## 2 Materials and Methods
### Research Methods
#### 2.3.2 Correlation Analysis  
The advantages of partial correlation analysis mainly lie in its ability to accurately reflect the intrinsic linear relationship between two variables and eliminate the interference of other variables on their relationship. For this study, using partial correlation analysis can result in intuitively recognizing the true linear relationship between NPP and temperature and NPP and precipitation, without being affected by other variables. In this study, an image-by-image metric partial correlation analysis was employed to explore the correlation levels between the annual average NPP and two meteorological factors. The partial correlation coefficient can be computed using the following formula:  
\[\mathrm{R_{xy,z}}=\frac{\mathrm{R_{xy}-R_{xz}\times R_{yz}}}{\sqrt{\left(1- \mathrm{R_{xz}}^{2}\right)}\sqrt{\left(1-\mathrm{R_{yz}}^{2}\right)}} \tag{5}\]  
\[\mathrm{R_{xz,y}}=\frac{\mathrm{R_{xz}-R_{xy}\times R_{yz}}}{\sqrt{\left(1- \mathrm{R_{yy}}^{2}\right)}\sqrt{\left(1-\mathrm{R_{yz}}^{2}\right)}} \tag{6}\]  
where \(R_{xy}\), \(R_{xz}\), and \(R_{yz}\) denote the correlation coefficients of annual mean NPP and annual mean temperature, annual mean NPP with annual precipitation, and between annual average temperature and annual precipitation, respectively. \(R_{xy,z}\) refers to the bias correlation coefficient among the annual mean NPP value and the annual mean temperature of northwest Liaoning when the annual precipitation remained constant, and the impact of the annual precipitation factor was excluded when analyzing the correlation of the mean annual temperature factor. \(R_{xz,y}\) is the correlation coefficient between the annual mean NPP and the annual mean temperature factor's independent effect. Additionally, a \(t\)-test with the following formula was used to determine the significance of the partial correlation coefficients, as follows:  
\[\mathrm{t}=\frac{\mathrm{R_{xy,z}}}{\sqrt{\left(1-\mathrm{R_{xy,z}}^{2}\right) }}\sqrt{\mathrm{n-m-1}} \tag{7}\]  
\[\mathrm{t}=\frac{\mathrm{R_{xz,y}}}{\sqrt{\left(1-\mathrm{R_{xz,y}}^{2}\right) }}\sqrt{\mathrm{n-m-1}} \tag{8}\]  
where \(R_{xyz,z}\) and \(R_{xz,y}\) are partial correlation coefficients, \(n\) is the number of samples (n = 21), and \(m\) is the number of independent variables. After using a partial correlation analysis, we introduced a compound correlation analysis to calculate the degree of correlation for multiple factors, taking into consideration the common effect of multiple influencing factors and the mutual influence and interconnection among factors. The advantages of complex correlation analysis mainly lie in its ability to comprehensively evaluate the complex relationships between one variable and multiple variables, providing more accurate and comprehensive evaluations and explanations. The vast majority of vegetation growth processes are influenced by both temperature and precipitation; therefore, it is crucial to consider the comprehensive impact of temperature and precipitation on NPP. The compound correlation between the annual mean NPP, annual precipitation, and annual mean temperature can be computed using the following formula:  
\[\mathrm{R_{xy,yz}}=\sqrt{1-\left(1-\mathrm{R_{xy}}^{2}\right)\left(1-\mathrm{ R_{xz,y}}^{2}\right)} \tag{9}\]Further, the complex correlation coefficients were examined for significance. The statistics were computed using the _F-test_ technique, as follows:  
\[\text{F}=\frac{\text{R}_{\text{xyz}}^{2}}{1-\text{R}_{\text{xyz}}^{2}}\times \frac{\text{n}-\text{k}-1}{\text{k}} \tag{10}\]  
where \(R_{xyz}\) is the coefficient of complex correlation between the annual mean NPP and the annual mean temperature and annual precipitation, \(n\) is the number of samples (n = 21), and \(k\) is the number of independent variables. NPP-Driven Partitioning  
The main meteorological factors controlling vegetation growth vary across regions.