Laplacian regularization of linear regression model for semi-supervised industrial soft sensor development

With more and more data collected from modern industrial processes, data-based soft sensor modeling methods have become popular for online prediction of those difficult-to-measure key variables. As one of the most widely used methods, the linear regression model has been studied for many years. In this paper, a new semi-supervised form of data-based soft sensor is developed upon this simple but commonly used linear regression model, in order to make it more engineering applicable. Particularly, a semi-supervised model structure is formulated by introducing a Laplacian regularization term, through which a lot more unlabeled data samples can be well incorporated for modeling. With an improved estimation for the input data distribution, the real data features can be better captured and thus the model parameters are well restricted, in order to provide a better performance. In addition, the basic Laplacian regularized linear regression model is further extended to nonlinear counterparts by the introductions of basis function and kernel trick in the input data space. For performance evaluation, two industrial case studies are provided, based on which both feasibility and effectiveness of the developed method are confirmed.