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MWRSPCA: online fault monitoring based on moving window recursive sparse principal component analysis

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Abstract

This paper proposes a moving window recursive sparse principal component analysis (MWRSPCA)-based online fault monitoring scheme, aim at providing an online fault monitoring solution for large-scale complex industrial processes (e.g., chemical industry processes) with time-varying and dynamically changing characteristics. It establishes a sparse principal component analysis (SPCA) model based on the sliding window block matrixes to perform process monitoring and incorporates normal process monitoring data set simultaneously to the model training set to update the monitoring model online, so that the process monitoring model has strong adaptability to time-varying processes. A recursive computing procedure of the corresponding sparse loading matrixes is derived based on a modified rank-one matrix approximation algorithm, so that the computational complexity of the process monitoring model is greatly decreased and the real-time monitoring capability can be guaranteed. The effectiveness of the proposed method is verified by the benchmark Tennessee-Eastman process. Compared with traditional fault monitoring methods, the proposed method can effectively improve the fault detection accuracies with lower false alarm rates, which is suitable for the fault monitoring of time-varying, long-term and continuous complex industrial processes.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 61971188, 61771492, the National Science Fund for Distinguished Young Scholars under grant No. 61725306 and the joint fund of NSFC and Guangdong provincial government under Grant No. U1701261, in part by the Hunan Natural Science Fund under Grant No. 2018JJ3349, project of Educational Commission of Hunan Province of China under Grant No. 19B364 and the postgraduate student research and innovation projects of Hunan Province under Grant No. CX20190415.

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Appendices

Appendix A: similarity analysis of moving window

Given the hybrid covariance matrix of two successive moving window monitoring matrixes, i.e., the new window and old window, as \({\mathbf{R}}_{*}\),

$$ {\mathbf{R}}_{*} { = }\left[ \begin{gathered} {\mathbf{R}}_{1} \hfill \\ {\mathbf{R}}_{2} \hfill \\ \end{gathered} \right] $$
(1)

where \({\mathbf{R}}_{1} \in {\mathbb{R}}^{{n_{1} \times m}}\) and \({\mathbf{R}}_{2} \in {\mathbb{R}}^{{n_{2} \times m}}\) are the covariance matrix of two windowed process monitoring data matrix (may have overlapping), by eigenvalue decomposition, the eigenvalue diagonal matrix \({{\varvec{\Lambda}}}_{*}\) and eigenvector matrix \({\mathbf{P}}_{*}\) can be obtained, satisfying the following equation.

$$ {\mathbf{R}}_{*} {\mathbf{P}}_{*} = {\mathbf{P}}_{*} {{\varvec{\Lambda}}}_{*} . $$
(2)

By using the transformation matrix \({\mathbf{P}}_{0}\), defined as,

$$ {\mathbf{P}}_{0} = {\mathbf{P}}_{*} {{\varvec{\Lambda}}}_{*}^{{{ - }\frac{{1}}{{2}}}} $$
(3)

We can make a transform of the covariance matrix \({\mathbf{R}}_{1}\) and \({\mathbf{R}}_{2}\), respectively

$$ \left\{ \begin{gathered} {\mathbf{R}}^{\prime}_{1} = {\mathbf{P}}_{0}^{T} {\mathbf{R}}_{1} {\mathbf{P}}_{0} \hfill \\ {\mathbf{R}}^{\prime}_{{\mathbf{2}}} = {\mathbf{P}}_{0}^{T} {\mathbf{R}}_{2} {\mathbf{P}}_{0} \hfill \\ \end{gathered} \right. $$
(4)

Then, the following equation can be derived,

$$ {\mathbf{P}}_{0}^{T} {\mathbf{R}}_{*} {\mathbf{P}}_{0} = {\mathbf{R}}^{\prime}_{1} + {\mathbf{R}}^{\prime}_{2} = {\mathbf{I}}. $$
(5)

By the application of eigenvalue decomposition to \({\mathbf{R}}^{\prime}_{1}\) and \({\mathbf{R}}^{\prime}_{2}\), we can achieve the eigenvalue of \({\mathbf{R}}^{\prime}_{1}\) and \({\mathbf{R}}^{\prime}_{2}\), which have the following relationship based on the Eq. (5),

$$ \gamma_{i}^{1} + \gamma_{i}^{2} = 1 $$
(6)

where \(\gamma_{i}^{1}\) and \(\gamma_{i}^{2}\) are the eigenvalues of \({\mathbf{R^{\prime}}}_{1}\) and \({\mathbf{R^{\prime}}}_{{\mathbf{2}}}\) respectively.

It can be seen that \(\gamma_{i}^{1}\) and \(\gamma_{i}^{2}\) are symmetric with respect to 0.5 and hence when the two successive windowed process monitoring data matrix are quite similar to each other, the eigenvalues \(\gamma_{i}^{1}\) or \(\gamma_{i}^{2}\) should be near 0.5. As pointed in (KANO et al. 2002), the following index \(\mu\) can be defined to evaluate the similarity of two data sets,

$$ \mu = 1 - \frac{{4\sum\limits_{i = 1}^{m} {\left( {\gamma_{i}^{1} - 0.5} \right)} }}{m}. $$
(7)

Here the similar index \(\mu\) changes between 0 and 1.

For the moving window-based process monitoring perceptive, the more similar of two successive process monitoring matrixes, the less variation information can be extracted and the more gradual changes in the model parameters, resulted in slowly adapting for the time-varying processes. Oppositely, if the similarity is smaller, the more variation information can be extracted and hence the process model can be adapted to the time-varying processes more effectively. In this work, the similar index is used for the empirical selection of the moving window size.

Appendix B: list of abbreviations and acronyms

PCA:

Principal component analysis.

SPCA:

Sparse principal component analysis.

RSPCA:

Recursive SPCA.

MWRSPCA:

Moving window RSPCA.

PLSA:

Partial least squares analysis.

KPCA:

Kernel principal component analysis.

ICA:

Independent component analysis.

LPPLS:

Locality-preserving partial least squares.

CIP:

Complex industrial process.

MSPM:

Multivariate statistical process monitoring.

FAR:

False alarm rate.

MDR:

Missing detection rate.

ACC:

Accuracy.

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Liu, J., Wang, J., Liu, X. et al. MWRSPCA: online fault monitoring based on moving window recursive sparse principal component analysis. J Intell Manuf 33, 1255–1271 (2022). https://doi.org/10.1007/s10845-020-01721-8

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