## 2 Materials and Methods
### Data Analysis Methods
#### 2.3.3 Ridge Regression Analysis  
Ridge regression, incorporating L2 penalty to manage predictor collinearity, was applied to isolate the effects of target climatic drivers on grassland GPP in arid Xinjiang by minimizing interference from correlated covariates. We employed ridge regression analysis in MATLAB 2024b to model the relationship between GPP and the four meteorological factors (VPD, PRE, TEM, SSR) simultaneously, thereby mitigating multicollinearity issues [28].  
This method has been widely applied in previous studies to explore the relationships between GPP and key meteorological factors within a multivariate framework, providing more robust coefficient estimates. Ridge regression effectively alleviates multicollinearity issues by introducing a regularization parameter \(\lambda\) when estimating the coefficients \(\beta\). All ridge regression computations were performed using MATLAB 2024b. The formula for ridge regression is as follows [29]:  
\[\hat{\beta}_{ridge}=\left(X^{T}X+\lambda I\right)^{-1}X^{T}y \tag{6}\]  
In the ridge regression formula, \(X\) is the matrix of independent variables, \(y\) is the dependent variable matrix, \(\lambda\) is the regularization parameter, and \(I\) is the identity matrix.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Category** & **Correlation (R)** & \(p\)**-Value Range** \\ \hline Extremely Significant Negative Correlation & \(R<0\) & \(p<0.01\) \\ Significant Negative Correlation & \(R<0\) & \(0.01\leq p<0.05\) \\ Not Significantly Negatively Correlated & \(R<0\) & \(p\geq 0.05\) \\ Not Significantly Positively Correlated & \(R>0\) & \(p\geq 0.05\) \\ Significant Positive Correlation & \(R>0\) & \(0.01\leq p<0.05\) \\ Extremely Significant Positive Correlation & \(R>0\) & \(p<0.01\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Classification of partial correlation significance.  
Prior to analysis, all variables were standardized in MATLAB 2024b so that variables with different units could be analyzed together:  
\[X_{m}=\frac{x-\min(x)}{\max(x)-\min(x)} \tag{7}\]  
where \(X_{m}\) is the normalized data of the climatic factors including VPD, PRE, TEM, and SSR.  
Normalization of each independent variable was essential to ensure that variables with different units could be appropriately compared and analyzed. The formula for normalization is as follows:  
\[Y_{m}=\sum_{i=1}^{n}a_{i}X_{im}+b \tag{8}\]  
In the equation, \(Y_{m}\) represents the normalized GPP, \(X_{im}\) denotes the normalized meteorological factors, and \(a_{i}\) are the regression coefficients.  
To better illustrate the impact of each climatic factor on GPP across different regions, this study simultaneously calculated the relative contribution rate using MATLAB 2024b, defined as follows:  
\[\eta_{c1}=a_{1}X_{trend} \tag{9}\]  
\[\eta_{rc1}=\frac{|\eta_{c1}|}{|\eta_{c1}|+|\eta_{c2}|+|\eta_{c3}|+|\eta_{c4}|} \tag{10}\]  
In these equations, \(\eta_{c1}\) represents the absolute contribution of a specific meteorological factor to GPP, while \(a_{1}\) is the regression coefficient of that climate factor obtained through ridge regression analysis. \(X_{trend}\) refers to the standardized trend of the climate factor, derived through normalization. \(\eta_{rc1}\) denotes the relative contribution rate of the meteorological factor to GPP, indicating the proportionate contribution of the meteorological factor to changes in GPP. This comprehensive contribution analysis was implemented in MATLAB 2024b. This metric was used for comparative evaluation of the relative importance of different factors.