Nonlinear probabilistic latent variable regression models for soft sensor application: From shallow to deep structure

Probabilistic latent variable regression models have recently caught much attention in the process industry, particularly for soft sensor applications. One of the main challenges for those models is how to effectively extract nonlinear features for latent variable regression. This paper proposes a nonlinear probabilistic latent variable regression (NPLVR) model based on the features extracted by variational auto-encoder. To extend the NPLVR model from shallow to deep structure, a hierarchical form of NPLVR model is proposed to extract deeper nonlinear information by stacking VAE. Under the same modeling framework, a semi-supervised version of hierarchical NPLVR model is further developed to handle the problem of scarce amount of labeled data samples, which is quite common in practical applications. Two industrial case studies are provided to demonstrate the effectiveness and superiority of the developed models.