## 2 Materials and Methods
### Methods
#### 2.3.4 Elastic Net Regression  
The variable selection method of elastic net was first proposed by [PERSON] and [PERSON] in 2005 [66]. The basic principle is to control the model learning process by adding a restriction term to the minimization loss function to prevent data overfitting when dealing with data with linear relationships [67]. The commonly used regularization methods are \(L_{1}\) regularization and \(L_{2}\) regularization. \(L_{1}\) regularization (lasso regression) enhances the generalization ability of the model by adding an \(L_{1}\) penalty to the coefficients, making it possible to perform both continuous compression and automatic selection of variables, which can make the coefficients of some features smaller. \(L_{2}\) regularization (ridge regression) prevents the problem of overfitting by finding the minimum of the loss function, and can deal well with the parameter estimation problem in the presence of multicollinearity in the independent variables, but does not refine the model and usually retains all predictor variables in the model. In contrast, elastic net regression is a combination of \(L_{1}\) regularization and \(L_{2}\) regularization, which essentially gives up the unbiasedness and partial accuracy of the least squares method and seeks a less effective but more realistic regression process, mainly by adding a penalty term to the loss function, so that the model can both feature selection to avoid data redundancy and achieve overfitting prevention while the model is stable. The mathematical expression of the elastic net regression theory model is defined as follows:  
\[y=\hat{\beta}_{0}+\hat{\beta}_{1}x_{1}+\hat{\beta}_{2}x_{2}+\cdots+\hat{\beta} _{k}x_{k}+\varepsilon \tag{9}\]  
where \(x\) represents each influencing factor leading to NPP change, \(y\) represents the percentage of NPP change per unit area, \(\hat{\beta}\) represents the model regression coefficient, and \(\varepsilon\) represents the random error of modeling. Based on the least squares principle, the elastic net regression minimization objective function can be solved by adding the \(L_{1}\) and \(L_{2}\) regularization ideas, and its specific mathematical expressions are defined as follows:  
The \(L_{1}\) regularization term is:  
\[L_{1}=\lambda\rho\|w\|_{1} \tag{10}\]  
The \(L_{2}\) regularization term is:  
\[L_{2}=\frac{\lambda(1-\rho)}{2}\|w\|_{2}^{2} \tag{11}\]  
The loss function is:  
\[J=\sum\limits_{i=1}^{N}\Bigl{(}y_{i}-w^{T}x_{i}\Bigr{)}^{2}+\lambda\rho\|w\|_ {1}+\frac{\lambda(1-\rho)}{2}\|w\|_{2}^{2} \tag{12}\]The expression of the regression coefficient is:  
\[\hat{\beta}=\text{argmin}\left[\frac{1}{2}\sum_{i=1}^{N}\left(y_{i}-w^{T}x_{i} \right)^{2}+\lambda\rho\|w\|_{1}+\frac{\lambda(1-\rho)}{2}\|w\|_{2}^{2}\right] \tag{13}\]  
where \(\{x_{i},y_{i}\}\) is the sampled data, \(x_{i}\) is the feature vector, \(y_{i}\) is the corresponding dependent variable, \(w\) is the feature weight vector, \(\lambda\) is the penalization factor, \(\rho\) is the elasticity factor. \(\rho\) can regulate the proportion of lasso regression and ridge regression. When \(\rho=0\) the above equation becomes lasso regression, when \(\rho=1\) the above equation becomes ridge regression. In this paper, the trial-and-error method is used to calculate the parameters \(\lambda\) and \(\rho\), which take values in the interval of 0.01-0.99. The results showed that the loss function \(J\) is minimized when \(\lambda=0.30\) and \(\rho=0.45\).