## 2 Methodology
### Analysis of Landslide Spatial Aggregation  
The frequency ratio (FR) method is one of the most widely used statistical models in LHA studies [5; 11; 18]. It has the advantages of reliable results and convenient use. The FR values indicate the spatial correlations between landslide occurrence and the predisposing factors and can quantitatively reflect the degree of influence of different landslide predisposing factors. It can be calculated by Equation (1):  
\[FR_{ij}=(l_{ij}/s_{ij})/(L/S) \tag{1}\]  
where \(FR_{ij}\) is the frequency ratio of class \(i\) under the \(j\) predisposing factor, \(l_{ij}\) is the number of occurred landslides in the above class \(i\), \(s_{ij}\) is the area of class \(i\), \(L\) is the total number of occurred landslides in the study area, and \(S\) is the total size of the study area. When the FR value is greater than 1, the relevant predisposing factors play a positive role in landslide occurrence. The higher the frequency ratio, the higher the probability of landslide occurrence.  
Some studies have found that the FR method has the limitation of not considering the spatial aggregation of landslides [19; 20; 22; 27; 28]. As shown in Figure 2, the size of the selected area in each case is \(S\), and the total number of landslides in it is \(L\). In the selected area of each case, class \(i\) is split into the same evaluation unit with an area of \(S_{u}\). Regarding case 1 and case 2, class \(i\) in each case has the same FR value of \((10/(5\cdot S_{u}))/(L/S)\). However, in case 1, landslides occur in only one evaluation unit, and the degree of landslide  
Figure 1: Flowchart of the methodology for landslide risk assessment.  
spatial aggregation is significantly higher. Thus, the FR value of class \(i\) in case 1 should be lowered by considering landslide spatial aggregation. Furthermore, the degree of landslide spatial aggregation is not only related to the proportion of evaluation units with occurred landslides but also to the average number of landslide points in each evaluation unit. For example, in case 2 and case 3, each evaluation unit of class \(i\) is connected with an occurred landslide. However, in case 2, each evaluation unit has 2 landslide points, while, in case 3, each evaluation unit has only 1 landslide point. This indicates that the degree of landslide spatial aggregation in case 3 is lower. Similarly, in case 4, only one evaluation unit has a landslide point, and the total number of landslides is 1. The proportion of evaluation units with a landslide in case 4 is the lowest, but its degree of spatial aggregation is also the lowest.  
According to the main cases of landslide spatial aggregation phenomena (Figure 2), we propose the DFR method to improve the FR method's analysis [20]. In this improved method, an index named \(LAI_{FR}\), reflecting the degree of landslide spatial aggregation, is defined as in Equation (2):  
\[LAI_{FR}=(U_{L}/U_{T})/(l_{ij}/U_{T}) \tag{2}\]  
where \(LAI_{FR}\) is the index denoting the landslide spatial aggregation degree, \(U_{L}\) is the number of evaluation units with occurred landslides under class \(i\) of predisposing factor \(j\), \(U_{T}\) is the total number of evaluation units under class \(i\) of predisposing factor \(j\), and \(l_{ij}\) is the number of occurred landslides in class \(i\).  
Usually, the higher the \(LAI_{FR}\) value is, the lower the spatial aggregation degree of landslides is in a certain class of a factor. When there is no landslide in a certain class, the \(LAI_{FR}\) value is equal to 0. Based on Equation (3), the FR value is optimized using \(LAI_{FR}\):  
\[DFR_{ij}=FR_{ij}\times LAI_{FR} \tag{3}\]  
Figure 2: The main cases of landslide spatial aggregation phenomena: (**a**) case 1; (**b**) case 2; (**c**) case 3; (**d**) case 4.  
where \(D\!F\!R_{ij}\) is the improved frequency ratio of a certain class \(i\) of factor \(j\). In this study, we also uniformly refer to the values of \(D\!F\!R\) and \(F\!R\) as FRs.