Soft-Sensor Development for Processes With Multiple Operating Modes Based on Semisupervised Gaussian Mixture Regression

Abstract

Gaussian mixture regression (GMR) is an effective tool in developing soft sensors for online estimating difficult-to-measure variables in industrial processes with multiple operating modes. However, the GMR usually requires a sufficient amount of labeled samples to guarantee accurate probability density function (PDF) estimations because of its supervised learning process. Unfortunately, in soft-sensor applications, labeled samples could be very infrequent due to technical or economic limitations, which may lead the GMR-based soft sensors to unreliable parameter estimation and model selection, resulting in poor prediction performance. To tackle this problem, a semisupervised GMR (S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> GMR) was proposed, where both labeled and unlabeled samples were effective. In the S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> GMR, the PDFs of Gaussian components in input space and the functional dependence between input and output variables were learned simultaneously based on the expectation-maximization algorithm. Moreover, the Bayesian information criterion was employed to automatically determine the number of Gaussians for the S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> GMR. The S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> GMR was first investigated by a numerical example, and then applied to a real-life ammonia synthesis process for estimating the oxygen concentration at the top of the primary reformer. The two case studies verified the effectiveness of the proposed method.