A probabilistic derivation of the partial least-squares algorithm

Traditionally the partial least-squares (PLS) algorithm, commonly used in chemistry for ill-conditioned multivariate linear regression, has been derived (motivated) and presented in terms of data matrices. In this work the PLS algorithm is derived probabilistically in terms of stochastic variables where sample estimates calculated using data matrices are employed at the end. The derivation, which offers a probabilistic motivation to each step of the PLS algorithm, is performed for the general multiresponse case and without reference to any latent variable model of the response variable and also without any so-called "inner relation". On the basis of the derivation, some theoretical issues of the PLS algorithm are briefly considered: the complexity of the original motivation of PLS regression which involves an "inner relation"; the original motivation behind the prediction stage of the PLS algorithm; the relationship between uncorrelated and orthogonal latent variables; the limited possibilities to make natural interpretations of the latent variables extracted.