###### Abstract
A few of the drawbacks of a univariate control system are that specific parameters can indeed be measured and manipulated for a singular process [1]. This provokes the elaboration of multivariate quality control approaches to address these drawbacks by testing multiple factors simultaneously [2]. Through multivariate methodology, professionals and manufacturing firms that track complicated systems can regulate their system's reliability. The kernel principal component analysis (KPCA) has become one of the increasingly prevalent nonlinear statistical techniques [3]. Statistical flaw detection methods focused on hypothesis tests, such as the Generalized Likelihood Ratio chart (GLR), seem useful calculation statistics for PCA and KPCA. The Generalized Likelihood Ratio (GLR) framework is a statistical hypothesis method that utilizes the principle of maximum likelihood estimation. Therefore, it is designed to have the optimal identification outcomes as it significantly increases the detection accuracy with a specific false warning rate [4]. It is one of the leading efficient strategies of univariate system identification. The GLR framework presently used throughout research (attributed to residues derived from the PCA concept) was developed to monitor the mean changes for numerous purposes, including ozone research [5] and chemical process data [6]. The foremost aim of this article is to analyze the sequential sampling GLR chart's efficiency under multiple fault situations through a nonlinear system identification algorithm that relies on KPCA-KDE. With such a short measurement of the SPE in future space, the KDE framework is used to generate UCLs for PCA and KPCA. In a linear approach such as PCA, the estimation of the upper control limits (UCLs) is dependent on the proposition that the state variables used in the measurements are Gaussian. Based on this statement, the specific model can be extracted analytically. The UCLs can therefore also be extracted analytically. Moreover, numerous actual manufacturing systems are nonlinear. Although the origin of such a system's unpredictability may be considered Gaussian, the variables used in the calculated results are non-Gaussian due to the intrinsic nonlinearities. Thus, the adoption of UCLs for fault diagnosis depending on multivariate Gaussian assumptions in such methods is improper, in addition to that it can contribute to imprecise outcomes. Current GLR charts research assesses the output utilizing ARL\({}_{1}\) measurements [4; 7; 8]. The next importance of this study is to investigate the SSGLR chart's efficiency using all necessary fault diagnosis conditions: false detection rate (FDR), false alarm rate (FAR), and ARL\({}_{1}\). Ref. [9] implemented a linear PCA-based GLR to identify faults, such that the dataset will be first designed using a linear PCA framework. Afterward, the flaws are identified through GLR statistics. The drawback of PCA is that it is not applicable in nonlinear situations and suggests that the interaction amongst variables is linear, thus might not necessarily be the most acceptable measurement form. In conclusion, several chemical mechanisms, such as fractional distillation, are typically represented as nonlinear systems. Numerous measures have been developed [10] ([PERSON] et al., [11] and [PERSON] and [PERSON] et al. [12]) to fix nonlinear PCA extensions and to produce virtuous results across linear models. KPCA is amongst the most renowned nonlinear mathematical techniques [11] ([PERSON] et al. (1997) [3], [PERSON] [13], and [PERSON] et al. [14]) which is a tool for performing a nonlinear type of principal component analysis that can effectively measure main components in elevated structures linked to input space through specific nonlinear modeling and assessment. As a result, in this study, we tackle the problem of nonlinear fault identification by first modeling the data with the KPCA method and then detecting the faults with SSGLR.