## 3 Data and Method
### Methodology
#### 3.2.2 Support Vector Machine (SVM)  
The support vector machine is a new generation of machine learning algorithms based on the nonlinear theory of covariate transformation to higher dimensional feature space and the principle of structural risk minimization, which was presented firstly by [PERSON] and [PERSON] in 1995 [PERSON] and [PERSON] (1995). It can solve nonlinear and high-dimensional pattern recognition problems with fewer samples. Generally, the characteristics of the SVM can be briefly stated as: (1) Significantly precise and robust; (2) able to model complex nonlinear decision boundaries [20]; (3) less prone to over fitting in comparison with other models such as artificial neural network (ANN) [25]; and (4) potential of implementation in pattern recognition, regression, and classification [20].  
In recent years, SVM modeling approaches have been implemented extensively in landslide automatic identification and landslide susceptibility assessment [25; 42]. The key to the application of the support vector machine is (1) to find an optimal classification hyperplane that is out of the data model; and (2) to use a reasonable kernel function to map linearly inseparable data into high-dimensional feature space, making it linearly separable [20].  
A set of linear separable training vectors \(x_{i}\) (\(i\) = 1, 2,..., n) was considered. The training vectors consist of two classes, which are defined as \(y_{i}\) = \(\pm\) 1. The purpose of support vector machine is to search an n-dimensional hyperplane differentiating the two classes by their maximum gap. Mathematically, it can be expressed as:  
\[\frac{1}{2}\|\,w\,\|^{2} \tag{3}\]  
The constraint is:  
\[y_{i}(w\bullet x_{i}+b)\geq 1 \tag{4}\]  
where \(w\) represent the norm of the normal of the hyperplane, \(b\) is a scalar base, and \(\bullet\) is the scalar product operation, \(x_{i}\) is \(i\)th sample value of the input vector, and \(y_{i}\) is the corresponding output.  
With the Lagrangian multiplier, the cost function is:  
\[L=\frac{1}{2}\,w^{2}-\sum_{i=1}^{n}\lambda_{i}(y_{i}((w\cdot x_{i})+b)-1) \tag{5}\]  
where \(\lambda_{i}\) is the Lagrangian multiplier, \(w\) and \(b\) can be achieved by the dual minimizing Equation (5).  
For a non-separable case, one can modify the constraints Equations (6) and (7):  
\[y_{i}(w\cdot x_{i}+b)\geq 1-\xi_{i} \tag{6}\]  
\[L=\frac{1}{2}w^{2}-\frac{1}{vn}\sum_{i=1}^{n}\xi_{i} \tag{7}\]  
where \(\xi_{i}\) is slack variables, and \(v\) (0, 1) represents the misclassification. In the present study, \(+\)1 and \(-\)1 indicate the failed and stable cases, respectively.  
There are four commonly used kernel function types including Linear, polynomial, radial basis, and sigmoid. Research shows that the radial basis function (RBF) is one of the most powerful kernel functions [43], which has been applied successfully to remote sensing classification and landslide susceptibility mapping. Therefore, this work chose the RBF in the assessment of earthquake landslide susceptibility. The mathematical expression of the RBF \(K_{RBF}\Big{(}x_{i}-x_{j}\Big{)}\) is  
\[K_{RBF}\Big{(}x_{i}-x_{j}\Big{)}=exp(-\gamma\|\,x_{i}-x_{j}\,\|^{2}) \tag{8}\]  
where \(\gamma\) is the parameter of the kernel functions.  
Before constructing LR and SVM models, we need to prepare the training and testing sample datasets which include both the presence and absence of landslide occurrences and associated controlling factors. These values were derived and rasterized by GIS. We extracted cell values at locations specified in a point feature class (training and testing samples we selected in Section 3.1.3) from \(30\times 30\) raster derived from the influence factor maps, and recorded the values to the attribute table of the point feature class.