## 2 Theoretical Background of the Methods  
### Logistic Model Tree  
Logistic Model Trees (LMTree), which is a relatively new machine learning algorithm, is developed based on the integration of tree induction algorithm and additive logistic regression [52]. The difference of LMTree when compared to the other decision tree algorithms is that the tree growing process is carried out using the LogitBoost algorithm [52; 55] and the tree pruning is performed using Classification And Regression Tree (CART) [56].  
Given a training dataset \(\mathrm{T}=(x_{i},y_{i})_{i=1}^{ds}\) with \(x_{i}\in R^{D}\) is the input vector, \(ds\) is the number of data samples, \(D\) is the dimension of the training dataset, and \(y_{i}\in(1,0)\) is the label class. In this research context, the input vector consists of eight variables (slope, aspect, elevation, land cover, soil type, lithology, distance to fault, and distance to river), whereas the label class contains two classes, landslide (LS) and non-landslide (Non-LS). The landslide class is coded as \"1\" and the non-landslide is coded as \"0\". The objective of LMTree is to construct a tree-like structure model that is capable of classifying the training dataset into the two above classes in term of probability. The predicted numeric value to the landslide class of sample is used as susceptibility index.  
Structurally, the LMTree model consists of a root node, a set of inner nodes, and a set of leaves. The aim of the training phase that includes the tree growing and the tree pruning processes is to determine the best tree structure with numbers of inner nodes and leaves. Accordingly, first, a logistic regression model Equation (1) is built at the root note using the binary LogitBoost algorithm [57] and the training dataset. In the next step, the training dataset at the root is split using the C4.5 splitting rule [58] in order to sort appropriate sub-datasets for the inner nodes, and then, logistic regression models Equation (1) for these inner nodes is built using their associated sorted datasets and the binary LogitBoost. The tree continues growing in the same procedure until it meets the stopping criterion of less than 15 samples at nodes. Finally, to prevent the LMTree model from over-fitting, the tree pruning is performed using the CART algorithm that is based on a combination of the model error and the model complexity [52].  
In the LMTree building process, the binary LogitBoost algorithm [57] is used to generate logistic regression models Equation (1) for all of the inner nodes and leaves, as follows.  
\[f_{\mathrm{LS,Non-LS}}(\mathbf{x})=\sum_{i=1}^{\mathrm{D}}\beta_{i}\mathbf{x}_{i}+ \beta_{0} \tag{1}\]  
where D is the total number of landslide input factors and \(\beta_{i}\) is the logistic coefficient.  
The membership probability [52] of the landslide class at the leaves of the LMTree model is posterior probabilities derived using Equation (2) and is used as landslide susceptibility index.  
\[p((\mathrm{LS,Non-LS})\left|\mathbf{x}\right.)=\frac{\exp f_{\mathrm{LS,Non-LS}}( \mathbf{x})}{\exp f_{\mathrm{LS}}(\mathbf{x})+\exp f_{\mathrm{Non-LS}}(\mathbf{x})} \tag{2}\]  
The complexity of the LMTree model could be estimated using the following equation [52]:  
\[\mathrm{MC}=\mathrm{O}(dept*ds*\log n+ds*D^{2}*dept+nt^{2}) \tag{3}\]  
where MC is the model complexity; _dept_ is the depth of the initial unpruned tree; \(nt\) is the number of nodes in the LMTree; \(ds\) is the number of training samples; and, \(D\) is the number of landslide predisposing factors.