## 2 Materials and Methods
### Attribution Analysis Methods  
In this study, a comprehensive analytical framework was employed to investigate air temperature factors influencing vegetation growth. This framework incorporates three attribution methods to circumvent the limitations of using any individual method. Firstly, a partial correlation analysis method was utilized to examine the positive or negative impacts of each air temperature indice on plant growth while controlling other variables. Subsequently, standardized multivariate linear regression was employed to identify the driving factors behind vegetation growth. The explanatory power of factors was used to assess the explanatory power of different air temperature factors on vegetation growth based on the GeoDetector. These methods enable a more comprehensive and nuanced understanding of the factors that contribute to plant growth. Partial correlation analysis, also known as net correlation analysis, analyzes the relationship between two variables when excluding the influence of other variables [30]. Before the analysis, we detrended the dataset to eliminate the influence of the trend [31]. The influences of a specified factor excluding other factors related to vegetation growth was found using the following formula:  
\[\rho_{l,k,m}=\frac{\rho_{lk}-\rho_{lm}\,\rho_{km}}{\sqrt{\big{(}1-\rho_{lm}^{2 }\big{)}\big{(}1-\rho_{km}^{2}\big{)}}}\, \tag{2}\]  
where \(\rho_{l,k,m}\) represents the partial correlation coefficient between \(l\) and \(k\), excluding the influence of \(m\). The Geographical Detector method is a relatively new statistical approach for detecting spatial heterogeneity. A Geographical Detector includes four detectors: a risk detector, a factor detector, an ecosystem detector, and an interaction detector [32]. Based on the factor detector, Y was employed as the vegetation indicator, and X was employed as each air temperature factor. The \(q\) value represented the level of explanatory power that factor X has on the spatial heterogeneity of factor Y:  
\[q=1-\frac{\sum_{h=1}^{L}N_{h}\sigma_{h}^{2}}{N_{\sigma}^{2}}=1-\frac{SSW}{SST} \tag{3}\]  
\[SSW=\sum_{h=1}^{L}N_{h}\sigma_{h}^{2},\ SST=N\sigma^{2} \tag{4}\]  
In the equation, \(h=1,\ldots,L\) represents the stratification of variable \(Y\) or factor \(X\), that is, classification or zoning; \(N_{h}\) and \(N\) are the number of units in layer \(h\) and the entire region, respectively; and \(\sigma_{h}^{2}\) and \(\sigma^{2}\) are the variances of \(Y\) in layer \(h\) and the entire region,respectively. \(SSW\) and \(SST\) are the sums of the within-layer variances and the total variance of the entire region, respectively. The value of \(q\) ranges from 0 to 1. The larger \(q\) illustrates the stronger influence of temperature determinants on vegetation and the more distinct spatial heterogeneity. A larger value of \(q\) indicates that the independent variable \(X\) has a stronger explanatory power for the attribute \(Y\) when the stratification is generated by \(X\). A \(q\) value of 1 means that \(X\) dominates the spatial and temporal variations of \(Y\), and \(100\times q\%\) illustrates the extent to which \(X\) explains \(Y\). Its significance can be tested in the calculation process [32]. Standardized multiple linear regression was utilized to explore the dominant regions where air temperature factors significantly influence vegetation growth. In the standardized multiple linear regression (SMLR) model, the coefficient before each variable denotes its significance in dividing the SMLR model. Prior to analysis, the variance inflation factor (VIF) between influencing factors was computed to detect multicollinearity.