## 2 Methodology
### Random Forest Regression Method  
Ensemble learning is a common technique that involves combining multiple learners to improve the performance of a machine learning model [29; 30; 31; 32; 33; 34; 35]. This approach is often divided into two categories based on whether the base learners are generated simultaneously or sequentially. The random forest regression model, which is based on the Bagging (bootstrap sampling) algorithm, exhibits the characteristics of a traditional parallel ensemble learning model, such as randomness in both feature selection and training set composition. Figure 5 illustrates the architecture. In random forest, bootstrapped datasets are created by randomly selecting samples from the training set with replacement, and these datasets are used to train multiple tree learners. The size of each bootstrapped dataset matches that of the original training set. Furthermore, instead of splitting nodes using all available attributes, decision trees in the random forest are constructed using random subsets of the features. A regression decision tree, which is built by partitioning the feature space into subgroups based on a set of splitting criteria, acts as the base learner for the random forest regression model. The training set \(D\) is represented by the following equation. \[\mathbf{D}=\{(x_{1},y_{1}),(x_{2},y_{2}),\ldots(x_{N},y_{N})\}, \tag{2}\]  
Each data point in the training set is represented by \(\mathrm{x_{i}}\), where \(\mathrm{x_{i}}\) is a feature vector (e.g., a set of measurements or attributes), and \(\mathrm{y_{i}}\) is the corresponding label or outcome associated with \(\mathrm{x_{i}}\) (e.g., the classification or predicted value). For example, in a dataset of weather data, \(\mathrm{x_{i}}\) could represent features like temperature and humidity, while \(\mathrm{y_{i}}\) could correspond to a label indicating whether it ra  
Figure 5: The random forest’s architecture. where \(N\) is the total number of data points in the training set. \(\mathbf{x}_{i}=\left(x_{i}^{1},x_{i}^{2},\ldots,x_{i}^{n}\right)\), where \(\mathbf{x}_{i}\) denotes the feature vector, and \(n\) represents the number of features. To generate multiple training sample sets, data is randomly sampled from the original dataset \(\mathbf{D}\) using the Bagging (bootstrap sampling) method. For each training sample set, a Classification and Regression Tree (CART) is constructed using the node random splitting technique, and the least mean square error principle is applied. For any partition feature j corresponding to a partition node \(s\), the dataset is divided into two subsets, \(\mathbf{R}_{1}\) and \(\mathbf{R}_{2}\). The partition feature \(j\) is determined by minimizing the mean square errors of \(\mathbf{R}_{1}\) and \(\mathbf{R}_{2}\), as well as the sum of these errors. The following are the resulting expressions:  
\[\min_{j,s}[\sum_{\mathbf{x}_{i}\in\mathbf{R}_{1}(j,s)}{(y_{i}-\mathbf{C}_{1})^{2}}+\sum_{ \mathbf{x}_{i}\in\mathbf{R}_{2}(j,s)}{(y_{i}-\mathbf{C}_{2})^{2}}], \tag{3}\]  
\[\mathbf{C}_{1}=\frac{1}{\mathbb{N}_{\mathbf{x}_{i}\in\mathbf{R}_{1}(j,s)}}y_{i}, \tag{4}\]  
\[\mathbf{C}_{2}=\frac{1}{\mathbb{N}_{\mathbf{2}_{x}\in\mathbf{R}_{2}(j,s)}}y_{i}, \tag{5}\]  
\(\mathbf{R}_{1}=\left\{x\Big{|}x_{i}^{(j)}\leq s\right\}\) and \(\mathbf{R}_{2}=\left\{x\Big{|}x_{i}^{(j)}>s\right\}\) are the split areas. Following the division of the feature space into \(M\) areas (\(\mathbf{R}_{1}\), \(\mathbf{R}_{2}\),..., \(\mathbf{R}_{M}\)), the process is repeated for each split area until the stop criterion is met. \(\mathbf{C_{1}}\) and \(\mathbf{C_{2}}\) represent the optimal output values of the two areas, respectively, and \(\mathbb{N_{1}}\) and \(\mathbb{N_{2}}\) are the number of samples in datasets \(\mathbf{R}_{1}\) and \(\mathbf{R}_{2}\), respectively. Consequently, a regression tree is designed as  
\[f(x)=\sum_{m=1}^{M}\mathbf{C}_{m}I(x\in\mathbf{R}_{m}), \tag{6}\]  
where the following equation provides the indicator function. \[I=\left\{