## 3 SAL: Proposed Self-Augmentation Labeling Approach
### Function Agreement  
In machine learning, we deal with two types of parameters: (i) machine learnable parameters and (ii) hyper-parameters [27]. The machine learning parameters are the ones which the algorithms learn/estimate on their own during the training for a given dataset. On the other hand, the hyper-parameters are the ones which the machine learning engineers or data scientists will assign specific values to, in order to control the way the algorithms learn, and also to tune the performance of the mode. Since the labeling process in the first stage is subject to strict constraints to ensure that the data are labeled as accurately as possible, we will define and explain two hyper-parameters that completely control the self-augmentation labeling process in our proposed model as follows:  
1. **Labeling threshold**. This threshold \(\alpha\) represents the minimum acceptable probability for labeling the observations. In order to improve the classification accuracy for the results, we will start with a very high value of \(\alpha\) to ensure that only the most accurately labeled observations are classified. The value of this threshold can be tolerated (decreased) by a fixed value which is denoted as \(\gamma\); this means that the threshold \(\alpha\) is decreased by \(\gamma\) when the value of learning rate meets the predefined criterion. 2. **Learning Rate**. The learning rate (LR) is a hyper-parameter that measures the rate or speed at which the model learns the values of the parameters. It evaluates how quickly the model adapts to the problem. In the proposed algorithm, the labeling threshold value is considered the most effective factor on LR. As a result, a high \(\alpha\) value leads to a smaller LR, implying that more training is required. On the other hand, as the value of the \(\alpha\) threshold is decreased, the LR increases, resulting in faster changes and less training. Let \(L_{i}\) denote a variable that represents the number of labeled observations at iteration \(i\), and \(U_{i}\) is a variable that represents the number of unlabeled observations at iteration \(i\). Typically, in our model, LR will be calculated for each iteration \(i\) as follows:  
\[\text{LR}_{i}=\frac{L_{i}}{U_{i}} \tag{5}\]  
Since the proposed approach aims to improve the accuracy as much as possible, which is influenced by the amount of labeled data, we will define the \(\beta\) threshold which denotes the maximum value of the learning rate that updates the \(\alpha\) threshold to ensure that the LR will not be stuck in a constant. So, any LR that exceeds this threshold will block the \(\alpha\) updating in the iteration. So, the LR value, along with \(\beta\) and \(\gamma\) thresholds, will guide the \(\alpha\) threshold decrement process as follows:  
\[\text{Decrement}(\alpha)= \begin{cases}\alpha=\alpha&,\text{If LR }>\beta\\ \alpha=\alpha-\gamma&,\text{If LR }<\beta\end{cases}  \tag{6}\]  
Figure 4: The proposed hybrid ensemble voting algorithm. The reason behind decreasing the \(\alpha\) threshold is because after some number of iterations, we will be unable to find observations that meet the criteria of a high value of \(\alpha\). Figure 5 shows the relation between the number of iterations with and without updating the \(\alpha\) threshold and it indicates that without this updating process, after a number of iterations, the LR will be a _constant function_. 3. **Illustrative Example**. An example is presented below to explain the proposed model's labeling process. Given three selected supervised classifiers (\(C=C1,C2,C3\)) that are trained on a small labeled dataset to provide decision models (\(M=M1,M2,M3\)), in this example, each unlabeled observation belongs to one of five possible labels as follows: \(L1,L2,L3,L4,L5\). Now, assume that \(U\) is the set of unlabeled observations. For a given unlabeled observation \(x_{u}\in U\) and with a given threshold value (\(\alpha=0.95\)), the labeling process is performed using the following steps:  
1.