## 2 Materials and Methods
### Statistical Model, Validation Tools, and Model-Building Strategies  
The multivariate adaptive regression splines (MARS; [45]) method was used for all modeling procedures as it has been confirmed to be very effective in modeling nonlinear components of the relationship between landslides and their causative factors [6; 46]. MARS is a nonparametric regression method that splits each independent variable into branches (optimizing their number based on the characteristics of the variable itself and the correlation with the distribution of other predictors). Each branch is defined by a hinge function (a function used for defining a nonlinear relationship between y and x) and the relative knot. The derived structures (hinge function and knots) identify a basis function that can take the shape of a simple linear regression (when the basis function corresponds to the model intercept, set to a constant value of 1) or more complex geometry (when the basis function is the product of one or more hinge functions associated with different covariates). \begin{table}
\begin{tabular}{c c c c} \hline
**Acronym** & **Description of Predictor** & **References** & **Potential Proxy Significance** \\ \hline ELE & Distribution of elevation & & Mean annual rainfall \\ \hline \multirow{3}{*}{LCL} & Morphological classification of the territory based on the variation in elevation with respect to the neighbouring areas & [37] & Morphological setting \\ \hline \multirow{3}{*}{SLO} & The first derivative of elevation & [38] & Speed of the water and potential underlying rupture surfaces [6; 27] \\ \hline \multirow{3}{*}{N} & Cosine of aspect (direction in which the slope degrades more rapidly) & [39] & Seasonal wet/dry cycles of soils [40] \\ \hline \multirow{3}{*}{E} & Sine of aspect (direction in which the slope degrades more rapidly) & [39] & Seasonal wet/dry cycles of soils [40] \\ \hline \multirow{3}{*}{PLN} & The second derivative of elevation, computed along the horizontal plane & [41] & Activation and propagation of landslides [42] \\ \hline \multirow{3}{*}{PRF} & The second derivative of elevation, computed along the direction of the highest slope gradient & [41] & The direction of flow [42] \\ \hline \multirow{3}{*}{TWI} & Calculated as \(\ln[\text{A}/\tan\beta]\), where A & \multirow{3}{*}{[43]} & \multirow{3}{*}{[43]} \\  & and \(\beta\), computed on each cell, corresponds to the area of upslope & & \\ \cline{1-1}  & & drained cells and the slope gradient, respectively & \\ \hline \multirow{3}{*}{SPI} & Natural logarithm of the catchment area multiplied by the tangent of the slope gradient & [44] & Proxy of the intensity of surface water erosion [6] \\ \hline \multirow{3}{*}{LITO} & Original geological map & Physical-mechanical properties of rocks [27] \\ \hline \multirow{3}{*}{USE} & \multirow{3}{*}{CORINE land cover (2018)} & Potential hydrological and surface \\  & & hydric erosion induced & disturbances [27] \\ \hline \end{tabular}
\end{table} 
Table 1: Details of the employed geo-environmental variables (modified from [7; 27]). In this way, hinge functions boost the maximum-likelihood-based adaptation skill of the logistic regression method, according to  
\[y=f(x)=\alpha+\sum_{i=1}^{N}\beta_{i}h_{i}(x) \tag{1}\]  
where \(y\) is the dependent variable (the outcome) predicted by the function _f(x)_, \(\alpha\) is the model intercept, and \(\beta_{i}\) is the coefficient of the _hi_ basis functions given the \(N\) number of base functions. For other information about the method, please refer to [6; 27; 47; 48; 49]. For this research, MARS analysis was performed using the \"earth\" R-package [50]. Due to the fact that the MARS method is based on a presence-absence approach, a random extraction of negative cases in the same number as the positive cases was carried out. The random selection of negative cases and the subsequent modeling was replicated one-hundred times to evaluate the independence of the results (resolution and precision) from the specific choice of the negative cases [6; 27].