## Theory
### Support Vector Machine  
A support vector machine (SVM) is a kind of supervised machine learning technique based on statistical learning theory, used for classification and regression analysis. The SVM constructs a set of hyperplanes, a concept of decision planes that define decision boundaries. For separable classification problems, the SVM algorithm finds an optimal separating hyperplane. The form of equation defining the decision surface that classifies is as follows:  
\[\mathbf{w}\cdot\mathbf{x}+b=0 \tag{2}\]  
which implies  
\[y_{i}(\mathbf{w}\cdot\mathbf{x}+b)\geq 1,\;\;i=1,\;2,\;\cdots,\;N \tag{3}\]  
where \(y_{i}\) is the sign of \(i\)-th data point, \(\mathbf{w}\) denotes the normal vector of the hyperplane, \(\mathbf{x}\) denotes the input data and \(b\) is bias [19].  
By maximizing the margin of separation subject to this constraint, the SVM tries to find an optimal hyperplane. Here, the margin of separation is the distance between the hyperplane and the support vector for a given weight vector \(\mathbf{w}\) and bias \(b\).  
If it is not possible to classify data linearly, the data can be classified by a kernel SVM. The use of the kernel function allows data to be separated without increasing the computational complexity, despite mapping into a high-dimensional space. The radial basis function (RBF) kernel is the most popular kernel which can be expressed as follows:  
\[Z=e^{-\frac{\left\|\mathbf{w}-X_{i}\right\|^{2}}{2\delta^{2}}} \tag{4}\]where \(\mathbf{x}\) is the input data and \(X_{j}\) is the center of the radial basis function. The classification performance of the SVM is influenced by two hyperparameters of the RBF. C is the SVM's penalty hyperparameter, and \(\delta\) is the kernel hyperparameter. A grid search is the most common approach for determining the two hyperparameters. The idea behind the grid search method is to find an optimal parameter by selecting discrete values at specified intervals. Basically, this method uses the pairs of (C, \(\delta\)) for cross-validation and selects the value of highest cross-validation accuracy. The exponentially increasing value of (C, \(\delta\)) is a way to find the optimal appropriate parameter.  
In this work, the appropriate values of C and \(\delta\) were changed to the form of \(2^{n}\) in order to find the appropriate values. Generally, the form of \(10^{n}\) is used. However, in the form of \(10^{n}\), the values of C and \(\delta\) vary greatly depending on \(n\). Hence, using a form of \(2^{n}\) was more suitable to get the appropriate values. With the selection of parameters, the classification with SVM is performed to find the parameter where the best classification performance is achieved. In this study, \(k\)-fold cross validation was used for the SVM-based classification.  
In \(k\)-fold cross validation, the original data are divided into \(k\) subsamples. A single subsample remains as the validation data for testing the model. The retained \(k\)-1 subsamples are used as training data. The cross-validation process is repeated \(k\) times until the \(k\) samples are used exactly once as the validation data. After that, the results are averaged to produce a single estimation. In this validation method, all subsamples are used once for training and validation, and each of them is used exactly once as the validation data.