## 3 Model Selection
### Regression Prediction  
In this paper, several regression prediction algorithms were employed, such as linear regression, ridge regression, lasso regression and decision tree regression. (1) Linear Regression [21]: Linear regression involves finding a line (or hyperplane in a multidimensional space) that minimizes the sum of squared differences between the observed data points and the predicted values. It is advantageous for making accurate predictions with small datasets. However, it is sensitive to data noise and may encounter issues when dealing with large amounts of data or high-dimensional feature spaces. The calculation formula is provided as follows:  
\[L=\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2}=\sum_{i=1}^{n}\left(y_{i}- \left(\beta_{0}+\beta_{1}x_{i1}+\ldots+\beta_{n}x_{in}\right)\right)^{2} \tag{3}\]  
where \(y_{i}\) is the actual value for the \(i\)-th observation; \(\hat{y}_{i}\) is the predicted value for the i-th observation, which is calculated using the linear model; \(x_{i1},x_{in}\) are the values of the predictors for the \(i\)-th observation; \(\beta_{0},\beta_{n}\) are the regression coefficients to be estimated; and \(n\) is the number of observations. The regression coefficient \(\hat{\beta}\) is solved by minimizing the loss function, where \(y_{i}\) represents the actual value and \(\hat{y}_{i}\) is the predictive value. (2) Ridge Regression [22]: Ridge regression addresses the issue of potential overfitting in linear regression by adding a regularization term to the loss function. This regularization term helps improve the model's ability to generalize and provides some robustness to data noise. One advantage of ridge regression is that it effectively tackles overfitting. However, a potential drawback is that tuning the ridge parameters may be necessary to achieve the best prediction performance. The calculation formula is provided as follows:  
\[L=\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2}+\lambda\sum_{j=1}^{p}\beta _{j}^{2} \tag{4}\]  
The regression coefficient \(\hat{\beta}\) is solved by minimizing the loss function, where \(\lambda\) represents the regularization parameter. (3) Lasso Regression [23]: Lasso regression is a similar process to ridge regression but uses L1 regularization instead, which can result in some coefficients being reduced to zero, effectively performing feature selection. One advantage of lasso regression is its ability to automatically select features relevant to the target variable. It addresses the fitting problem and improves the model's generalization ability. However, a drawback is that tuning the lasso parameters may be necessary to achieve the best prediction, which can be computationally inefficient for large-scale datasets. The calculation formula is provided as follows:  
\[L=\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2}+\lambda\sum_{j=1}^{p}\left| \beta_{j}\right| \tag{5}\]  
where \(y_{i}\) represents the actual value, \(\hat{y}_{i}\) is the predictive value, \(\lambda\) represents the regularization parameter, and \(\beta_{j}\) represents the parameter vector that needs to be estimated. (4) Decision Tree Regression [24]: In decision tree regression, data are divided into subsets based on a series of rules, and each subset is then predicted using a simple model, typically the average. This segmentation process allows decision tree regression to handle nonlinear relationships well and to be robust to missing values and outliers. However, it is prone to overfitting during training, and the noise present in the training dataset can impact the prediction results. Decision tree regression involves recursively partitioning the data and constructing leaf nodes for each subset, resulting in a tree structure. When given a new input sample, the feature values are evaluated starting from the root node and traversing down to a leaf node, whose value is then used as the prediction result.