Probabilistic multivariate statistical process control via kernel parameter uncertainty propagation

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.

Figures (10)

• Figure 1: Workflow for the generation of probabilistic control charts. Deterministic kernel calibration provides prior means, Bayesian sampling yields posterior draws of kernel parameters, and posterior propagation through K-PCA produces probabilistic monitoring statistics and contribution diagnostics. Here, is $\mathbf{X}$ a matrix of process observations (timeline) as rows, and process variables as columns, and y is the process state (either '0' if the process is in normal operation or '1' if the process is faulty).
• Figure 2: Workflow for generation of probabilistic control charts when no fault data are initially available. Deterministic unsupervised kernel calibration is followed by provisional chart-based label assignment and subsequent Bayesian posterior inference.
• Figure 3: Deterministic kernel parameter optimization (a) loss trace and (b) kernel parameter convergence for fault F01. All four converged in a perfect fault detection in the control charts.
• Figure 4: K-PCA MSPC control charts calibrated with the GPC-L-BFGS convergence kernel parameters, for a SE kernel.
• Figure 5: MCMC (a) chains for the GPC-based log-likelihood, with L-BFGS-GPC convergence values as prior mean and (b) the posterior distributions.