Robust semi-supervised mixture probabilistic principal component regression model development and application to soft sensors

• Robust semi-supervised probabilistic method is proposed to deal with the soft sensing problem.
• Semi-supervised mixture model is used for modeling both labeled and unlabeled data.
• The Expectation-Maximization algorithm is employed for parameter learning.
• Bayes soft alignment method is developed for online soft sensing.
• The superiority of the developed method is tested on the Tennessee Eastman process.

Traditional data-based soft sensors are constructed with equal numbers of input and output data samples, meanwhile, these collected process data are assumed to be clean enough and no outliers are mixed. However, such assumptions are too strict in practice. On one hand, those easily collected input variables are sometimes corrupted with outliers. On the other hand, output variables, which also called quality variables, are usually difficult to obtain. These two problems make traditional soft sensors cumbersome. To deal with both issues, in this paper, the Student's t distributions are used during mixture probabilistic principal component regression modeling to tolerate outliers with regulated heavy tails. Furthermore, a semi-supervised mechanism is incorporated into traditional probabilistic regression so as to deal with the unbalanced modeling issue. For simulation, two case studies are provided to demonstrate robustness and reliability of the new method.