Scaled Predictor Envelopes and Partial Least-Squares Regression

Abstract

Partial least squares (PLS) is a widely used method for prediction in applied statistics, especially in chemometrics applications. However, PLS is not invariant or equivariant under scale transformations of the predictors, which tends to limit its scope to regressions in which the predictors are measured in the same or similar units. Cook, Helland, and Su (2013) built a connection between nascent envelope methodology and PLS, allowing PLS to be addressed in a traditional likelihood-based framework. In this article, we use the connection between PLS and envelopes to develop a new method—scaled predictor envelopes (SPE)—that incorporates predictor scaling into PLS-type applications. By estimating the appropriate scales, the SPE estimators can offer efficiency gains beyond those given by PLS, and further reduce prediction errors. Simulations and an example are given to support the theoretical claims.

• Dimension reduction
• Envelope model
• Grassmann manifold
• Scale invariance

Acknowledgments

We are grateful to the editor, the associate editor and two referees for their insightful suggestions and comments that helped us improve the article. Research for this paper was supported in part by grant DMS-1007547 and DMS-1407460 from the United States National Science Foundation.