Towards a complete identification of orthogonal variation in multiple regression from a PLS1 modeling point of view: including OPLS by a change of orthogonal basis

Abstract

It is well known that the predictions of the single response orthogonal projections to latent structures (OPLS) and the single response partial least squares regression (PLS1) regression are identical in the single‐response case. The present paper presents an approach to identification of the complete y‐orthogonal structure by starting from the viewpoint of standard PLS1 regression. Three alternative non‐deflating OPLS algorithms and a modified principal component analysis (PCA)‐driven method (including MATLAB code) is presented. The first algorithm implements a postprocessing routine of the standard PLS1 solution where QR factorization applied to a shifted version of the non‐orthogonal scores is the key to express the OPLS solution. The second algorithm finds the OPLS model directly by an iterative procedure. By a rigorous mathematical argument, we explain that orthogonal filtering is a ‘built‐in’ property of the traditional PLS1 regression coefficients. Consequently, the capabilities of OPLS with respect to improving the predictions (also for new samples) compared with PLS1 are non‐existing. The PCA‐driven method is based on the fact that truncating off one dimension from the row subspace of X results in a matrix Xorth with y‐orthogonal columns and a rank of one less than the rank of X. The desired truncation corresponds exactly to the first X deflation step of Martens non‐orthogonal PLS algorithm. The significant y‐orthogonal structure of X found by PCA of Xorth is split into two fundamental parts: one part that is significantly contributing to correct the first PLS score toward y and one part that is not. The third and final OPLS algorithm presented is a modification of Martens non‐orthogonal algorithm into an efficient dual PLS1–OPLS algorithm. Copyright © 2014 John Wiley & Sons, Ltd.