Mixture semisupervised probabilistic principal component regression model with missing inputs

Principal component regression (PCR) has been widely used as a multivariate method for data-based soft sensor design. In order to take advantage of probabilistic features, it has been extended to probabilistic PCR (PPCR). Commonly, industrial processes operate in multiple operating modes. Moreover, in most cases, outputs are measured at a slower rate than inputs, and for each sample of input variable, its corresponding output may not always exist. These two issues have been solved by developing the mixture semi-supervised PPCR (MSPPCR) method. In this paper, we extend this developed model to the case of simultaneous missing data in both input and output. Missing data in multidimensional input space constitutes a significantly more challenging problem. Missing input data occurs frequently in industrial plants because of sensor failure and other problems. We develop and solve the MSPPCR model by using the expectation-maximization (EM) algorithm to deal with missing inputs, in addition to missing outputs and multi-mode conditions. Finally, we present two case studies to demonstrate its performance.