Semi-supervised mixture of latent factor analysis models with application to online key variable estimation

Data-driven virtual sensors have extensive applications in industrial processes for online estimating those important but difficult-to-measure variables. In the virtual sensor application, labeled samples could be very infrequent due to technical or economical limitations, thereby virtual sensors developed upon insufficient labeled samples may not be well trained, which leads to poor estimation performance. In addition, industrial processes are inherently stochastic and the vast majority of them present nonlinear and non-Gaussian characteristics. To cope with these issues, this paper proposes a semi-supervised mixture of latent factor analysis models (S(2)MLFA). In the S(2)MLFA, the insufficiency of labeled samples is remedied by exploiting both labeled and unlabeled data sets, while the nonlinear and non-Gaussian characteristics are handled by the mixture model structure. Moreover, the process uncertainties are modeled by the probabilistic model formulation. An efficient expectation-maximization-based learning algorithm is developed for training the S(2)MLFA, and a modified Akaike information criterion is presented for model selection. The S(2)MLFA is investigated by a numerical example and two real-world industrial processes, through which the effectiveness and feasibility of the proposed schemes are verified.