模拟研究表明，所提出的方法在显著降低计算负担的同时，实现了准确的估计和有效的推断。该方法受一项针对老年人的身体活动干预试验启发，在该试验中，高维可穿戴设备数据在多次访视中纵向采集。我们的应用表明，使用人际干预策略（而非个体内策略）可显著增加早晨的身体活动量。所提出的方法已在R软件包中实现。