Optimising Kernel-based Multivariate Statistical Process Control

TL;DR

The paper tackles the limitation of linear MSPC in capturing nonlinear process relationships by extending MSPC to Kernel MSPC and optimising kernel parameters with Kernel Flows (KF). It demonstrates that learning per-variable kernel parameters and potentially combining kernels improves fault detection on the Tennessee Eastman Process, including faults not detected in prior studies, while offering interpretability about variable contributions. The main contributions are (i) applying KF to jointly optimise kernel type and per-variable kernel parameters for Kernel MSPC, (ii) enabling per-variable kernel learning for enhanced detection of hard-to-catch faults, and (iii) showing faster convergence and better performance than traditional optimisation methods. The approach promises improved, data-driven process monitoring with clearer insights into variable importance, applicable to complex industrial dynamics.

Abstract

Multivariate Statistical Process Control (MSPC) is a framework for monitoring and diagnosing complex processes by analysing the relationships between multiple process variables simultaneously. Kernel MSPC extends the methodology by leveraging kernel functions to capture non-linear relationships between the data, enhancing the process monitoring capabilities. However, optimising the kernel MSPC parameters, such as the kernel type and kernel parameters, is often done in literature in time-consuming and non-procedural manners such as cross-validation or grid search. In the present paper, we propose optimising the kernel MSPC parameters with Kernel Flows (KF), a recent kernel learning methodology introduced for Gaussian Process Regression (GPR). Apart from the optimisation technique, the novelty of the study resides also in the utilisation of kernel combinations for learning the optimal kernel type, and introduces individual kernel parameters for each variable. The proposed methodology is evaluated with multiple cases from the benchmark Tennessee Eastman Process. The faults are detected for all evaluated cases, including the ones not detected in the original study.

Figures (12)

• Figure 1: Proposed fault detection workflow.
• Figure 2: Explained variance in the regular PCA model of the normal-functioning data.
• Figure 3: The T$^2$ and $\mathrm{SPE_x}$ control chart calibration with regular PCA and normal-functioning process data.
• Figure 4: Capturing the faults using the regular PCA control charts. While the "easy" E1 fault is captured in both T$^2$ (a) $\mathrm{SPE_x}$ (d) control charts, the "difficult" fault is not captured in any of them. Increasing the number of PCs in the model (e) is not helpful in fault identification.
• Figure 5: (a) The variation explained by PCs in K-PCA is dependent on the shape of the kernel. In the showcased methods, the kernel function utilised is a Gaussian one. (b) The loss value for line-search of the Gaussian kernel parameter, for both E1 and D1 faults.