A physical causality-informed generative latent variable modeling paradigm for industrial virtual metrology

Generative latent variable models (GLVMs) have played an important role and been attracting widespread interest in industrial virtual metrology for predicting key variables in real-time, due to their outstanding capabilities of handling correlations, high dimensionality, uncertainties, and missing values. However, there is an overlooked issue associated with the GLVMs. That is, the existing GLVMs develop predictive models by establishing correlations between process variables, ignoring the causal dependence, which impairs the interpretability and generalization performance of the GLVMs because it is nontrivial to capture the true correlations. In view of such limitation of the GLVMs, with the aid of process knowledge for causal analysis, a novel physical causality-informed (PCI) modeling paradigm for the GLVMs, named PCI-GLVM, is proposed in this paper. The PCI-GLVM paradigm is further instantiated using a semi-supervised probabilistic principal component analysis (SsPPCA) model, for which a highly-efficient training algorithm based on the expectation–maximization algorithm is developed. Comprehensive performance evaluations of the PCI-SsPPCA are conducted on a numerical example and two industrial processes, validating the superiorities of the PCI-SsPPCA over state-of-the-art benchmark models.