## 2 Materials and Methods
### Methods  
Three supervised machine learning classifiers (support vector machine, artificial neural network, and K-nearest neighbor) were evaluated for the research analysis. Artificial neural networks are used as machine learning classifiers modeled after how the nervous system and brain work. This classifier was chosen because it offers a method of simulating non-linear connections between systems. According to [33], the multi-layered feed-forward neural network is commonly adopted to analyze pattern recognition tasks. The layers are three stages that receive information; the output is where the processing results are given and hidden, and the layers are between input and output. Each layer comprises a central unit called the perceptron unit, modeled by the McCulloch-Pitts [33] equation, given below. \[q^{k}(r)=\varnothing^{k}\Big{(}\sum_{i=1}^{p}\omega_{ik}\;r_{i}+b_{k}\Big{)} \tag{3}\]  
where \(r\) corresponds to the input vector, \(b_{k}\) is a bias, \(\omega_{ik}\) is a weight coefficient, and \(\varnothing^{k}\) is a non-linear activation function. The hyperparameter settings for this classifier comprise four hidden layers with a corresponding number of 192, 32, 24, and 1 neurons per layer, chosen to increase the model's performance. The batch size of the neural network, in addition to the layers of the neurons, is 512, with an optimal learning rate of 0.01 and an epoch of 20. \begin{table}
\begin{tabular}{c c} \hline \hline
**Zone** & **Candidate Leak Nodes in the Network for Each Zone** \\ \hline Zone 1 & \\ (8 candidate leak nodes) & \(Nd_{4}\), \(Nd_{5}\), \(Nd_{3}\), \(Nd_{7}\), \(Nd_{9}\), \(Nd_{8}\), \(Nd_{11}\), \(Nd_{13}\) \\ \hline Zone 2 & \\ (7 candidate leak nodes) & \(Nd_{14}\), \(Nd_{17}\), \(Nd_{12}\), \(Nd_{15}\), \(Nd_{16}\), \(Nd_{10}\), \(Nd_{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Candidate leak zones used for the generation of the dataset for model training. The support vector machine is a supervised learning technique widely used for classification and regression based on structural risk minimization. This technique creates a decision boundary between classes by mapping the training data onto a higher-dimensional space and then obtaining the maximum margin hyperplane within that space [34]. The objective of the techniques is achieved by locating the optimal separating hyper plane that maximizes the margin between the closest sample points in the training dataset, which are called support vectors [33]. \(X\in\left[k_{i},y_{i}\right]^{n}\) represents a dataset with two classes, and \(k\in R^{p}\) denotes the measured variables. In order to determine the hyper plane g(r), \(n\) is the number of the training samples, while the label vector is \(Y\in\left[1,-1\right]\) where the classes are separated by g(r) = \(W^{t}r+b\). The position of a separating hyperplane is determined by \(W\) and \(b\) (bias). The geometric separation of the mapped data in the high-dimensional space is defined by the term \(\gamma\) in this context. Discriminant techniques are used to extend two-class classification problems to multi-class classification problems. In the present study, the hyperparameter for the support vector machine model is given Equation (4), which represents the radial basis function kernel that was selected because of its generality and successful results in fault diagnosis applications. \[K\big{(}r_{i},r_{j}\big{)}=exp\Big{(}-\gamma\big{|}\big{|}r_{i}-r_{j}\big{|} \big{|}^{2}\Big{)} \tag{4}\]  
The k-NN algorithm is based on the principle that the most similar samples belonging to the same class have a high probability [35]. Using the the K-NN algorithm, the Euclidean distance function is applied to calculate the similarity or difference between classes. The features of the system consist of measurement, which represents the class.