Process monitoring based on factor analysis: Probabilistic analysis of monitoring statistics in presence of both complete and incomplete measurements

• We derive the distribution of factors and residuals.
• We perform probabilistic analysis on the monitoring statistics.
• We propose novel monitoring statistics by considering the covariance of factors and residuals.
• We extend the probabilistic analysis to the process monitoring with incomplete measurements.
• The proposed method is illustrated by its application in two cases.

In generic process monitoring approaches based on probabilistic latent variable models, such as probabilistic principal component analysis (PPCA) or factor analysis (FA) model, the online score and residual are characterized by probability distributions. However, only their expectations are involved in the calculations of monitoring statistics, square prediction error (SPE) and Hotelling T2, which ignore the information of their variances and may result in missed fault alarms. Based on the FA model, this paper investigates the probabilistic uncertainties of monitoring statistics arising from both inherent nature and missing measurements of the process data. The proposed method derives the distributions of both the online factor and residual at each sampling instant, and then transforms generic monitoring statistics into general quadratic forms. As a result, novel monitoring statistics are developed based on the probabilistic uncertainties of the generic statistics. In addition, the proposed monitoring statistics are extended to the case of incomplete measurements, in which the conditional distributions of the online measurement, factor and residual are computed and used to construct the statistics for process monitoring. Simulation examples illustrate the feasibility of the proposed method and demonstrate its effectiveness.