Envelopes: A new chapter in partial least squares regression

Abstract

Partial least squares (PLS) regression has been a very popular method for prediction. The method can in a natural way be connected to a statistical model, which now has been extended and further developed in terms of an envelope model. Concentrating on the univariate case, several estimators of the regression vector in this model are defined, including the ordinary PLS estimator, the maximum likelihood envelope estimator, and a recently proposed Bayes PLS estimator. These are compared with respect to prediction error by systematic simulations. The simulations indicate that Bayes PLS performs well compared with the other methods. The model for partial least squares is presented in 5 ways. Three estimators in the model are introduced and compared through simulations. The ordinary partial least‐squares estimator does well, but the newly introduced Bayes estimator does better in many respects.