X‐tended target projection (XTP)—comparison with orthogonal partial least squares (OPLS) and PLS post‐processing by similarity transformation (PLS + ST)

Abstract

Target projection (TP) also called target rotation (TR) was introduced to facilitate interpretation of latent-variable regression models. Orthogonal partial least squares (OPLS) regression and PLS post-processing by similarity transform (PLS R ST) represent two alternative algorithms for the same purpose. In addition, OPLS and PLS R ST provide components to explain systematic variation in X orthogonal to the response. We show, that for the same number of components, OPLS and PLS þ ST provide score and loading vectors for the predictive latent variable that are the same as for TP except for a scaling factor. Furthermore, we show how the TP approach can be extended to become a hybrid of latent-variable (LV) regression and exploratory LV analysis and thus embrace systematic variation in X unrelated to the response. Principal component analysis (PCA) of the residual variation after removal of the target component is here used to extract the orthogonal components, but X-tended TP (XTP) permits other criteria for decomposition of the residual variation. If PCA is used for decomposing the orthogonal variation in XTP, the variance of the major orthogonal components obtained for OPLS and XTP is observed to be almost the same, showing the close relationship between the methods. The XTP approach is tested and compared with OPLS for a three-component mixture analyzed by infrared spectroscopy and a multicomponent mixture measured by near infrared spectroscopy in a reactor.

OnPLS—a novel multiblock method for the modelling of predictive and orthogonal variation

“…Subsequently, the POSC method was reported by other authors, e.g. in Reference [43]. Numerous alternatives to and variations on OPLS and POSC exist, some of which are equivalent to one of these methods and others similar to one or both [44][45][46].…”

Interpretation of partial least squares regression models by means of target projection and selectivity ratio plots

“…Since the predictive vector from OPLS [4] and ST þ PLS [7] are the same as the TP component except for a scaling factor [8], our results are valid also for those algorithms.…”

“…Ergon proved that the score and loading vectors obtained from OPLS and PLS þ ST were identical. Kvalheim et al [8] recently proved that the predictive component for both OPLS and PLS þ ST was identical to the predictive component produced by TP except for a scaling factor.…”

History, philosophy and mathematical basis of the latent variable approach – from a peculiarity in psychology to a general method for analysis of multivariate data

“…OPLS represents a way to partition the variation in a data matrix into systematic variation patterns unrelated to the response (orthogonal variation) and the predictive pattern covarying with the response. Similar division can be achieved by extending the TP model by using various LV methods to study the orthogonal part …”