Mixture Bayesian Regularization of PCR Model and Soft Sensing Application

In this paper, a Bayesian regularization mechanism is provided for automatically determining the number of latent variables in the probabilistic principal component regression (PPCR) model. Different from the unsupervised principal-component-analysis model, the response variable is incorporated for the supervision of selecting latent variables in the PPCR model. By introducing two hyperparameter vectors, the effectiveness of each latent variable can be well measured and controlled. For the mixture form of the PPCR model, a corresponding mixture Bayesian regularization strategy is further developed to control the dimensionality of latent variables. The expectation-maximization algorithm is employed for the parameter learning of both single and mixture Bayesian regularization models. Two probabilistic soft sensors are then developed for the online estimation of key variables in industrial processes, the performances of which are evaluated through two case studies. Compared to the single Bayesian regularization model, the mixture model shows stronger soft sensing abilities in nonlinear and multimode processes.