🤖 AI Summary
This study addresses regression problems involving functional predictors and multivariate responses by proposing a novel coefficient function decomposition method that explicitly leverages the interdependencies among response variables. By integrating the functional predictor structure with response correlations, the approach introduces a joint smoothing-and-sparsity penalty strategy that enhances both curve selection and estimation accuracy across settings ranging from small to large-scale scenarios—with up to thousands of functional predictors. Theoretical analysis and extensive numerical experiments demonstrate that the proposed method substantially outperforms existing alternatives. An efficient implementation is provided in the R package FRegSigCom, enabling scalable and high-dimensional functional regression modeling.
📝 Abstract
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.