## 3 Methodology
### Frequency Ratio Methods  
#### 3.2.1 General Development  
Before analysing the susceptibility maps, the true impact of each variable should be individually evaluated by means of bivariate statistical method and then classified to have a direct or inverse relationship with landslide occurrence. Such a method reveals that landslide occurrence is dependent on a single variable. [PERSON] theorem is applied to obtain the conditional probability (Equation (2)):  
\[P\bigg{(}\frac{\text{landslide}}{X}\bigg{)}=\frac{P(\text{landslide in X})}{P(X)} \tag{2}\]  
In this study, the conditional probability is equivalent to factor class weights directly determined by means of FR. This way of working is based on determining the relationship between the landslide distribution and each of the factors involved in the phenomenon. Thus, FR is the ratio between the area affected by landslides and the total area, as well as the probability of landslides occurring or not for each factor. FR is determined from the ratio of landslides in the inventory and the attributed factors. Thus, for each class j of factor i:  
\[FR_{ij}=\frac{\frac{A_{ij}}{B}}{\frac{C_{ij}}{D}} \tag{3}\]  
where:  
A\({}_{\text{ij}}\): number of pixels with landslide for each factor i and class j  
B: total number of pixels with landslides in the study area  
C\({}_{\text{ij}}\): number of pixels in the class area j of the factor i  
D: total number of total pixels in the study area  
The FR value may be higher than 1 and this further indicates that the ith class of factor F (Fi) favours landslide occurrence. Based on the inventory data, FR values were obtained according to Equation (3), and then normalised in the 0-1 range and multiplied by 1000 (FRn) for the discrete variables. This is the most used methodology to obtain the FR data, either for discrete variables (non-numerical) or continuous variables. For discrete variables, the intervals are established by the nature of the available data. For continuous variables, constant intervals ('equal interval') are usually used without the intervention of special adjustments. In this article, this procedure of dividing a continuous variable into constant intervals is referred to as the 'general frequency ratio (GFR)' method. [PERSON] et al. (2017) [15] and [PERSON] et al. (2020) [31] developed a proposed improvement to this general method called the modified frequency ratio. This is an optimisation for continuous variables that sets smaller and smaller normalised intervals until the best answer is found. The optimisation is developed by means of an iterative algorithm applied to each of the involved variables, the objective factor being the probability of success according to the ROC curves. This method has the disadvantage of not adapting well to hybrid models, which combine discrete variables with continuous variables. That the working intervals are not balanced means that the perspective of adequately weighting each of these variables is lost. These hybrid models, such as the case presented in this paper, are the most common and would require a specific optimisation procedure. For this reason, the modified frequency ratio is not used in this study. This paper considers that it is possible to establish classes in continuous variables in a more objective manner that considers the nature and characteristics of the data used. An initial option is to obtain these classes by means of classical cluster data analysis. The procedure consists of calculating the FR for a wide series of intervals, for example, 40\({}^{\circ}\) or more, in intervals of 2\({}^{\circ}\) by 2\({}^{\circ}\). These intervals are clustered using the k-means clustering technique, with a Euclidean similarity (or closeness) metric. The goal of this cluster type is to divide n observations into k clusters, where each observation belongs to the cluster with the closest mean and acts as a representative of that cluster. This creates a division of the data space into Voronoi cells.