## 3 Methodology
### Semiparametric GWR Method  
The Geographically Weighted Regression (GWR) model is a local spatial modelling technique and an extension of the traditional linear regression model, which primarily considers spatial heterogeneity when analysing the spatial relationship among geographical objects [31]. The GWR model can be expressed as:  
\[y_{i}=\beta_{0}(u_{i},v_{i})+\sum_{k=1}^{p}\beta_{k}(u_{i},v_{i})x_{ik}+ \varepsilon_{i} \tag{8}\]  
where \((u_{i},v_{i})\) represents the geographic location of the ith point, \(y_{i}\) represents the fitting value of dependent variable at location \(i\), \(\beta_{k}(u_{i},v_{i})\) represents the local parameter estimate for the independent variable \(k\) at location \(i\), \(\beta_{0}(u_{i},v_{i})\) and \(\varepsilon_{i}\) represent the estimated intercept and error term for location \(i\), respectively, and \(\beta_{k}(u_{i},v_{i})\) is estimated by the weighted least square method and expressed in matrix form as follows:  
\[\hat{\beta}(u_{i},v_{i})=\left[X^{T}W(u_{i},v_{i})X\right]^{-1}X^{T}W(u_{i},v _{i})Y \tag{9}\]  
where \(W(u_{i},v_{i})\) is a spatial weight matrix, and \(X\) and \(Y\) are the independent variable and dependent variable matrixes, respectively.  
The simultaneous spatial heterogeneity and homogeneity of the impact factors of vegetation cover dynamics are difficult solve using only the local parameters included in the spatial regression model performed by the GWR [32; 33]. This study used a mixed semi-parametric geographically weighted regression (SGWR) model--an extension of traditional GWR local space modelling--to study the relationship between fractional vegetation cover change and impact factors, and contains local and global coefficients to simultaneously analyse the spatial stationarity and non-stationarity of the impact factors. The SGWR model can be expressed as follow:  
\[y_{i}=\beta_{0}(u_{i},v_{i})+\sum_{l=1}^{j}\alpha_{l}x_{il}+\sum_{k=j+1}^{p} \beta_{k}(u_{i},v_{i})x_{ik}+\varepsilon_{i} \tag{10}\]  
where \(\alpha_{l}\) represents the global regression parameter estimation of independent variable \(x_{il}\). The kernel function is one of the key parameters in the SGWR model, which includes the Poisson function, Gaussian function, and logistic regression function methods. The Gaussian function is usually chosen as the kernel function of spatial weight:  
\[w_{ij}=exp\left[-(\frac{d_{ij}}{b})^{2}\right] \tag{11}\]  
where \(d_{ij}\) is the distance between location point \(j\) and \(i\); \(b\) is the bandwidth, a non-negative attenuation parameter describing the functional relationship between weight and distance.  
A geographic variability test (GVT) is used to determine local and global variables in the SGWR model based on the comparison of multiple GWR models. The Akaike information criterion (AIC) values of two GWR models were compared to evaluate the geographic variability of the kth independent variable. The standard deviation of the geographic variability test (DIFF of criterion) is introduced to test the geographic variability of the \(k\)th variable. A positive value of DIFF of criterion indicates that the corresponding variable does not have geographic variability, whereas a negative value indicates that the variable has geographic variability in space.  
For comparison purposes, we fitted the ordinary least squares model (OLS) and the GWR regression using with the same dataset and variables. The three models were compared using the Akaike information criterion (AIC), residual sum of squares, and adjusted R\({}^{2}\) values [34; 35]. The model with a lower AIC value means better fits the true value. The sum of the squares of the residuals is the sum of the squares of the differences between the actual value and the model simulation value; therefore, smaller sums denote higher fitting accuracy. The adjusted R-square value better represents the fitting degree of the model to the data for multiple regression analysis models.