## 3 Methods
### Negative Sample Selection Method  
#### 3.3.1 Geographical Information Similarity  
This study employs the FR method for discrete evaluation factors and the fuzzy membership method for continuous evaluation factors to calculate geographical information similarity [21; 22]. The FR method, which is a standard bivariate analysis method, identifies which classification of discrete factors is more likely to contribute to landslides, as detailed in Equation (1). To ensure consistency with continuous factors, the values are normalized using Equation (10), scaling them to a range of 0-1, thereby establishing the similarity between the discrete factor classification and the factor typical classification of landslide.  
\[S_{ij}=\frac{FR_{ij}-min(FR_{i})}{max(FR_{i})-min(FR_{i})} \tag{10}\]  
where \(S_{ij}\) denotes the similarity between the \(j\)-th classification of the \(i\)-th factor and the factor typical classification, \(FR_{ij}\) denotes the FR for the \(j\)-th classification of the \(i\)-th factor, and \(max(FR_{i})\) and \(min(FR_{i})\) denote the maximum and minimum values of \(FR_{i}\), respectively.  
The fuzzy membership function measures the membership level of an element in a fuzzy set [23; 24]. The primary advantage of the function is its ability to assign precise membership weights to fuzzy events based on the frequency of occurrence of certain events. Let there be n landslide events with the following evaluation factor values: \(x_{1},x_{2},x_{3},\ldots,x_{n}\). The fundamental definition of the membership function for evaluation factor \(x\) is as follows:  
\[F(A_{ij})=\frac{A_{ij}-A_{imin}}{A_{imax}-A_{imin}},i=1,2,\ldots,n \tag{11}\]  
where \(\big{(}A_{ij}\big{)}\) denotes the frequency value of landslides under the classification \(A_{ij}\) and \(A_{ij}\) denotes the number of landslides associated with the j-th classification of the i-th factor. In contrast, \(A_{imax}\) and \(A_{imin}\) represent the maximum and minimum values of \(A_{ij}\), respectively.  
In this study, to more accurately characterize the nonlinear relationship between the evaluation factors and the landslide occurrence frequency (i.e., the relative proportion of landslides within each factor classification interval), we did not directly apply the linear normalization formula (Equation (9)). Instead, the functional relationship \(f(x)\) between the evaluation factor values and the corresponding landslide frequency was derived using the curve fitting tools in Origin software.  
\[f(x)=y_{0}+\frac{A}{w\sqrt{\pi/2}}\text{exp}(-2(\frac{x-x_{c}}{w})^{2}) \tag{12}\]  
where \(f(x)\) represents the predicted landslide frequency value corresponding to the evaluation factor \(x\), whose mathematical expression is determined by the landslide frequency distribution pattern; \(y_{0}\) denotes the baseline offset parameter, \(A\) represents the peak area, \(x_{c}\) denotes the peak position, and \(w\) represents the peak width parameter. By applying Equation (11) to normalize the \(f(x)\) function, the membership degree \(S_{x}\) for a single evaluation factor is obtained as follows:  
\[S_{x}=\frac{f(x)-min(f(x))}{max(f(x))-min(f(x))} \tag{13}\]  
By integrating the similarity value of both discrete and continuous factors, the final geographic information similarity can be obtained as follows:  
\[S=f(S_{1},S_{2},\cdots,S_{i},\cdots,S_{n}) \tag{14}\]where \(S\) represents the geographical information similarity at a given location, \(n\) represents the number of evaluation factors, and \(f()\) denotes the integrated method. This study employs weighted synthesis methods.