Bayesian Nonlinear Gaussian Mixture Regression and its Application to Virtual Sensing for Multimode Industrial Processes

Abstract

Virtual sensors have established themselves as effective tools in process industries for online estimating variables that are crucial but difficult to measure. However, multimode industrial processes render developing high-accuracy virtual sensors quite challenging. The main difficulties lie in that, in multimode processes, the distributions of process data are strongly non-Gaussian and the mathematical relationships between the explanatory and primary variables are highly nonlinear. Even within one operating mode, the primary variables could depend on the explanatory variables in nonlinear ways. In order to address these issues, this article proposes a virtual sensing approach named Bayesian nonlinear Gaussian mixture regression (BNGMR) with the aid of single-hidden layer feedforward neural networks (SLFNs). In the BNGMR, a fully Bayesian model structure that absorbs the merits of SLFNs and the mixture models is designed. In addition, we develop a training algorithm for the BNGMR to realize predictive virtual sensor development based on variational inference. Extensive assessments of the performance of the BNGMR are carried out using both artificial example and real-world industrial processes. The experiments have demonstrated the predictive advantage of the BNGMR over several benchmark methods and also have provided practitioners with good illustrations. Note to Practitioners —Although the case studies on two real-world industrial processes have verified the effectiveness and feasibility of the Bayesian nonlinear Gaussian mixture regression (BNGMR) for industrial applications, it is still worth emphasizing that before training the BNGMR, some important works on data preprocessing should be done, which include: 1) careful selection of explanatory variables that are closely related to primary variables; 2) time alignment matching between primary and explanatory variables; and 3) dealing with samples with missing values (for example, removing them) and elimination of outliers. We suggest to complete 1) and 2) based on process mechanisms and to pay particular attention to removing the outlier, as the presence of outliers could skew the estimations of Gaussian components and result in significant performance deterioration.