Hybrid Input–Output Probabilistic Slow Feature Analysis for adaptive process monitoring

Industrial process data are usually dynamic due to closed-loop control systems. Current dynamic latent-variable methods generally assume that the dynamics of the process are fixed. This assumption has two implications. First, the system is not influenced by external inputs. Second, the system parameters remain time-invariant. However, in real industrial scenarios, systems are often regulated by manipulated variables and their parameters may drift over time. Failure to account for these time-varying factors will result in an increasing disparity between existing models and the actual system, ultimately leading to unreliable monitoring results. To address this issue, a Hybrid Input–Output Probabilistic Slow Feature Analysis (H-IOPSFA) model is proposed along with an adaptive process monitoring approach. The H-IOPSFA model is designed to account for the directed effect of the manipulated variables on the system dynamics and process variables in the presence of continuous and binary variables. A recursive model updating method is then introduced to accommodate normal process changes, offering significantly faster convergence than training from scratch. Additionally, by simultaneously monitoring dynamic and static variations, an adaptive monitoring strategy is developed to effectively differentiate between real faults and operating condition changes. Finally, the H-IOPSFA model and the adaptive monitoring method are applied to the TE process and a practical industrial process. Compared with classical dynamic monitoring methods, the proposed method achieves the highest fault detection rate (98.63% on the TE and 96.61% on the practical process) while realizing an acceptable fault alarm rata (8.23% on the TE and 7.33% on the practical process), which demonstrates its superior performance. © 2025 Elsevier Ltd