On the Equivalence Between Null Space and Orthogonal Space in Latent Variable Regression Modeling

The concepts of null space and orthogonal space have been developed in independent contexts and with different purposes: the former arises in the inversion of partial least-squares (PLS) regression models, and the latter in orthogonal PLS (O-PLS) modeling. In this study, we bridge PLS model inversion and O-PLS modeling by mathematically proving that the null space and the orthogonal space are the same space. We also provide a graphical interpretation of the equivalence between the two spaces, using both a simulated and a real case study.