Concurrent probabilistic PLS regression model and its applications in process monitoring

•We propose a concurrent PPLS (CPPLS) method to perform further decomposition on the PPLS model.•This proposed model has the advantages of both general probabilistic models and the concurrent PLS model.•Based on this concurrent probabilistic PLS model, monitoring statistics are constructed for evaluation of five subspaces.•The proposed method is illustrated by its application in the TE process.

The probabilistic PLS (PPLS) algorithm derives the latent variables by maximizing the likelihood of input scores and quality scores, but imposes no constraint on the input residuals and the quality residuals, which implies that residuals may contain large information. Motivated by the concurrent PLS method, this paper proposes a concurrent PPLS (CPPLS) method to perform further decomposition of these residuals, and then two more subspaces are obtained. In this method, the maximum-likelihood method along with the expectation-maximization (EM) algorithm are employed to develop the model, in which the variance of each variable explained by latent variables is introduced to determine the number of latent variables. Based on the CPPLS model, five monitoring statistics all based on Mahalanobis norm are constructed for the evaluation of five subspaces decomposed by CPPLS, respectively.