# Monitoring Vegetation Change and Its Potential Drivers in Inner Mongolia from 2000 to 2019
## 2 Materials and methods
### Methods
#### 2.3.5 GeoDetector Model  
GeoDetector is a statistical method used to explore spatial differentiation and quantify drivers [69]. In this study, four detector tools of the geographical detector model were used to quantify the drivers of NDVI in Inner Mongolia. The dependent variable \(Y\) is the NDVI value for Inner Mongolia and the independent variable \(X\) includes natural and social drivers. (1) The factor detector focuses on exploring the spatial heterogeneity of the dependent variable \(Y\) and the extent to which the independent variable \(X\) explains the spatial heterogeneity of \(Y\). The expressions are:  
\[q=1-\frac{\sum\limits_{h=1}^{L}N_{h}\delta^{2}}{N\delta^{2}}=1-\frac{SSW}{SST} \tag{13}\]  
\[SSW=\sum\limits_{h=1}^{L}N_{h}\delta_{h}^{2},SST=N\delta^{2} \tag{14}\]  
where \(q\) represents the explanatory power of the metric factors on \(Y\). Specifically, a larger \(q\) indicates that the explanatory power of each metric factor is stronger, indicating that it has more influence on the spatial distribution of the NDVI, and the range of \(q\) values is {0,1}. Note that \(h=1\), \(2\dots\), \(L\) is the classification or partition of \(Y\) or \(X\); \(N_{h}\) and \(N\) represent the number of cells in layer \(h\) and the entire area, respectively; \(\delta_{h}^{2}\) and \(\delta^{2}\) are the variances of layer \(h\) and the entire area, respectively; and \(SSW\) and \(SST\) are the sum of the within-layer variance and total variance of the entire area, respectively. (2) An ecological detector was used to determine if a significant difference existed between the effects of two drivers on the spatial distribution of the dependent variable, as measured by the F-statistic:  
\[F= \frac{N_{x1}\times(N_{x2}-1)\times SSW_{x1}}{N_{x2}\times(N_{x1}-1) \times SSW_{x2}} \tag{15}\]  
where \(N_{x1}\) and \(N_{x2}\) denote the sample sizes of the two drivers, and \(SSW_{x1}\) and \(SSW_{x2}\) denote the sum of the within-layer variance for the stratum formed by the two drivers. (3) An interaction detector was used to identify interactions between the different drivers by assessing whether the explanatory power of factors \(X1\) and \(X2\) increases or decreases for the dependent variable \(Y\) when they function together [70]. It is also possible that the effects of these factors on the dependent variables were independent. The results of this assessment were obtained by comparing the \(q\)-values \(q\)(\(X1\)) and \(q\)(\(X2\)) for the two factors on \(Y\), with the \(q\)-values on \(Y\) after the interaction between the two factors. (4) A risk detector was used to detect the appropriate extent or type of vegetation cover impact by different drivers, and uses the \(t\)-statistic as follows:  
\[t= \frac{\overline{Y}_{h=1}-\overline{Y}_{h=2}}{\left[\frac{Var(\overline{Y}_{ h=1})}{n_{h}=1}+\frac{Var(\overline{Y}_{h=2})}{n_{h}=2}\right]^{\frac{1}{2}}} \tag{16}\]  
where \(\overline{Y_{h}}\) is the mean value of the NDVI attributes in sub-region \(h\), \(n_{h}\) is the number of samples in sub-region \(h\), and \(Var\) denotes variance. GeoDetector was used to analyze variables that must be discretized for continuous types of data. Additionally, the natural intermittency grading method was used to maximize the differences between the groups. This method considers the large latitudinal span of Inner Mongolia, the diverse climate, temperature, soil, etc., and examines the representativeness, typicality, and scientificity of various driving factors. Nine representative and available datasets were selected as drivers for exploring the distribution and changes in the NDVI in Inner Mongolia, including five natural factors: Precipitation, temperature, elevation, soil type, and slope; and four social factors: Population, GDP, land use type,and livestock density. Based on the closest distance method, the average NDVI and nine drivers during 2000-2019 were resampled using ArcGIS 10.6, and the spatial resolution of the raster data was unified to 10 \(\times\) 10 km.