Unified Low-Dimensional Subspace Analysis of Continuous and Binary Variables for Industrial Process Monitoring

Industrial data often consist of continuous variables (CVs) and binary variables (BVs), both of which provide crucial information about process operating conditions. Due to the coupling between industrial systems or equipment, these hybrid variables are usually high-dimensional and highly correlated. However, existing methods generally model hybrid variables directly in the observation space and assume independence between the variables to overcome the curse of dimensionality. Thus, they are ineffective at capturing dependencies among hybrid variables, and the effectiveness of process monitoring will be compromised. To overcome the limitations, this study proposes to seek a unified subspace for hybrid variables using the probabilistic latent variable (LV) model. By introducing a low-dimensional continuous LV, the proposed method can avoid the curse of dimensionality while capturing the dependencies between hybrid variables. Nevertheless, the inference of LV is analytically intractable and thus time-consuming due to the heterogeneity of CVs and BVs. To accelerate offline learning and online inference procedures, this study originally derives an analytical Gaussian distribution to approximate the true posterior distribution of the LV, based on which an efficient expectation-maximization algorithm is developed for parameter estimation. The Gaussian approximation is simultaneously optimized with the latest parameters to achieve a high approximation accuracy. The LV is then estimated by the posterior mean of the Gaussian approximation. By mapping the heterogeneous variables into a unified subspace, the proposed method defines three monitoring statistics, which are physically interpretable and thoroughly evaluate the probability of hybrid variables being normal. The effectiveness of the proposed method in detecting anomalies in CVs and BVs is shown through a numerically simulated case and a real industrial case.