## 3 Materials and Methods
### Stacking Ensemble  
Stacking ensemble involves constructing models using different learning algorithms and then training a combiner algorithm to make final predictions based on the outputs of the base algorithms. This combiner can be any ensemble technique [36]. In this study, a stacking ensemble approach was implemented, incorporating three backbone models (Informer, MLR, and SVR) with various feature combinations, resulting in twelve base models (INF, INF-RB, INF-UB, INF-RB-UB, MLR, MLR-RB, MLR-UB, MLR-RB-UB, SVR, SVR-RB, SVR-UB, SVR-RB-UB), and one top-level model (MLR). MLR was selected as the meta-learner for our stacking ensemble approach because of its computational efficiency, ease of interpretation, and its role in mitigating overfitting risks. In contrast to more intricate models, MLR offers transparent regression coefficients, which explicitly reveal the influence of each base model on the ultimate forecast. This clarity enhances our understanding of the prediction process. Furthermore, the straightforward nature of MLR is particularly beneficial in scenarios with smaller sample sizes, as it helps to maintain the model's stability and its ability to generalize, thereby ensuring robust and reliable predictions from our stacked model. The architecture of the scheme used is depicted in Figure 4.  
The stacking ensemble model consists of two layers: a base learner layer and a meta-learner layer. The working principles of each layer are outlined using mathematical formulas. **Base learner layer:** Let the training dataset be represented as \((X,Y)\), where \(X\) is divided into \(X_{basic}\), \(X_{region}\), \(X_{usage}\) as input feature matrices, and \(Y\) denotes the output target. Twelve different base learners were selected for training. Each base learner \(f_{m}(X_{i})\) was trained on the training set to produce predictions \(\hat{Y}_{m}\). \[\hat{Y}_{m}\ =\ f(X_{i}),\quad m\ =\ INF,INF-RB,\dots,SVR-RB-UB \tag{17}\]  
where \(X_{i}\in\{X_{basic},X_{basic}+X_{region},X_{basic}+X_{usage},X_{basic}+X_{ region}+X_{usage}\}\). These predictions from the base learners are combined to form a new feature matrix \(Z\), where each column represents the prediction of one base learner:  
\[Z\ =\ [\hat{Y}_{INF},\hat{Y}_{INF-RB},\dots,\hat{Y}_{SVR-RB-UB}] \tag{18}\]  
**Meta-learner layer:** The meta-learner uses the prediction results \(Z\) from the base learners as input and the true target \(Y\) for training. The meta-learner \(g(Z)\) is trained to learn how to best combine the outputs of the base learners to generate the final prediction \(\hat{Y}\). \[\hat{Y}\ =\ g(Z)\ =\ g([\hat{Y}_{INF},\hat{Y}_{INF-RB},\dots,\hat{Y}_{SVR-RB- UB}]) \tag{19}\]  
Specifically, the meta-learner is a multiple linear regression model:  
\[\hat{Y}\ =\ \beta_{0}+\beta_{1}\hat{Y}_{INF}+\beta_{2}\hat{Y}_{INF-RB}+\dots+ \beta_{12}\hat{Y}_{SVR-RB-UB} \tag{20}\]  
where \(\beta_{0}\) denotes the intercept, and \(\beta_{1},\beta_{2},\cdots,\beta_{12}\) are the regression coefficients, determined by minimizing the least squares error function. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline kernel & Type of kernel function & ’rbf’ (Radial Basis Function) \\ C & Regularization parameter & \(10^{0}\), \(10^{1}\), \(10^{2}\), \(10^{3}\) \\ gamma & RBF kernel parameter & np.logspace (\(-2\), \(2\), \(5\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: SVR model training parameters. The detailed steps for stacking prediction are as follows:  
1. Data preprocessing: Split the original dataset \((X,Y)\) into the training set \((X_{train},Y_{train})\) and test set \((X_{test},Y_{test})\);
2. Base learner training: Train each base learner on the training set to generate predictions \(\hat{Y}_{m,train}\). Simultaneously, perform predictions on the test set to generate \(\hat{Y}_{m,test}\);
3. Secondary training set generation: Combine the predictions of the base learners on the training set to form a new feature matrix \(Z_{train}\).