Semi-supervised PLVR models for process monitoring with unequal sample sizes of process variables and quality variables

As the key indicators of chemical processes, the quality variables, unlike process variables, are often difficult to obtain at the high frequency. Obtaining the data of quality variables is expensive, so the data are only collected as a small portion of the whole dataset. It is common to see in both continuous and batch processes that the sample sizes of process variables and quality variables are unequal. To effectively integrate two different observation sources, including quality variables collected at a low frequency and process variables sampled at a high rate, a semi-supervised probabilistic latent variable regression model (SSPLVR) is proposed in this article. It enhances the performance monitoring of the variations of process variables and quality variables. The proposed semi-supervised model is applied to continuous and batch processes respectively. The SSPLVR model calibrated by the expectation-maximization algorithm is derived and the corresponding statistics is also systematically developed for the fault detection. Finally, two simulated case studies, TE benchmark for a continuous process problem and the penicillin fermentation for a batch process problem, are presented to illustrate the effectiveness of the proposed method.