Constrained numerical optimization of PCR/PLSR predictors

Assuming a fully known latent variables (LV) model, the optimal multivariate calibration predictor is found from Kalman filtering theory. From this follows the best possible column space for a loading weight matrix Wopt. in a predictor based on the latent variables, and thus the optimal factorization of the regressor matrix X. Although the optimal predictor cannot be directly determined in a practical case, we may still make an attempt to find it. The paper presents a simple algorithm for a constrained numerical search for a Wopt. matrix spanning the optimal column space, using a principal component analysis (PCR) or a partial least squares (PLS) factorization as a starting point. The constraint is necessary in order to avoid overfitting, and it is based on an assumption of a smooth predictor. A simulation example and data from a metal ion mixture experiment are used to demonstrate the feasibility of the proposed method.