Generalized partial least squares regression based on the penalized minimum norm projection

A regression methodology motivated by latent root regression has recently been proposed based on constrained principal component analysis (RM-CPCA). Currently, this approach is only applicable to situations where the number of observations exceeds the number of variables. The aim of this paper is to extend the RM-CPCA algorithm to the situation where the number of observations is less than the number of predictor variables. This is achieved by introducing the idea of penalized least squares regression into the minimum norm projection. Thus by appropriately modifying the RM-PCA algorithm, a stable projection results, that is able to deal with rank deficiency and multi-collinearity. The proposed methodology, the penalized minimum norm (PMN) projection algorithm, is illustrated by application to near infrared data and the results are compared with principal component regression (PCR) and partial least squares (PLS).