Nonlinear industrial soft sensor development based on semi-supervised probabilistic mixture of extreme learning machines

Soft sensors play an important role in process industries for monitoring and control of key quality variables, and calibration of analyzers. Owing to the merits of fast learning speed and good generalization performance, extreme learning machines (ELMs) have been widely accepted to develop soft sensor models for nonlinear industrial processes. However, there still exist some challenges in developing high-accuracy ELM-based soft sensors. Specifically, ELMs with shallow networks seem to have inadequate representation capabilities for complex nonlinearities, while ELMs with deep networks have difficulties in determining the number of hidden layers and hidden nodes for each layer which readily results in overfitting. In addition, in soft sensor applications, labeled samples are usually limited due to technical or economical reasons, which adds obstacles to model training. To deal with these issues, we propose a semi-supervised probabilistic mixture of ELMs (referred to as the '(SPMELMs)-P-2'). In the (SPMELMs)-P-2, localized ELMs are trained and combined, which are completed in a unified probabilistic way such that process nonlinearities and uncertainties can be accommodated. Moreover, based on the variational Bayes expectation-maximization algorithm, we develop a training algorithm for the (SPMELMs)-P-2, where unlabeled samples are able to be exploited and the regularization parameter for each ELM can be adaptively determined. The performance of the (SPMELMs)-P-2 is evaluated through two real-world industrial processes, and the results demonstrate the advantages of the proposed method in contrast with several state-of-the-art relevant soft sensing approaches.