Semisupervised Bayesian Gaussian Mixture Models for Non-Gaussian Soft Sensor

Soft sensors have been widely accepted for online estimating key quality-related variables in industrial processes. The Gaussian mixture models (GMM) is one of the most popular soft sensing methods for the non-Gaussian industrial processes. However, in industrial applications, the quantity of samples with known labels is usually quite limited because of the technical limitations or economical reasons. Traditional GMM-based soft sensor models solely depending on labeled samples may easily suffer from singular covariances, overfitting, and difficulties in model selection, which results in the performance deterioration. To tackle these issues, we propose a semisupervised Bayesian GMM (S(2)BGMM). In the S(2)BGMM, we first propose a semisupervised fully Bayesian model, which enables learning from both the labeled and unlabeled datasets for remedying the deficiency of infrequent labeled samples. Subsequently, a general framework of weighted variational inference is developed to train the S(2)BGMM, such that the rate of learning from unlabeled samples can be controlled by penalizing the unlabeled dataset. Case studies are carried out to evaluate the performance of the S(2)BGMM through a numerical example and two real-world industrial processes, which demonstrate the effectiveness and reliability of the proposed approach.