Abstract
This paper presents a new semi-supervised probabilistic density-based regression approach, called Semi-supervised Weighted Gaussian Regression (SWGR), for the soft sensing of nonlinear and multimode industrial processes given a limited number of labeled data samples. In SWGR, different weights are assigned to each training sample based on their similarities to a query sample. Then a local weighted Gaussian density is built for capturing the joint probability of historical samples around the query sample. The training process of parameters in SWGR incorporates both labeled and unlabeled data samples via a maximum likelihood estimation algorithm. In this way, the soft sensor model is able to approximate the nonlinear mechanics of input and output variables and remedy the insufficiency of labeled samples. At last, the output prediction as well as the uncertainty of prediction can be obtained by the conditional distribution. Two case studies validate that the proposed semi-supervised soft sensing method outperforms some recent methods.