Development of Semi-supervised Multiple-output Soft-sensors with Co-training and Tri-training MPLS and MRVM

Abstract Soft sensors are the most commonly used tools to estimate the hard-to-measure variables in the chemical processes and other industries, mainly due to unknown mechanism, significant measurement delay and highly unacceptable costs. However, a small number of labeled data and a large number of unlabeled data are not fully investigated and coordination of them to train the models and to improve the prediction accuracy is even rare. Multiple tasks learning or multiple outputs learning adds more complexity to this problem. In this light, this paper proposed semi-supervised multiple-output learning soft sensor models with co-training MPLS (Multiple-output partial least squares), co-training MRVM (Multiple-output relevance vector machines), tri-training MPLS and tri-training MRVM. Co-training MPLS model is developed by extending the traditional co-training PLS model. Co-training MRVM is promoted by replacing MPLS with MRVM. Tri-training MPLS and tri-training MRVM are built by combining tri-training algorithm with MPLS and MRVM. The proposed four models can make full use of appropriate unlabeled data to optimize the regression model, and then to directly strengthen multiple-output variables prediction. These models were firstly demonstrated by a numerical example, then accessed by a wastewater plant (WWTP) simulated with well-established WWTP validation platform, Benchmark Simulation Model No. 1 (BSM1). The results proved that the proposed models were able to significantly improve the prediction performance and efficiency.