Soft-Sensor Development for Processes With Multiple Operating Modes Based on Semisupervised Gaussian Mixture Regression

Gaussian mixture regression (GMR) is an effective tool in developing soft sensors for online estimating difficult-to-measure variables in industrial processes with multiple operating modes. However, the GMR usually requires a sufficient amount of labeled samples to guarantee accurate probability density function (PDF) estimations because of its supervised learning process. Unfortunately, in soft-sensor applications, labeled samples could be very infrequent due to technical or economic limitations, which may lead the GMR-based soft sensors to unreliable parameter estimation and model selection, resulting in poor prediction performance. To tackle this problem, a semisupervised GMR (S
²
GMR) was proposed, where both labeled and unlabeled samples were effective. In the S
²
GMR, the PDFs of Gaussian components in input space and the functional dependence between input and output variables were learned simultaneously based on the expectation-maximization algorithm. Moreover, the Bayesian information criterion was employed to automatically determine the number of Gaussians for the S
²
GMR. The S
²
GMR was first investigated by a numerical example, and then applied to a real-life ammonia synthesis process for estimating the oxygen concentration at the top of the primary reformer. The two case studies verified the effectiveness of the proposed method.