Envelope Model for Function-on-Function Linear Regression

Abstract

The envelope model is a recently developed methodology for multivariate analysis that enhances estimation accuracy by exploiting the relation between the mean and eigenstructure of the covariance matrix. We extend the envelope model to function-on-function linear regression, where the response and the predictor are assumed to be random functions in Hilbert spaces. We use a double envelope structure to accommodate the eigenstructures of the covariance operators for both the predictor and the response. The central idea is to establish a one-to-one relation between the functional envelope model and the multivariate envelope model and estimate the latter using an existing method. We also developed the asymptotic theories, confidence and prediction bands, an order determination method along with its consistency, and a characterization of the efficiency gain by the proposed model. Simulation comparisons with the standard function-on-function regression and data applications show significant improvement by our method in terms of cross-validated prediction error. Supplementary materials for this article are available online.

KEYWORDS:

• Confidence and prediction bands
• Covariance operator
• Gaussian process in Hilbert space
• Karhunen-Loeve expansion
• Lattice
• Order determination

Supplementary Materials

The supplementary materials contain proof of theorems, lemmas, and propositions, details on estimation and asymptotics as well as additional simulations.

Acknowledgments

The authors are grateful for the careful review and helpful suggestions from two anonymous reviewers and Associate Editor.

Additional information

Funding

The research of Zhihua Su is partly supported by Simon’s Foundation grant 632688. The research of Bing Li supported is partly by National Science Foundation grant DMS-2210775.