# Machine-Learning-Based Classification Approaches toward Recognizing Slope Stability Failure
## 2 Data Collection and Methodology
#### 2.1.4 Improved Support Vector Machine using Sequential Minimal Optimization Algorithm (SMO-SVM)  
Support Vector Machine (SVM)  
Insulating different classes is the pivotal norm during the performance of the support vector machine (SVM). This happens through detecting the decision boundaries. SVMs have been shown to have a suitable accuracy in various classification problems when dealing with high-dimensional and non-separable datasets [54; 55; 56]. Theoretically, SVMs act according to statistical theories [57] and aim to transform the problem from non-linear to linear future spaces [58]. More specifically, assessing the support vectors that are set at the edge of the module descriptors will lead to locating the most appropriate hyperplane between the modules. Note that guiding cases that could not be a support vector are known as neglected vectors [59]. The graphical description of the mentioned procedure is depicted in Figure 7:  
Figure 6: The structure of Radial basis function. In the SVMs, the formulation of the hyperplane (i.e., the decision surface) is as follows:  
\[\sum\limits_{j=1}^{V}l_{j};f_{j};K\Big{(}x,x_{j}\Big{)}=0 \tag{11}\]  
where \(l_{j}\), \(x\), and \(f_{j}\) represent the Lagrange coefficient, the vector that is drawn from the input space, and the \(j\)th output, respectively. Also, \(K(x,x_{i})\) is indicative of the inner-product kernel (i.e., the linear kernel function) and operates as given in Equation (12):  
\[K\Big{(}x,x_{j}\Big{)}=x\cdot x_{j} \tag{12}\]  
where \(x\) and \(x_{j}\) stand for the input vector and input pattern related to the \(j\)th instance, respectively. The vector \(w\) and the biasing threshold \(b\) are obtained from Equations (13) and (14) [60]:  
\[w=\sum\limits_{i=1}^{V}l_{j};f_{j};x_{j} \tag{13}\]  
\[b=w\cdot x_{k}-f_{k} \tag{14}\]  
Also, \(Q(l)\), which is a quadratic function (in \(l_{i}\)), is given using the following formula:  
\[ \begin{array}{l}minQ(l)=min\frac{1}{2}\sum\limits_{j=1}^{V}\sum\limits_{h=1 }^{V}f_{j}l_{h}K\Big{(}x_{j},x_{h}\Big{)}l_{j}l_{h}-\sum\limits_{j=1}^{V}l_{j }\\ \text{where,}\\ 0\leq l_{j}\leq C\\ \sum\limits_{j=1}^{V}l_{j};f_{j}=0\end{array}  \tag{15}\]  
Sequential minimal optimization (SMO) is a simple algorithm that was used in this study to improve the performance of the SVM. When corresponding the linear weight vector to the optimal values of \(l_{i}\) using SMO, adding a matrix repository and establishing a repetitive statistical is not required. Every pair of Lagrange multiplier, such as \(l_{1}\) and \(l_{2}\) (\(l_{1}<l_{2}\)), vary from 0 to \(C\). During the SMO performance, the \(l_{2}\) is first computed. The extent of the transverse contour fragment is determined based on the obtained \(l_{2}\)[60]. If the targets \(f_{1}\) and \(f_{2}\) are equal, the subsequent bounds apply to  
Figure 7: The graphical depiction of the mechanism of an SVM. (Equation (10)); otherwise, they apply to \(l_{2}\). Equation (16) defines the directional minimum along with the contour fragment acquired using SMO:  
\[l_{2}{{}^{\prime}}=l_{2}+\frac{f_{2}(E_{1}-E_{2})}{\lambda} \tag{16}\]  
where \(E_{j}\) is the difference between the \(j\)th SMV output and the target. The parameter \(\lambda\) stands for an unbiased function and is given by the following equation:  
\[\lambda=K(x_{1},x_{1})+K(x_{2},x_{2})-2K(x_{1},x_{2}) \tag{17}\]  
In the following, the \(l_{2}{{}^{\prime}}\) clipped to the range \([A,B]\):  
\[A=max(0,l_{2}-l_{1})\quad B=min(C,C+l_{2}-l_{1}) \tag{18}\]  
\[A=max(0,l_{2}+l_{1}-C)\quad B=min(C,l_{2}+l_{1}) \tag{19}\]  
\[l_{2}{{}^{\prime\prime}}=\left\{ \begin{array}{ll}B&\text{if $l_{2}{{}^{\prime}} \geq B$}\\ l_{2}{{}^{\prime}}&\text{if $A<l_{2}{{}^{\prime}}<B$}\\ A&\text{if $l_{2}{{}^{\prime}}\leq A$}\end{array} \right. \tag{20}\]  
\[l_{1}{{}^{\prime}}=l_{1}+v(l_{2}-l_{2}{{}^{\prime\prime}}),\quad v=f_{1}f_{2} \tag{21}\]  
Finally, \(l_{1}\) and \(l_{2}\) are moved to the bound with the lowest unbiased function.