The objective function of partial least squares regression

Abstract

A simple objective function in terms of undeflated X is derived for the latent variables of multivariate PLS regression. The objective function fits into the basic framework put forward by Burnham et al. (J. Chemometrics, 10, 31–45 (1996)). We show that PLS and SIMPLS differ in the constraint put on the length of the X-weight vector. It turns out that PLS does not penalize the length of the part of the weight vector that can be expressed as a linear combination of the preceding weights, whereas SIMPLS does. By using artificial data sets, it is shown that it depends on the data which of the two methods explains the larger amount of variance in X and Y. The objective function framework adds insight to the nature of PLS and SIMPLS and how they relate to other methods. In addition, we present an implicit deflation algorithm for PLS, explain why PLS and SIMPLS become equivalent when Ychanges from multivarite to univariate, and list some geometrical results that may also prove useful in the study of other latent variable methods. © 1998 John Wiley & Sons, Ltd.