Hybrid Probabilistic Slow Feature Analysis of Continuous and Binary Data for Dynamic Process Monitoring

Industrial process data are usually high-dimensional with dynamic characteristics, and a mix of continuous and binary quantities. However, current dynamic latent variable (DLV) methods primarily focus on analyzing continuous variables (CVs), overlooking the prevalence and significance of binary variables (BVs). BVs often serve as control references, indicating operating conditions or specific states and influencing the behavior of CVs. Integrating BVs into DLV models is crucial for elucidating the correspondence between CVs and BVs and uncovering the real operating patterns of the system. The main challenge lies in effectively accommodating the statistical heterogeneity exhibited by CVs and BVs, while comprehensively investigating their contemporaneous and temporal dependencies. To address this challenge, this study proposes a novel DLV model called hybrid probabilistic slow feature analysis (HPSFA). The HPSFA algorithm is specifically designed to extract slow features (SFs) from CVs while incorporating supervision from BVs. To efficiently infer posterior distributions of SFs, a variational recursive filter (VRF) is developed using the local approximation method, providing closed-form posterior estimations. Leveraging the VRF, an efficient expectation-maximization algorithm is proposed for parameter estimation. For process monitoring, three statistics are designed based on prediction or reconstruction errors, which are separated from dynamic variations and exhibit reduced variability. This reduction in variability enables the definition of narrower control regions while maintaining the desired confidence level. The HPSFA method is thoroughly evaluated through both simulated and real industrial case studies to demonstrate its validity and superior performance over existing approaches. The experimental results show that HPSFA timely detects both static and dynamic anomalies of the hybrid variables, and achieves the highest-fault detection rate (85.89%) while maintaining a considerably low-false alarm rate (2.67%) in the practical industrial case.