## 2 Data and Methods
### Statistical Analysis Strategy
Furthermore, to maximize the correlation, we performed the following process [42; 43; 44] when obtaining the partial correlation coefficient between NDVI\({}_{\text{max}}\) and a factor: First, we obtained the maximum value of the NDVI for every year; then, we determined the month in which the NDVI reached its maximum value; and finally, we obtained the median as the final result month for this cell. If NDVI\({}_{\text{max}}\) occurred in June, we calculated the partial correlation coefficients between NDVI\({}_{\text{max}}\) and SM in June and the average SM in June-May, June-April, June-March, June-February, and June-January. The control variable in the partial correlation was the average VPD of the same period. If the absolute value of the partial correlation coefficient between NDVI\({}_{\text{max}}\) and the average SM for April to June was the highest, then the presesacon length of the pixel SM was 2 months (zero in the current month); this represented the ultimate partial correlation coefficient between NDVI\({}_{\text{max}}\) and SM for the pixel. In addition to calculating the partial correlation coefficients of NDVI\({}_{\text{max}}\) with SM and VPD, we also calculated the partial correlation coefficients of NDVI\({}_{\text{max}}\) with three other meteorological factors. To better elucidate the impact of each meteorological factor on peak vegetation growth, we determined the strongest correlated meteorological factor for each pixel location using the maximum absolute value of the partial correlation coefficients. In addition, this study also employed ridge regression analysis due to the collinearity issues among the climate factors [45; 46; 47]. This is a biased estimation regression method specifically designed for analyzing collinear data. We first normalized the data and input them into the ridge regression model. Then, we used generalized cross-validation to select the optimal parameters for the ridge regression model [45; 48]. The explanatory variables in this study are the climate factors on an annual scale, including SM, VPD, temperature, precipitation, and radiation, while the dependent variable is NDVI\({}_{\text{max}}\). The coefficient of the ridge regression is used as the intensity of the sensitivity [40; 49]. By identifying the results with the largest absolute values of the ridge regression coefficients, we obtained the climate factor with the greatest sensitivity to NDVI\({}_{\text{max}}\) for each pixel location. \[\theta(\alpha)=\ \left(X^{T}X+\alpha I\right)^{-1}X^{T}y \tag{8}\]  
\[X_{norm}=\frac{x-min(x)}{max(x)-min(x)} \tag{9}\]  
\[Y_{norm}=\sum_{i=1}^{n}a_{i}x_{norm}+b \tag{10}\]  
where \(\theta(\alpha)\) represents the analytical solution for the ridge regression coefficients, allowing for the direct calculation of the regression coefficients. \(a_{i}\) is a regularization term. Typically, the appropriate \(\alpha\) value is selected through generalized cross-validation to find the best model performance. \(X_{norm}\) represents the standardization of the data, including vegetation peak, NDVI\({}_{\text{max}}\), and meteorological factors such as SM, VPD, temperature, precipitation, and radiation. \(Y_{norm}\) represents the standardized NDVI\({}_{\text{max}}\), \(x_{norm}\) represents the standardized climate factors, and \(a_{i}\) represents the ridge regression coefficients.