Recently, regression analysis has become a popular tool for face recognition. Most existing regression methods use the one-dimensional, pixel-based error model, which characterizes the representation error individually, pixel by pixel, and thus neglects the two-dimensional structure of the error image. We observe that occlusion and illumination changes generally lead, approximately, to a low-rank error image. In order to make use of this low-rank structural information, this paper presents a two-dimensional image-matrix-based error model, namely, nuclear norm based matrix regression (NMR), for face representation and classification. NMR uses the minimal nuclear norm of representation error image as a criterion, and the alternating direction method of multipliers (ADMM) to calculate the regression coefficients. We further develop a fast ADMM algorithm to solve the approximate NMR model and show it has a quadratic rate of convergence. We experiment using five popular face image databases: the Extended Yale B, AR, EURECOM, Multi-PIE and FRGC. Experimental results demonstrate the performance advantage of NMR over the state-of-the-art regression-based methods for face recognition in the presence of occlusion and illumination variations.

A novel dimensionality reduction algorithm named "global−local preserving projections" (GLPP) is proposed. Different from locality preserving projections (LPP) and principal component analysis (PCA), GLPP aims at preserving both global and local structures of the data set by solving a dual-objective optimization function. A weighted coefficient is introduced to adjust the trade-off between global and local structures, and an efficient selection strategy of this parameter is proposed. Compared with PCA and LPP, GLPP is more general and flexible in practical applications. Both LPP and PCA can be interpreted under the GLPP framework. A GLPP-based online process monitoring approach is then developed. Two monitoring statistics, i.e., D and Q statistics, are constructed for fault detection and diagnosis. The case study on the Tennessee Eastman process illustrates the effectiveness and advantages of the GLPP-based monitoring method.

A novel method named tensor global–local structure analysis (TGLSA) is proposed for batch process monitoring. Different from principal component analysis (PCA) and locality preserving projections (LPP), TGLSA aims at preserving both global and local structures of data. Consequently, TGLSA has the ability to extract more meaningful information from data than PCA and LPP. Moreover, the tensor-based projection strategy makes TGLSA more applicable for the three-dimensional data than multiway-based methods, such as MPCA and MLPP. A TGLSA-based online monitoring approach is developed by combining TGLSA with a moving window technique. Two new statistics, i.e., SPD and R 2 statistics, are constructed for fault detection and diagnosis. In particular, the R 2 statistic is a novel monitoring statistic, which is proposed based on a support tensor domain description method. The effectiveness and advantages of the TGLSA-based monitoring approach are illustrated by a benchmark fed-batch penicillin fermentation process.

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A novel fuzzy phase partition method and a hybrid modeling strategy are proposed for quality prediction and process monitoring in batch processes with multiple operation phases. The fuzzy phase partition method is proposed on the basis of a sequence-constrained fuzzy c-means (SCFCM) clustering algorithm. It divides the batch process into several fuzzy operation phases by performing the SCFCM algorithm on trajectory data of phase-sensitive process variables. This SCFCM-based partition method not only has high computation efficiency and good partition accuracy but also is easy to implement and popularize. In addition, it generates “soft” partition results, where a “transition” phase exists between two adjacent “steady” operation phases. A hybrid modeling strategy is developed to build appropriate models for all operation phases according to their own characteristics. Phase-based multiway PLS models are built for regular steady phases that have longer durations and stable process behaviors. Just-in-time PLS models are built for those phases with shorter durations but time-varying or nonlinear process behaviors, including all transition phases and several irregular steady phases. This hybrid modeling strategy significantly enhances the modeling accuracy, resulting in better quality prediction and process monitoring performance. Advantages of proposed methods are illustrated by case studies in a fed-batch penicillin fermentation process.

Integrated phase partition, online phase identification, and phase-based monitoring methods are proposed for multiphase batch processes with uneven durations. A new phase partition method is developed based on the warped K-means (WKM) clustering algorithm, which divides the entire batch into several operation phases by clustering the trajectory data of phase-sensitive process variables. This WKM-based phase partition method can efficiently cope with the sequentiality of batch data and, thus, ensures a reasonable phase partition result. Besides, because only phase-sensitive variables are used for phase partition, the phase partition accuracy is improved. An online phase identification method is proposed to identify the corresponding operation phase of a new sample according to a phase identification combination index (PICI). PICI quantifies the correlation of a new sample with each operation phase by calculating distance and time difference between the sample and the phase center. The PARAFAC2 and unfolded principal component analysis (uPCA) methods are applied to build monitoring models from the uneven-length batch data in each phase. T 2 and SPE statistics are constructed for fault detection. The contribution plot of T 2 statistic is used for fault diagnosis. The effectiveness and advantages of proposed methods are illustrated by the case study in a fed-batch penicillin fermentation process.

A novel process monitoring method is proposed based on sparse principal component analysis (SpPCA). To reveal meaningful variable correlations from process data, the SpPCA is developed to sequentially extract a set of sparse loading vectors from process data. To build a high-performance monitoring model, a fault detectability matrix is applied to select the sparse loading vectors used for process modeling from all sparse loading vectors obtained by SpPCA. The fault detectability matrix ensures that the faults related to any monitored process variable are detectable in the principal component subspace and no overlapped (or redundant) loading vectors are involved in the monitoring model. Moreover, the selected sparse loading vectors classify all process variables into nonoverlapping groups according to variable correlations. Two-level contribution plots, which consist of group-wise and group-variable-wise contribution plots, are used for fault diagnosis. The first-level group-wise contribution plot describes the individual contribution of each variable group to the fault. The second-level group-variable-wise contribution plot reflects the individual contribution of each process variable to the fault. The two-level contribution plots not only utilize meaningful correlations between process variables in the same group, but also effectively remove the interference from process variables in other groups. Therefore, the fault diagnosis reliability and accuracy are significantly improved. The implementation, performance, and advantages of the proposed methods are illustrated with an industrial case study.

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