In an industry process, FR lower than 0. The data sets of two faults in mode 2 were tested, and the MR were listed in Table 4. In the table, the smallest missed detection rates are shown in bold. Fault 1 is a bias in cooling water temperature. Due to the control loop in the CSTR process, these would be a bias in outlet temperature , and then the cooling water flow rate would increase. The reason is that the correct classification for each subgroup by using E-M algorithm and the PCs selected by SPCS are discriminative and could construct the subspace that contains the important fault information for abnormal data.
Fault 2 is a bias in inlet solute concentration. Then, due to the control loop in the CSTR process, there would be biases in outlet concentration and outlet temperature. In Figures 8 c and 8 d , we could hardly see which algorithm is better. Given that the modern industrial processes typically have multiple operating modes, BIP is utilized to compute the posterior probabilities of each monitored sample belonging to the multiple components and derive an integrated global probabilistic index for fault detection of multimode processes.
In each submode, we use the sparse principal component selection to select the key PCs that have the best relation with fault. This algorithm constructs an elastic net regression between all PCs and each sample and then selects PCs according to the nonzero regression coefficients which indicate the discriminative expression of the sample. The authors declare that there is no conflict of interests regarding the publication of this paper. Mathematical Problems in Engineering. Indexed in Science Citation Index Expanded. Journal Menu. Special Issues Menu. Subscribe to Table of Contents Alerts.
Table of Contents Alerts. Abstract According to the demand for diversified products, modern industrial processes typically have multiple operating modes. Introduction Over the past two decades, with the development of complex chemical processes and the growing demand of plant safety and stable product quality, timely process monitoring is gaining importance. Preliminaries 2. Principal Component Analysis Principal component analysis is a multivariate statistical analysis which is widely used in chemical process monitoring, fault detection, and so forth [ 36 — 38 ].
Construction of Finite Gaussian Mixture Model Based on EM For the process running at multiple operating condition, owing to the mean shifts or covariance changes, the assumption of multivariate Gaussian distribution becomes invalid [ 21 , 22 ]. Bayesian Inference-Based Probability In the previous section, the FGMM has been constructed, and it is essential to further derive the confidence boundary around the normal operating regions for process monitoring and fault detection.
In the proposed monitoring approach, given an arbitrary monitored sample belonging to each Gaussian component, Bayesian inference strategy is used to calculate the posterior probability as follows: which can also be formulated as Given that each component follows a unimodal Gaussian distribution, the squared Mahalanobis distance of from the center of follows distribution, provided that belongs to , Under the assumption that and has degree of freedom, denotes the squared Mahalanobis distance between and the mean center of.
For the monitored sample , a local Mahalanobis distance-based probability index relative to each Gaussian component can be defined as or Given the appropriate degree of freedom, can be computed by integrating the probability density function. Sparse Principal Component Selection Sparse representation has proven to be an extremely powerful tool for acquiring, representing, and compressing high-dimensional data [ 41 — 43 ].
For any nonnegative and , the elastic net estimates are given by In brief, it is expected that the elastic net is used to group a set of sparse coefficients to construct the sparse alignment matrices, in which the sparse representation information or the potential discriminative information is encoded to enhance the discriminative ability in an unsupervised manner. References J. Lee, C. Yoo, S. Choi, P. Vanrolleghem, and I. Ge and Z. Kim and I. Jin, Y. Lee, G. Lee, and C. Zhao and F. Tong, A. Palazoglu, and X. Liu and D. Ge, Z. Song, and F.
Wang, U. Kruger, and B. Hu, H. Ma, and H. Ma and H. Lee, I. He, S.
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