Some theoretical aspects of partial least squares regression

We give a survey of partial least squares regression with one y variable from a theoretical point of view. Some general comments are made on the motivation as seen by a statistician to study particular chemometric methods, and the concept of soft modelling is criticized from the same angle. Various aspects of the PLS algorithm are considered and the population PLS model is defined. Asymptotic properties of the prediction error are briefly discussed and the relation to other regression methods are commented upon. Results indicating positive and negative properties of PLSR are mentioned, in particular the recent result of Butler, Denham and others which seem to show that PLSR can not be an optimal regression method in any reasonable way. The only possible path left towards some kind of optimality, it seems, is by first trying to find a good motivation for the population model and then possibly finding an optimal estimator under this model. Some results on this are sketched.