## 3 Machine Learning Technique  
### Support Vector Machine (SVM)  
In general, an SVM comprises a hyperplane or set of hyperplanes that can be used for classification or regression analysis. Intuitively, if the hyperplane has a large difference from the closest training data point, the classifier error is small; thus, a good classification requires finding the hyperplane that is farthest from the closest training data for any classified point. In general, the initial problem is dealt with in a finite dimensional space, which often results in the data not being linearly separated. To solve this problem, a method has been proposed to facilitate the separation by matching from the finite level to the higher level of the initial problem [25; 29]. The linear classification using SVM is shown in Figure 5, and it is calculated as follows:  
First, the given training dataset \(D\) (the set of \(n\) points) is defined as follows:  
\[D=\left\{\left(\frac{(xi,yi)}{xi}\right)\in\mathbb{R}^{p},yi\in\{-1,1\}_{i=1 }^{n}.\right. \tag{2}\]  
Figure 4: Feature extraction from the FRF of experimental data. Each \(xi\) is a \(p\)-dimensional real vector, and \(yi\) is 1 or \(-1\), indicating which class \(xi\) belongs to. The linear separation of the above training dataset according to the value of \(yi\) is called the hyperplane and can be represented as a set of points \(X\) satisfying the following conditions:  
\[w\cdot x-b=0 \tag{3}\]  
where is the dot product, \(w\) is the normal vector of the hyperplane, and b is the deflection constant that fixes the hyperplane off-set in the \(p\)-dimensional space. The support vector \((X^{+},\ X^{-})\) of a given hyperplane is defined as follows:  
\(X^{+}\): the data closest to the hyperplane among the data of \(yi=+1\);  
\(X^{-}\): the data closest to the hyperplane among the data of \(yi=-1\). A hyperplane passing through \(X^{+}\) with a legal vector equal to a given plane can be represented as \(w\cdot x-b=+1\), and a hyperplane passing through \(X^{-}\) in a similar manner is represented as \(w\cdot x-b=-1\). The margin of the hyperplane is the distance between the hyperplane passing through each support vector. Geometrically, the distance between two hyperplanes, or margin, is \(\frac{2}{\|w\|}\). The SVM is an algorithm that maximizes the margin. Because there should be no data points between the hyperplanes passing through the support vector, the following equation is established:  
\[\left\{ \begin{array}{l}w\cdot x-b\geq+1,\ if\ yi=+1\\ w\cdot x-b\leq-1,\ if\ yi=-1\end{array} \right.. \tag{4}\]  
The above two expressions can be expressed as follows:  
\[yi(w\cdot xi-b)\geq 1,\ \text{for all }1\leq i\leq n. \tag{5}\]  
The SVM problem that follows the condition of the hyperplane and attempts to find the maximum value of the margin can be expressed as the following optimization problem:  
\[argmin\|w\| \tag{6}\]  
[PERSON] proposed a nonlinear classification by applying kernel tricks to the maximum margin hyperplane problem [29]. The form of the nonlinear classification algorithm is similar to the existing linear classification algorithm; however, the dot product is replaced by the nonlinear kernel function. This solves the maximum margin hyperplane problem of the transformed feature space. The transformation here may be nonlinear or may raise the dimension. That is, the classifier is a linear hyperplane in a high dimensional feature space but a nonlinear hyperplane in a conventional dimensional space. The kernel functions used for nonlinear classification are as follows [30]:  
Figure 5: Binary linear separation with a support vector machine (SVM).