## 2 Materials and Methods
### Methods
#### 2.3.4. Geographically Weighted Regression (GWR) and Multiscale GWR (MGWR)  
Variance inflation factor (VIF) is employed to detect the severity of multicollinearity in the ordinary least square (OLS) regression analysis. VIF is calculated as [61]:  
\[VIF=\frac{1}{1-{R_{i}}^{2}} \tag{2}\]  
where \({R_{i}}^{2}\) represents the unadjusted coefficient of determination for regressing the \(i\)-th independent variable on the remaining ones. The reciprocal of VIF is known as tolerance. In general, a VIF above 4 or a tolerance below 0.25 suggests the presence of multicollinearity and further investigation is required. When the VIF is greater than 10 or the tolerance is less than 0.1, significant multicollinearity must be rectified. The geographically weighted regression (GWR) was first proposed by [PERSON] et al. (1996) [62] to investigate the local non-stationarity of independent variables in space. Due to the improved fit of the simulation model, GWR has been increasingly utilized in spatial regression than general linear regression models. As GWR is ineffective in diagnostic analysis, OLS regression is essential to validate the model's accuracy prior to GWR analysis [63]. OLS is a global linear regression model in which all independent variables in each region are used to estimate a dependent variable. The basic principle is to minimize the sum of the squares of the differences between the observed dependent variable in the given dataset and those predicted by the linear function of the independent variable. The GWR model derives separate regression coefficients for each grid, unlike the OLS model, which only measures global or average significance. The GWR model is structured as follows [64; 65]:  
\[y_{i}=\beta_{0}(u_{i},v_{i})+\sum_{j=1}^{k}\beta_{\text{bwj}}(u_{i},v_{i})x_{ ij}+\varepsilon_{i} \tag{3}\]  
where (\(u_{i}\), \(v_{i}\)) is the central coordinate at position \(i\); \(y_{i}\) is the attribute value at \(i\); bwj represents the bandwidth used by the regression coefficient of the \(j\)-th variable; \(\beta_{\text{bwj}}(u_{i},v_{i})\) is the regression coefficient for the \(j\)-th variable at \(i\); \(\beta_{0}(u_{i},v_{i})\) and \(\varepsilon_{i}\) are the intercept and error term at \(i\), respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Trajectory** & **State** & **Performance** & **Integrated Degradation Types** \\ \hline Improvement & Improvement & Stable & Improving \\ Improvement & Improvement & Degradation & Improving \\ Improvement & Stable & Stable & Improving \\ Improvement & Stable & Degradation & Improving \\ Improvement & Degradation & Stable & Improving \\ Improvement & Degradation & Degradation & Stable \\ Stable & Improvement & Stable & Stable \\ Stable & Stable & Degradation & Stable but stressed \\ Stable & Degradation & Stable & Early signs of decline \\ Stable & Degradation & Degradation & Declining \\ Degradation & Improvement & Stable & Declining \\ Degradation & Improvement & Degradation & Declining \\ Degradation & Stable & Stable & Declining \\ Degradation & Stable & Degradation & Declining \\ Degradation & Degradation & Stable & Declining \\ Degradation & Degradation & Degradation & Declining \\ \hline \hline \end{tabular} Note: Integration rules are referred to as “SDG Indicator 15.3.1” by Trends.Earth from Conservation International, 2018 ([[https://trends.earth/docs/en/background/understanding_indicators15.html](https://trends.earth/docs/en/background/understanding_indicators15.html)]([https://trends.earth/docs/en/background/understanding_indicators15.html](https://trends.earth/docs/en/background/understanding_indicators15.html)), accessed on 25 May 2021). Copyright 2017–2021 by Conservation International.  
\end{table} 
Table 2: Sub-indicators of land productivity and their integration. In common with the GWR model, the regression coefficient (\(\beta_{\text{bwj}}\)) of the multiscale GWR (MGWR) model is calculated based on local regression, as well as kernel functions and bandwidth selection criterion.