Probabilistic contribution analysis for statistical process monitoring: A missing variable approach

Root cause analysis in multivariate statistical process monitoring: Integrating reconstruction-based multivariate contribution analysis with fuzzy-signed directed graphs

“…There are 500 samples under normal operation, which are used to build a PPCA model (Kim & Lee, 2004;Chen & Sun, 2009) with 14 principal components, which explained 75.98% variance. Following (Chen & Sun, 2009), the single likelihood-based control limit is set to be the 99%-fractile of the distribution with degrees of degree of n=38 (the number of process variables), i.e.…”

“…it is a generic framework that is applicable to any process monitoring model, provided that a variable reconstruction algorithm can be developed for that particular model. Since the primary focus of this study is not fault detection but diagnosis, a simple probabilistic PCA (PPCA) is used as the process monitoring model (Kim & Lee, 2003;Chen & Sun, 2009) to demonstrate how the proposed method can help identify the root cause. Clearly, diagnosis is only possible for the faults that can be detected by the monitoring model.…”

Reconstruction-based multivariate contribution analysis for fault isolation: A branch and bound approach

“…When data are not Gaussian distributed, notable solutions to MSPM include kernel density estimation [8], Gaussian mixture model (GMM) [9,10,11], independent component analysis [12], one-class SVM model [13], among others. The probabilistic PCA (PPCA) mixture model is an extension of GMM by incorporating a probabilistic version of PCA, and it has been shown to be effective for MSPM [14,15]. The probabilistic formulation used by the PPCA mixture model provides a unified likelihood-based statistic that offers clearer monitoring result.…”

“…replacing these variables with the reconstructed values would bring the process back to normal. Similarly, a missing variable contribution analysis was proposed for probabilistic formulation of PCA and the PPCA mixture model [14]. In this method, each variable is treated as if it was missing and the expectation (with respect to the missing variable) of monitoring statistic is re-calculated.…”

“…Specifically, the contribution of multiple process variables to a detected fault is evaluated by following the reconstruction and missing variable approach [14,19,20,21]. In order to find the combination of variables that are the most responsible to the fault, a combinatorial optimization problem is formulated, followed by the development of a feasible solution set and bounds to enable the application of BAB.…”