## 3 Materials and Methods
### Landslide Susceptibility Assessment
#### 3.3.2 Modeling Techniques  
To address the critical concern of overfitting, especially given our limited dataset of 204 instances (102 landslide, 102 non-landslide) and 12 geological indicators, we implemented tailored regularization strategies for each model: Naive Bayes (NB), Support Vector Machine (SVM), Random Forest (RF), and XGBoost. We employed 5-fold stratified cross-validation to conduct both hyperparameter tuning and evaluate overall model performance. A brief overview of each technique is provided below. * Naive Bayes (NB)  
Naive Bayes is a probabilistic classifier based on [PERSON] theorem with the assumption of feature independence. Given a feature vector \(\mathrm{x}=(x_{1},x_{2},\ldots,x_{n})\) and class label \(\mathrm{C}_{k}\), the posterior probability is calculated as follows [38]:  
\[P(\mathrm{C}_{k}\mid x)\ =\ \frac{P(\mathrm{C}_{k})\prod_{i=1}^{n}P(x_{i}\mid \mathrm{C}_{k})}{P(x)} \tag{1}\]  
where \(P(\mathrm{C}_{k})\) is the prior probability of class \(\mathrm{C}_{k}\), and \(P(x_{i}\mid\mathrm{C}_{k})\) is the likelihood of feature \(x_{i}\), given class \(\mathrm{C}_{k}\). The classifier assigns the class with the highest posterior probability. For Naive Bayes, traditional regularization techniques like L2 regularization or Dropout were not applicable, as it is a probabilistic model without iterative training or adjustable weights. Additive smoothing was used to prevent zero probabilities for unseen feature combinations, which was crucial for robust performance on small datasets. We tuned the alpha parameter for MultinomialNB or BernoulliNB using cross-validation. The optimal alpha value was 1.0. * Support Vector Machine (SVM)  
Support Vector Machine is a supervised learning algorithm that seeks to find the optimal hyperplane that maximizes the margin between two classes [39]. Given a training dataset \(\{(x_{i},y_{i})\}_{i=1}^{n}\), where \(x_{i}\in R^{d}\) and \(y_{i}\in\{-1,1\}\), SVM solves the following optimization problem:  
\[\min_{w,b}\frac{1}{2}|\mathbf{w}\,|^{\,2}\ \text{subject to}\ \ y_{i}\Big{(}w^{ \intercal}x_{i}+b\Big{)}\geq 1 \tag{2}\]  
\(\mathbf{w}\) is the weight vector that defines the orientation of the decision hyperplane. Minimizing \(|\mathbf{w}\,|^{\,2}\) maximizes the margin between classes, improving the model's generalization. SVM inherently regularizes by maximizing the classification margin. This regularization was controlled by the C parameter and, when using the RBF kernel, the gamma parameter. A smaller C value led to stronger regularization by allowing more misclassifications to achieve a larger margin, beneficial for our small dataset. Similarly, a smaller gamma with the RBF kernel resulted in a simpler, more generalized decision boundary. We used a logarithmic grid search for both parameters. The optimal C value was 0.1, and the optimal gamma (RBF kernel) was 0.001. Dropout and Early Stopping were not applicable to SVMs. * SVM-RF Hybrid Modeling Framework (SVM-RF)  
The SVM-RF model is a hybrid approach that combines the strengths of Support Vector Machine (SVM) and Random Forest (RF). While SVM excels at finding the optimal decision boundary in the feature space, Random Forest provides robust feature selection and reduces overfitting by aggregating predictions from multiple decision trees [40]. The integration of these methods has been shown to enhance classification accuracy in landslide susceptibility studies by leveraging the strengths of both techniques. Each model was individually optimized using distinct strategies. For SVM, we employed the same optimization method as with our standalone SVM model, tuning its box constraint (C) and kernel scale (gamma) within the ranges [0.1, 100] and [0.001, 10], respectively. For random forest, it naturally resisted overfitting through its ensemble nature, employing bootstrap aggregating and random feature selection during tree construction. Our regularization strategy constrained the complexity of individual trees and enhanced ensemble diversity.