We consider the functional regression model with multivariate response and functional predictors. Compared to fitting each individual response variable separately, taking advantage of the correlation between the response variables can improve the estimation and prediction accuracy. Using information in both functional predictors and multivariate response, we identify the optimal decomposition of the coefficient functions for prediction in population level. Then we propose methods to estimate this decomposition and fit the regression model for the situations of a small and a large number $p$ of functional predictors separately. For a large $p$, we propose a simultaneous smooth-sparse penalty which can both make curve selection and improve estimation and prediction accuracy. We provide the asymptotic results when both the sample size and the number of functional predictors go to infinity. Our method can be applied to models with thousands of functional predictors and has been implemented in the R package FRegSigCom.

翻译：本文研究具有多元响应变量与函数型预测变量的函数回归模型。相较于单独拟合每个响应变量，利用响应变量间的相关性可提升估计与预测精度。通过综合函数型预测变量与多元响应变量的信息，我们在总体层面识别出用于预测的系数函数最优分解形式。随后，我们分别针对函数型预测变量数量$p$较小与较大的情况，提出了估计该分解并拟合回归模型的方法。对于较大的$p$，我们提出了一种同时具备平滑性与稀疏性的惩罚项，既能实现曲线选择，又能提高估计与预测精度。我们给出了样本量与函数型预测变量数量均趋于无穷时的渐近理论结果。本方法可应用于具有数千个函数型预测变量的模型，并已实现于R软件包FRegSigCom中。