### Literature Review  
Synthetic nonlinear data can be simulated through the following. In 1924, [PERSON] [15] initially introduced an \(\overline{X}\)-chart to track output metrics online in an attempt to identify deficiencies in the system in the right direction. However, only recent observations were taken into account in the \(\overline{X}\)-chart. Hence, it is inefficient to identify minor and medium mean changes in the procedure. The Cumulative Sum (CUSUM) defined by [PERSON] [16] and the Exponentially Weighted Moving Average (EWMA) suggested by [PERSON] [17] take into consideration previous results to detect slight mean changes in the procedure. As a result, investigations have shown that both techniques are inadequate at detecting substantial system alterations. This prompts the development of multivariate quality control systems to solve these flaws by concurrently assessing numerous aspects [2]. Experts and industrial organizations that monitor complex processes can control their system's dependability using the multivariate approach. Chi-square and Hotelling T\({}^{2}\) charts [18] are the most prominent multivariate control charts. In particular, these models are an expansion of the univariate Shewhart-type control chart. Thus, they are generally prone to minor and medium system changes. [PERSON] [19] suggested a multivariate CUSUM (MCUSUM) monitoring plan resulting through a Sequential Probability Ratio Test (SPRT). The MCUSUM successfully identifies changes all along the pre-specific path defined by the out-of-control mean curve. This indicates that MCUSUM may be unsuccessful if the system requires turning in an unexpected direction. [PERSON] et al. [20] described a multivariate EWMA (MEWMA) process chart that has been irreducible in the context of the change. Its efficiency has been perceived to be at a minimum as high as MCUSUM. [PERSON] and [PERSON] [21] further developed a multivariate GLR monitoring model and demonstrated that the MGLR chart may provide significantly superior total efficiency in predicting a considerable variation of changes. The other option of advancing performance is using the GLR (Generalized Likelihood Ratio) control charts based on Sequential Probability Ratio measures. The literature has too many GLR statistic discoveries to track: Bernoulli/binomial models and procedures, GLR charts for geometric observation, auto-correlated structures, and univariate standard processes. For reference,see [PERSON] et al. [22], [PERSON] et al. [7], [PERSON] and [PERSON] [8], [PERSON] and [PERSON] [4], [PERSON] and [PERSON] [23], and [PERSON] [24]. There seems to be a growing concern in multivariate statistical process monitoring (MSPM) across both scientific and corporate systems. Specific renowned data-focused fault analysis approaches have been established comprising independent component analysis (ICA) [25], principal component analysis (PCA), partial least squares (PLS) [26; 27], and canonical variate analysis (CVA) [28]. PCA-based instability analysis and evaluation is an appealing factual domain and has received substantial interest among numerous researchers in the disciplines ([PERSON] and [PERSON] [29]). PCA ([PERSON] [30]) is a multivariate statistical process control (MSPC) methodology used to diagnose irregularities in the evaluation of heavily reliant factors. The PCA's central research goal is to quantify observations and factors. It also minimizes the uncertainty of original data (X) by moving data selection to a restricted dataset and compiling specific parameters to protect many current dataset descriptions. The PCA concept describes a dataset (consisting of many system factors) using a smaller range of preserved variables identified as the key elements representing much of the model uncertainties. This improves computing performance, reducing the time to locate and transmit flaws. These properties make it a prominent option for surveillance of the manufacturing processes ([PERSON] et al. [31]).