Selection of principal variables through a modified Gram–Schmidt process with and without supervision

In various situations requiring empirical model building from highly multivariate measurements, modelling based on partial least squares regression (PLSR) may often provide efficient low‐dimensional model solutions. In unsupervised situations, the same may be true for principal component analysis (PCA). In both cases, however, it is also of interest to identify subsets of the measured variables useful for obtaining sparser but still comparable models without significant loss of information and performance. In the present paper, we propose a voting approach for sparse overall maximisation of variance analogous to PCA and a similar alternative for deriving sparse regression models influenced closely related to the PLSR method. Both cases yield pivoting strategies for a modified Gram–Schmidt process and its corresponding (partial) QR‐factorisation of the underlying data matrix to manage the variable selection process. The proposed methods include score and loading plot possibilities that are acknowledged for providing efficient interpretations of the related PCA and PLS models in chemometric applications. The article discusses two variable selection methods ‐ Principal variable selection (PVS) for unsupervised learning, and Principal regression variables (PRV) for supervised learning. Both methods select variables in a greedy manner based on a voting procedure, and yield pivoting strategies for a modified Gram‐Schmidt process and its corresponding (partial) QR‐factorisation of the underlying data matrix. The proposed methods include score‐ and loading plot possibilities.