Supervised Principal Component Regression for Functional Responses with High Dimensional Predictors

Abstract

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.

KEYWORDS:

• Functional data
• Nonasymptotic error bound
• Sparsity
• Supervised principal subspace

Acknowledgments

The authors thank the Editor, Associate Editor, and referees for their constructive comments and suggestions.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

Sun’s research is partially supported by the Natural Sciences and Engineering Research Council of Canada, a New Frontiers in Research Fund NFRFE-2019-00603 and a Data Sciences Institute Catalyst Grant. Kong’s research was partially supported by the Natural Sciences and Engineering Research Council of Canada and a Data Sciences Institute Catalyst Grant.