Optimal Penalized Function-on-Function Regression Under a Reproducing Kernel Hilbert Space Framework

ABSTRACT

Many scientific studies collect data where the response and predictor variables are both functions of time, location, or some other covariate. Understanding the relationship between these functional variables is a common goal in these studies. Motivated from two real-life examples, we present in this article a function-on-function regression model that can be used to analyze such kind of functional data. Our estimator of the 2D coefficient function is the optimizer of a form of penalized least squares where the penalty enforces a certain level of smoothness on the estimator. Our first result is the representer theorem which states that the exact optimizer of the penalized least squares actually resides in a data-adaptive finite-dimensional subspace although the optimization problem is defined on a function space of infinite dimensions. This theorem then allows us an easy incorporation of the Gaussian quadrature into the optimization of the penalized least squares, which can be carried out through standard numerical procedures. We also show that our estimator achieves the minimax convergence rate in mean prediction under the framework of function-on-function regression. Extensive simulation studies demonstrate the numerical advantages of our method over the existing ones, where a sparse functional data extension is also introduced. The proposed method is then applied to our motivating examples of the benchmark Canadian weather data and a histone regulation study. Supplementary materials for this article are available online.

Keywords:

• Function-on-Function regression
• Minimax convergence rate
• Penalized least squares
• Representer theorem
• Reproducing kernel Hilbert space

Supplementary Materials

The supplementary materials contain proofs of Theorem 1, Theorem 2, and Theorem 3.

Acknowledgments

The authors are grateful to Dr. Xiaoyu Zhang who kindly provided the histone regulation data and pertinent explanations of the experiments. The authors thank the associate editor and the reviewers for their insightful comments that have significantly improved the article.

Additional information

Funding

Sun and Ma’s research was supported by U.S. National Science Foundation under grants DMS-1440037, DMS-1440038, and DMS-1438957 and by U.S. National Institute of Health under grants R01GM122080 and R01GM113242. Du’s research was supported by U.S. National Science Foundation under grant DMS-1620945. Wang’s research was supported by U.S. National Science Foundation under grant DMS-1613060.