Functional regression with multivariate responses

TL;DR

The paper addresses functional regression with multivariate responses by formulating a coefficient-surface decomposition and showing how exploiting response correlations improves estimation and prediction. It introduces an optimal decomposition framework based on generalized eigenvalue problems for both single and multiple functional predictors, with regularized estimation that enforces smoothness and, in high-dimensional settings, sparsity. The authors develop two regimes (small and large $p$) and provide asymptotic results, along with a practical cross-validated mechanism implemented in the R package FRegSigCom. Through extensive simulations and applications to corn NIR data and simultaneous EEG-fMRI, the method demonstrates superior predictive performance and meaningful curve selection, even in ultra-high-dimensional settings. The work advances scalable, information-rich functional regression with thousands of predictors by leveraging multivariate response structure and flexible regularization.

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.

Figures (5)

• Figure 1: Two examples for the relative approximation errors of the optimal decomposition of $\mathbf{b}(t)$ versus the number $K$ of components, where different correlation levels indicated by $\rho$ are considered. The left plot is for the covariance matrix of $\boldsymbol{\mu}_{\mathbf{Y}|\mathbf{X}}$ with the $(i,j)$-th entry equal to $\rho^{|i-j|}$; the right plot is for the covariance matrix with diagonal elements equal to 1 and off-diagonal elements equal to $\rho$.
• Figure 2: Comparison of the components $\alpha_k(t)$ obtained by 3 methods: FPCA, FPLS and our optimal decomposition. For comparison, all components are scaled so that $\int_0^1\int_0^1\alpha(s)\Sigma(s,t)\alpha(t)dsdt=1$. Top-left: the first component $\alpha_1(t)$. Top-right: the second component. Bottom-left: the third component. Bottom-right: the relative approximation error $\boldsymbol{\mu}_{\mathbf{Y}|\mathbf{X}}$ versus the number $K$ of components for the three methods.
• Figure 3: Plots of $\log_{10}(\|\alpha_j\|_{L^2})$ versus $j$ for the three cases: $\|\alpha_j\|_{L^2}^2=0.001, c_1/j, c_2/j^2$, respectively, and $c_1$ and $c_2$ are constants such that $\|\boldsymbol{\alpha}\|_{L^2,2}=1$.
• Figure 4: The original and centered NIR spectra curves of 80 samples in the corn data.
• Figure 5: The estimates $\widehat{\alpha}_k(t)$, $1\le k\le 4$, for the corn data.