Two Industrial Twin Soft Sensing Methods With Estimation Interval Based on Symmetric Skewed Distributions and Combined Weights

Abstract

Various disturbances and uncertainties existing in actual industrial production environments can degrade the measurement accuracy of soft sensors based on single-value deterministic estimation. In addition, outliers caused by operational errors or recording mistakes may affect the generalization ability of soft sensors. Inspired by this, two Bayesian twin extreme learning machines based on symmetric skewed distributions, BTELM-ALD and BTELM-STD, are proposed. Both soft sensing methods perform parameter learning in a Bayesian framework and train a pair of twin models based on combined weights to provide estimation intervals for key indicators. They use skewed heavy-tailed distributions to model the residuals, which enhances robustness to outliers. BTELM-ALD uses an asymmetric Laplace distribution (ALD) instead of Gaussian distribution and constructs a pair of twin models based on the combined weights ( $p$ , $1-p$ ). The introduction of suitable surrogate functions makes the posterior distribution and marginal likelihood easy to solve. In BTELM-STD, a univariate skewed t-distribution (STD) is presented and written as a hierarchical representation. The corresponding twin models are constructed based on the combined weights ( $s$ , $-s$ ), and then variational inference and the Newton method are used to optimize the parameters. Experimental results on several cases including an actual PTA oxidation process illustrate the validity and advantages of our proposed methods.