# Data Mining Approaches for Landslide Susceptibility Mapping in Unyeonsan, Seoul, South Korea
## 4 Methodology
### Support Vector Machine  
#### 4.1.1 Basic Principles of SVM  
The SVM method is a training algorithm based on nonlinear transformations that uses a classification based on the principle of structural risk minimization, which has performed well in the test phase [51]. When training data are represented in multidimensional space, several hyperplanes can be used to distinguish the data into two classes; however, there is only one optimum linear separating hyperplane [52]. This optimal hyperplane should maximize the distance between the data closest to the hyperplane. Thus, the objective of the SVM method is to determine a multi-dimensional hyperplane that differentiates the two classes with the maximum margin. The SVM model performs this process according to three main concepts: margin, support vector, and kernel.  
The margin is the distance between a specific data point and the hyperplane that separates the data patterns. It is the shortest distance from the data vector of each class to the given hyperplane. The larger the margin, the more accurately the SVM model performs the classification; hence, the optimal hyperplane maximizes the margin when \(\overrightarrow{\omega}\) is the normal vector of the hyperplane, and is the norm of the normal of the hyperplane. \(||\overrightarrow{\omega}||\) is the norm of the normal of the hyperplane.  
\[margin=\ \frac{2}{||\overrightarrow{\omega}||} \tag{5}\]  
The hyperplane, which defines the two classes that maximize the margins when the data are linearly separable, can be defined by Equation (6). Under these conditions, the training data on these two hyperplanes constitute the support vector. Therefore, the support vector maintains the shortest distance to the hyperplane. The support vectors are dependable on the boundary of the hyperplane and they are time-efficient when a new dataset is tested with a support vector. The dataset of linear separable training vectors \(x_{i}\) (\(i=1\), \(2\), \(\dots\), \(n\)) and the training vectors consisting of two classes are represented by \(y_{i}\) as the value of \(\pm 1\)[53; 54].  
\[y_{i}\Big{(}(\overrightarrow{\omega}\cdot\overrightarrow{x_{i}})+b\Big{)}-1\geq 0 \tag{6}\]  
where \(b\) is the scalar base, and the scalar product operation is denoted by (\(\cdot\)).  
Figure 6: Flowchart of the spatial prediction of landslide susceptibility.  
In addition, because the two hyperplanes must maximize the margin between them, the optimization of objective Equation (7) with the constraints of Equation (6) yields the planes of the two classes:  
\[\min(\frac{1}{2}||\vec{\omega}||^{2}) \tag{7}\]  
However, most cases do not satisfy the constraint as the data are not linearly separable. A formula including the slack variable \(\xi_{i}\) indicates the distance from the hyperplane to the data located on the wrong side [52]; the penalty term C introduced to account for misclassification [55; 56] was developed to address this problem.  
\[y_{i}\Big{(}(\overrightarrow{\omega}\cdot\overrightarrow{x_{i}})+b\Big{)} \geq 1-\xi_{i} \tag{8}\]  
\[\min(\frac{1}{2}||\vec{\omega}||^{2}+\ C\sum_{i=1}^{n}\xi_{i}) \tag{9}\]