Probabilistic multivariate statistical process control via kernel parameter uncertainty propagation

2026-03-19

AI Summary

This work proposes a probabilistic kernel-based multivariate statistical process control (K-MSPC) framework to address the overconfidence and instability in conventional kernel methods, which typically ignore uncertainty in kernel parameters. The approach employs a two-stage procedure: first calibrating kernel parameters in a supervised or unsupervised manner, then performing Bayesian inference using an automatic relevance determination (ARD) kernel combined with Markov chain Monte Carlo (MCMC) sampling. This propagates parameter uncertainty into monitoring statistics, yielding T² and prediction error control charts—as well as contribution plots—equipped with credible intervals. To the best of our knowledge, this is the first method to explicitly quantify and propagate kernel parameter uncertainty in nonlinear process monitoring. Evaluation on the Tennessee Eastman process demonstrates that posterior mean–based monitoring outperforms deterministic alternatives, while credible bands widen significantly during faults, effectively capturing epistemic uncertainty; furthermore, the ARD kernel reduces posterior uncertainty.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Data Mining & Knowledge Management: Anomaly/Outlier Detection

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Kernel-based multivariate statistical process control (K-MSPC) extends classical monitoring to nonlinear industrial processes. Its performance depends critically on kernel parameters such as lengthscales and variance terms. In current practice these parameters are typically selected by heuristics or deterministic optimisation, and then treated as fixed, despite being inferred from finite and noisy data. This can lead to overconfident control limits and unstable alarm behaviour when the kernel choice is uncertain. This work proposes a probabilistic K-MSPC framework that quantifies and propagates kernel parameter uncertainty to the monitoring statistics. The approach follows a two-stage workflow: (i) deterministic kernel calibration using supervised or unsupervised models, and (ii) Bayesian inference of kernel parameters via Markov chain Monte Carlo. Posterior samples are propagated through kernel Principal Component Analysis to produce probabilistic $T^2$ and squarred prediction error control charts, together with uncertainty-aware contribution plots. The framework is evaluated on the Tennessee Eastman Process benchmark. Results show that posterior-mean monitoring often improves fault detection compared to deterministic prior-mean charts for the squared exponential kernel, while credible bands remain narrow in-control and widen under faults, reflecting amplified epistemic uncertainty in abnormal regimes. The automatic relevance determination kernel reduces posterior uncertainty and yields performance close to the deterministic baseline, whereas unsupervised calibration produces wider posterior bands but still robust fault detection.

Problem

• kernel parameter uncertainty
• multivariate statistical process control
• probabilistic monitoring
• control limits
• fault detection

Innovation

• probabilistic K-MSPC
• kernel parameter uncertainty
• Bayesian inference
• uncertainty propagation
• kernel PCA