We propose a supervised principal component regression method for relating functional responses with high dimensional predictors. Unlike the conventional principal component analysis, the proposed method builds on a newly defined expected integrated residual sum of squares, which directly makes use of the association between the functional response and the predictors. Minimizing the integrated residual sum of squares gives the supervised principal components, which is equivalent to solving a sequence of nonconvex generalized Rayleigh quotient optimization problems. We reformulate the nonconvex optimization problems into a simultaneous linear regression with a sparse penalty to deal with high dimensional predictors. Theoretically, we show that the reformulated regression problem can recover the same supervised principal subspace under certain conditions. Statistically, we establish non-asymptotic error bounds for the proposed estimators when the covariate covariance is bandable. We demonstrate the advantages of the proposed method through numerical experiments and an application to the Human Connectome Project fMRI data.

翻译：我们提出了一种监督主成分回归方法，用于将功能性响应与高维预测变量相关联。与传统的分析不同，该方法基于新定义的期望积分残差平方和，直接利用了功能性响应与预测变量之间的关联。最小化积分残差平方和得到监督主成分，这等价于求解一系列非凸广义瑞利商优化问题。我们将这一非凸优化问题重新表述为带稀疏惩罚的同步线性回归，以处理高维预测变量。理论上，我们证明在某些条件下，重新表述的回归问题能够恢复相同的监督主子空间。统计上，当协变量协方差矩阵为带状可估时，我们为所提出的估计量建立了非渐近误差界。通过数值实验以及人类连接组项目fMRI数据的应用，我们展示了所提方法的优势。