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The main challenge in developing soft sensors in process industry is the existence of irregularity of data, such as measurement noises, outliers, and missing data. This paper is concerned with a comparative study among various data-driven soft sensor algorithms and the Bayesian methods. The algorithms to be considered for a comparative study in this paper include ordinary least-squares, robust regression, error-in-variable methods, partial least-squares, and the Bayesian inference algorithms. Methods for handling irregular data are reviewed. An iterative Bayesian algorithm for handling measurement noise and outliers is proposed. Performance of the Bayesian methods is compared with other existing methods through simulations, a pilot-scale experiment, and an industrial application.

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A variational Bayesian approach to robust identification of switched auto-regressive exogenous models is developed in this paper. By formulating the problem of interest under a full Bayesian identification framework, the number of local-models can be determined automatically, while accounting for the uncertainty of parameter estimates in the overall identification procedure. A set of significance coefficients is used to assign proper importance weights to local-models. By maximizing the marginal likelihood of the identification data, insignificant local-models will be suppressed and the optimal number of local-models can be determined. Considering the fact that the identification data may be contaminated with outliers, t distributions with adjustable tails are utilized to model the contaminating noise so that the proposed identification algorithm is robust. The effectiveness of the proposed Bayesian approach is demonstrated through a simulated example as well as a detailed industrial application.

Just-in-time (JIT) learning methods are widely used in dealing with nonlinear and multimode behavior of industrial processes. The locally weighted partial least squares (LW-PLS) method is among the most commonly used JIT methods. The performance of LW-PLS model depends on parameters of the similarity function as well as the structure and parameters of the local PLS model. However, the regular LW-PLS algorithm assumes that the parameters of the similarity function and structure of the local PLS model are known and do not fully utilize available knowledge to estimate the model parameters. A Bayesian framework is proposed to provide a systematic way for real-time parameterization of the similarity function, selection of the local PLS model structure, and estimation of the corresponding model parameters. By applying the Bayes' theorem, the proposed framework incorporates the prior knowledge into the identification process and takes into account the different contribution of measurement noises. Furthermore, Bayesian model structure selection can automatically deal with the model complexity problem to avoid the overfitting issue. The advantages of this new approach are highlighted through two case studies based on the real-world near infrared data.