Supervised Principal Component Regression for Functional Responses with High Dimensional Predictors
提出一种监督主成分回归方法，用于分析高维预测变量与函数响应之间的关系，通过最小化集成残差平方和得到监督主成分，并转化为带稀疏惩罚的线性回归问题，在人类连接组项目fMRI数据上展示了优势。
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 nonasymptotic 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. Supplementary materials for this article are available online.