Orthogonal PLS (O-PLS) and related algorithms

The concept of orthogonalized partial least squares regression or, better, as it was originally named, orthogonalized projection to latent structures (O-PLS) was first introduced in 2001 by Johann Trygg and Svante Wold, as a way to deal with the large amount of variation in predictor matrices for multivariate calibration (and classification), not correlated to the responses. In this context, O-PLS operates by partitioning the systematic variance in the X block into a Y relevant and an orthogonal data sets, both having a bilinear structure. During the years, there has been a large debate on the OPLS algorithm itself, to highlight its real peculiarities, whether it really had unique features or, on the other hand, to understand whether it should just be seen as one out of the many possibilities of identifying sources of common and distinctive variation among two or more blocks. Starting from these considerations, in this special section of the issue, four papers representing the different aspects of how the research and the debate around O-PLS and related issues have evolved so far, are collected.