Rotational symmetry, model reduction and optimality of prediction from the PLS population model

The PLS model can be defined through a simple restriction in the PLS population algorithm. Since the corresponding restriction does not hold for the sample estimates, sample PLS cannot be optimal in any strict sense. One purpose of this paper is to show that the PLS model itself can be connected to a certain optimality property related to rotational symmetry. Almost all known prediction methods behave as we expect under the group of rotations, so this is a natural symmetry group to look at in general. The theoretically best possible prediction methods under rotational symmetry can be defined in principle. Considering the freedom of choice we are left with from this when doing model reduction, we are led in a natural way to the PLS population model. Prediction methods arising from this are complicated but can be studied.