# Data Mining Approaches for Landslide Susceptibility Mapping in Unyeonsan, Seoul, South Korea
## 4 Methodology
### Support Vector Machine
#### 4.1.2 SVM for Nonlinear Classification  
In the preceding case of nonlinear boundary data, the SVM model is unable to generate an appropriate hyperplane. Therefore, the data can be classified into a linear problem as a hyperplane by mapping them to a high-dimensional space value from the low-dimensional data that cannot be linearly separated. To solve Equations (8) and (9) when the dataset is linearly separable, the training data are expressed as a vector inner product in the optimization process. This enables the inner product in the training algorithm to be expressed by the inner product of a function in a feature space when nonlinear data in the input space are mapped to a high-dimensional feature space by a specific function.  
The hyperplane of the data with nonlinear data patterns can be derived by transforming the data space, but \(x\) cannot be computed directly; an arbitrary kernel function should be used to satisfy \(x\)[53]. The kernel functions, K(\(x_{i},x_{j}\)), convert the original data into linearly separable data in high-dimensional feature space [52]. The kernel function should satisfy [PERSON]'s condition. The following four functions are the main kernel functions used in the SVM model:  
\[\text{Linear : K}(x_{i},\ y_{i})=x_{i}{}^{T}\cdot x_{j} \tag{10}\]  
\[\text{Polynomial : K}(x_{i},\ y_{i})=\Big{(}\gamma\cdot\Big{(}x_{i}{}^{T}\cdot x _{j}\Big{)}+r\Big{)}^{d},\ \gamma>0 \tag{11}\]  
where \(\gamma\) and \(d\) are parameters of the kernel functions.  
\[\text{Radial basis function : K}(x_{i},\ y_{i})=\mathrm{e}^{-\gamma||x_{i}-x_{j}||^{2}}, \ \gamma>0 \tag{12}\]  
\[\text{Sigmond : K}(x_{i},\ y_{i})=\tanh\Big{(}\gamma\cdot x_{i}{}^{T}\cdot x _{j}+r\Big{)} \tag{13}\]  
In radial basis function (RBF) kernel, \(||x_{i}-x_{j}||\) is the squared Euclidean distance between the vectors, and \(\gamma\) is defined as follows:  
\[\gamma=\ \frac{1}{2\sigma^{2}} \tag{14}\]  
where \(\sigma\) is a freely adjustable parameter that controls the performance of the kernel, and \(r\) is a parameter of the sigmoid kernel.  
In this study, the Environment for Visualizing Images (ENVI 5.0, Exelis Visual Information Solutions, Boulder, CO, USA) software program was used for the application of the SVM model (Boulder, CO, USA). The software provides four types of kernels in the SVM classifier as mentioned above: linear, polynomial, RBF, and sigmoid. Each kernel function has its own parameters, which should be adjusted appropriately according to the characteristics of the purpose. The default kernel of the ENVI software is the RBF kernel, which is known to provide more accurate prediction results in most classifications, particularly in nonlinear problems. After applying the kernel and classifying each input factor, the SVM model results were mapped using the process of overlapping and summation.  
\[SVM_{all}=\sum_{i=1}^{n}SVM_{i}\left(n\ =\ \text{number of input factors}\right) \tag{15}\]  
\(SVM_{all}\) is the sum of all classification results by the SVM model for all variables, and \(SVM_{i}\) is the single result of a given variable; in this study, the total number of input factors, \(n\) was 17. The results were mapped and validated using the landslide locations not used for training.