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. INDEX TERMS Principal component analysis, random Fourier features, local and global structure analysis, fault detection, fault identification.

A Review of Kernel Methods for Feature Extraction in Nonlinear Process Monitoring

“…This is because both approaches entail differentiating the statistical index, which is difficult if the chain involves a kernel function [86]. Nevertheless, many researchers have derived analytical expressions for either kernel contributions-based diagnosis [66,79,81,83,87,94,119,127,133,136,146,150,156,157,162,164,194,213,241,268,275,276,278,279,288,289,293] or kernel reconstructions-based diagnosis [86,117,140,155,161,163,176,217,236,254,265,285]. However, most derivations are applicable only when the kernel function is the RBF, Equation (5).…”

“…It was adopted recently by Yu et al [277] for kernel CCA. Meanwhile, random Fourier features was adopted by Wu et al [279] for kernel PCA. This scheme exploits Bochner's theorem [59,279], in which the kernel mapping is approximated by passing the data through a randomized projection and cosine functions.…”

“…The projections from GLPP are in the middle of those from LPP and PCA because the local (LPP) and global (PCA) structures are simultaneously preserved. Other works in this regard can be found in [204,215,222,279,282]. To learn more about manifold learning, we refer the reader to a comparative review of dimensionality reduction methods by Van der Maaten et al [320].…”

Reliable Method to Detect Alloy Soldering Fractures under Accelerated Life Test

“…A ductile fracture occurs when the soldering alloy is exposed to plastic deformation due to stress caused by damage including excessive tensions such as thermal variation or mechanical effort. In some research [6][7][8][9][10][11][12][13], it is mentioned that mechanical effort causes parallel cracks, and thermal variation generates a network of multiple fissures. Furthermore, there have been fracture analyses using measures of resistance and visual inspection.…”

“…Due to this, there have been development tests that determine the capacity of a soldering alloy to resist thermal fatigue. Li et al [7], Wu et al [8], and Cheg [14] mentioned that electronic systems are a dynamic object susceptible to vibration. Their investigations demonstrated that a soldering alloy is a place of failure caused by vibrations.…”