## 3 SSGLR-Based Monitoring  
Hypothesis assessment procedures such as the Generalized Likelihood Ratio (GLR) have acknowledged significant consideration in earlier research [9; 12]. Inference methodological processes utilize precise mathematical assumptions to decide whether the data are compatible with or divergent from a prescribed mechanism. The null hypothesis matches a negative distortion, termed an alternate concept. For efficiency evaluation methods, the shortcomings of contingent and self-reliant variable assessments are explicitly defined using examples from standard and irrational operating techniques [1]. The Generalized Probability Ratio model can be utilized for this reason and is discussed in the following section. The Generalized Likelihood Ratio (GLR) procedure with sequential sampling identifies replacement theories based on parameters that support infinite values and is commonly alluded to as a composite theory. The sequential sampling considered here refers to the situation in which observations are taken one by one. A reasonable quality estimation method that uses the Maximum Likelihood Estimates (MLEs) concept has been used to estimate the appropriate criteria. The SSGLR chart follows the idea of maximum probability projections to optimize the identification frequency with a set untrue warning level. The SSGLR mechanism is carried out in the subsequent section:  
1. The zero and alternative assumptions are established, and their corresponding probability features are obtained. 2. All uncertain dimensions in the alternative hypothesis are calculated from the test results through the MLE. 3. The log probability proportion of alternative to null hypotheses is then estimated, and its maximal probability is determined that significantly increases the measurement distance. Consider at sampling point k we have the data X\({}_{1}\),... X\({}_{k}\), where \(i=1,2,\ldots k-1\), \(X_{i}=X_{i1},\ldots X_{ini}\) and there are \(l\) observations in sample k. The null assumption once there is no alteration in the mean can be interpreted as:  
\[L(\infty,\mu_{0}|X_{1},X_{2},\ldots,X_{k})=\left(2\pi\delta_{0}^{2}\right)^{ -\frac{N_{k,l}}{2}}exp\left[-\frac{1}{2\delta_{0}^{2}}\sum_{i=1}^{k-1}\sum_{j =1}^{n_{i}}\left(X_{ij}-\mu_{0}\right)^{2}+\sum_{j=1}^{l}\left(X_{kj}-\mu_{0} \right)^{2}\right]\]  
Assume that a distinct source produces an abrupt change in the procedure amongst the observation \(\tau\) and \(\tau+1\) then parameter \(\mu\) moves to \(\mu_{1}\) (\(\mu_{1}\
eq\mu_{0}\)) from \(\mu_{0}\), i.e., samples \(X_{\tau+1}\), \(X_{\tau+2}\),... are strained from a procedure that is out-of-control with the parameter \(\mu_{1}\). Then, the likelihood function can be presented as follows:  
\[L(\tau,\mu_{1}|X_{1},X_{2},\ldots,X_{k})=\left(2\pi\delta_{0}^{2}\right)^{ \frac{N_{k,l}}{2}}exp\left[-\frac{1}{2\delta_{0}^{2}}\sum_{i=1}^{\tau}\sum_{j =1}^{n_{i}}\left(X_{ij}-\mu_{0}\right)^{2}+\sum_{i=\tau+1}^{k-1}\sum_{j=1}^{n _{i}}\left(X_{ij}-\mu_{1}\right)^{2}+\sum_{j=1}^{l}\left(X_{kj}-\mu_{1}\right) ^{2}\right]\]We can attain the maximum likelihood estimation of \(\mu_{1}\), i.e., \(\hat{\mu}_{1}\), through the maximization of \(L(\tau,\mu_{1}|X_{1},X_{2},\ldots,X_{k})\), which is:  
\[\hat{\mu}_{1,\tau,k}=\frac{1}{N_{1,k,l,\tau}}{\sum_{i=\tau+1}^{k-1}}{\sum_{j=1 }^{n_{i}}}{X_{ij}}+{\sum_{j=1}^{l}}{X_{kj}} \tag{15}\]  
Subsequently, a log likelihood ratio statistic for defining whether there has been a change in the procedure mean in the previous \(k\) trials is:  
\[R_{k,l}=\ln\max_{0\leq\tau\leq k}\frac{L(\tau,\hat{\mu}_{1}|X_{1},X_{2},\ldots,X_{k})}{L(\mu_{0}|X_{1},X_{2},\ldots,X_{k})}\]  
\[R_{k,l}=\max_{0\leq\tau\leq k}\frac{N_{1,k,l,\tau}}{2\delta_{0}^{2}}(\hat{\mu} _{1,k,l,\tau}-\mu_{0})^{2}\]  
In calculating \(R_{k,l}\), we must track all previous data points from samples 1 to \(k\) in addition to finding the maximum, which entails large computation when k is big. A window of size m pertaining to the maximization in \(R_{k,l}\) is occupied only over the past m samples.