Local and Global Randomized Principal Component Analysis for Nonlinear Process Monitoring

Abstract

Kernel principal component analysis (KPCA) has been widely used in nonlinear process monitoring since it can capture the nonlinear process characteristics. However, it suffers from high computational complexity and poor scalability while dealing with real-time process monitoring and large-scale process monitoring. In this paper, a novel dimension reduction technique, local and global randomized principal component analysis (LGRPCA), is proposed for nonlinear process monitoring. The proposed LGRPCA method first maps the input space onto a feature space to reveal nonlinear patterns through random Fourier features. With the aid of random Fourier features, the proposed LGRPCA method is scalable and with much lower computational and storage costs. To exploit the underlying local and global structure information in the feature space, local structure analysis is integrated into the framework of global variance information extraction. The resulting LGRPCA can provide an improved representation of input data than the traditional KPCA. Thus, the proposed LGRPCA method is quite suitable for real-time process monitoring and large-scale process monitoring. $T^{2}$ and squared prediction error (SPE) statistic control charts are built for fault detection using the proposed LGRPCA method. Furthermore, contribution plots to LGRPCA-based $T^{2}$ and SPE ( $Q$ ) statistics are established to identify the root cause variables through a sensitivity analysis principle. The superior performance of the proposed LGRPCA-based nonlinear process monitoring method is demonstrated through a numerical example and the comparative study of the Tennessee Eastman benchmark process.