Exploring the latent variable space of PLS2 by post-transformation of the score matrix (ptLV)

Projection to Latent Structures (PLS) regression is largely applied in chemometrics. The most used algorithm for performing PLS is probably PLS2. PLS2 solves the problem of redundancy and collinearity in complex data sets and produces a small set of latent variables that can be used to investigate complex phenomena. However, the presence of specific cluster structures or trends in the data can drive PLS2 towards wrong directions and a redundant number of latent variables is generated. To overcome this unexpected behaviour, OSC-based methods were developed. The main idea was to use the concept of orthogonality to identified two different type of sources of structured variation which are modeled into two different subspaces: the non-predictive subspace described by latent variables orthogonal to the Y-response and the predictive subspace related to the Y-response. OSC-based methods work on the variable space producing suitable weight vectors to project the data. In this study, a new post-transformation method, called post-transformation of the Latent Variable space (ptLV), is introduced. The method generates a latent space isomorphic to that discovered by PLS2 where the non-predictive data variation is separated from the predictive one. It works on the score space and can be applied also to kernel-PLS2 (KPLS2). The relationships with post-transformation of PLS2 (ptPLS2) are investigated and a real and two simulated data sets are used to illustrate how ptLV works in practice.