PLS post-processing by similarity transformation: a simple alternative to OPLS

Abstract

Several methods for orthogonal signal correction (OSC) based on pre-processing of the modeling data have been developed in recent years, and OPLS (orthogonal projections to latent structures) is a well known algorithm. The main result from these methods is a reduction in the number of …nal components in partial least squares (PLS) regression, while the predictions are virtually unchanged (identical for OPLS). This raises the question whether the same or similar results can be obtained in a more direct way using an ordinary PLS model as starting point, and as shown in the present paper this can indeed be done by use of a simple similarity transformation. This post-processing PLS+ST method is compared with OPLS, assuming a single response variable. The PLS+ST factorization of the data matrix X is just a similarity transformation of the non-orthogonalized PLS factorization, while OPLS is a similarity transformation of the orthogonalized PLS factorization. The predictions are therefore identical, but the residuals are somewhat di¤erent. A theoretically founded modi…cation of the orthogonalized PLS factorization, and a corresponding modi…cation of OPLS, leads to identical factorizations for all these methods, within similarity transformations. The PLS+ST vs. OPLS comparison also leads to an alternative post-processing method, using the ordinary PLS algorithm twice, with predetermined and permuted loading weights vectors in the second step. A limited comparison with post-processing using principal components of predictions (PCP) or canonical correlation analysis (CCA) is included.