Subsequently, Rato and Reis proposed sensitivity enhancing transformations and defined new statistics to improve fault detection capabilities. They also used the residuals between the estimated PCs and the actual projected PCs to establish statistics for strengthening the sensitivity to faults. Using DPCA, Rato and Reis predicted the current principal components (PCs) on the basis of the past observation data. Rato and Reis proposed a method to determine the time-lagged number of the DPCA model and analyzed its effect on process monitoring. In the modeling stage, the dynamic characteristics of the process data were described by an augmented matrix with several time-lagged samples. first proposed dynamic PCA (DPCA) to improve the monitoring performance for dynamic processes. Therefore, the process data are time series, which mean that the current sample shows the correlation with past samples. In modern industrial processes, feedback control systems are generally adopted. However, the above methods are all performed under the assumption that no dynamic characteristics exist in industrial processes. constructed two sub-models based on KPCA and kernel local-nonlocal preserving discriminant analysis for fault detection. Incorporating prior fault information, Deng et al. proposed a method for monitoring independent and related variables. Using KPCA and support vector data description (SVDD), Huang et al. established several local KPCA models by combining mutual information (MI) and spectral clustering. To enhance the monitoring performance for the nonlinear plant-wide process, many modified methods based on KPCA were proposed. Then, PCA is performed in high-dimensional space. KPCA first projects original data into high-dimensional features through nonlinear mapping. To further deal with the nonlinear characteristics among the variables, kernel-based MSPM methods have been widely researched, such as kernel PCA (KPCA). However, PCA and PLS can only obtain linear relationships among process data. Two statistics are constructed to monitor variation in these two spaces. For process monitoring, PCA and PLS project original variables into the low-dimensional feature space consisting of latent variables and the residual space consisting of the original variables and the reconstruction vectors. PLS can be used for predictive and descriptive modeling as well as for the discriminative variable selection. PCA realizes dimensionality reduction based on the variance information of the original variables, and can retain the main information of the original variables. Principal component analysis (PCA) and partial least squares (PLS) are two commonly used methods. This condition significantly promotes the progress of data-driven methods, especially that of multivariate statistical process monitoring (MSPM),. Meanwhile, thanks to the continuous development of industrial informatization, a large amount of process data are stored. The need for factory safety and high quality production gives a major push to the development of process monitoring, which has attracted wide attention in academia and industrial companies, ,. Results show that the proposed method is superior to and more effective than other advanced dynamic process monitoring methods. Three cases are used to verify the performance of the novel approach. A support vector data description model is established to monitor the independent variables. To monitor the dynamic processes, kernel principal component analysis model is constructed on the basis of the residuals, where the residuals are obtained by comparing measured values of instruments with the predicted values of KPCR models. Third, corresponding KPCR models are established to describe the dynamic relationships with the selected dynamic related variables as input variables and with response variables as output variables. Second, process variables are divided into response and independent variable sets. First, dynamic nonlinear related variables are selected for each variable through mutual information by considering variables with different time delays. Therefore, a novel dynamic nonlinear process monitoring method based on dynamic nonlinear feature selection and kernel principal component regression (KPCR) is proposed in this study. To further improve the monitoring performance for dynamic nonlinear processes, establishing a nonlinear filtering model for each variable is necessary. In addition, there are usually complex nonlinearities among variables. Each process variable may have strong autocorrelation and cross-correlation with other variables with different delays. In actual industrial processes, data are usually time series.
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