## 2 Methods
### Window Removal and Identification Method  
Unlike windows, which only contain a normal signal, windows containing noise or distorted signals may have a different DCT value [18]. Due to the repetitive feature of ECG signals, the DCT energy of normal signals is reflected to a certain extent, but the DCT energy for signals other than normal ones tends to reduce rapidly. Such energy changes can be used to remove unnecessary signals. Therefore, in this study, DCT coefficients are used to remove windows that are difficult to be identified due to the noise or distorted signals. Window removal is defined as Equation (6), and the threshold is determined as the value that can obtain the best result through the experiment. First, the unnecessary windows are removed by applying Equation (6) in training windows. Thereafter, all of the test windows are checked by applying Equation (6) with the remaining training windows. The absolute value of the difference between the DCT coefficient's maximum and minimum value of the test window is more than 1.1 times the average value of the difference between the maximum and minimum value of the training windows. Additionally, the standard deviation of a test window is more than 1.1 times the average standard deviation of training windows. A test window that meets the above conditions is removed. The flow chart of the window removal method is shown in Figure 2.  
It is possible to derive various identification results based on the classifier. NN, SVM, and LDA classifiers are used in various fields, not only the ECG field, but also mechanical systems, EEG and vision. These classifiers are performed simply and quickly [26; 27; 28]. In this study, Euclidean-based NN, SVM, and LDA classifiers are applied to analyze the individual identification result according to the classifier. In order to perform classification, half of each of the subjects of the NSR, PTB, and QT DB are set as the training data, and the remaining data is set as test. Euclidean distance, as a method of obtaining the distance between two points, is defined as Equation (7) and SVM, as a method of determining the optimal classification plane with minimum error, is defined as Equation (8). Furthermore, LDA, which involves determining the \(W\) that maximizes the rate between classes and minimizes the rate within each class is defined as Equations (10)-(12).  
\[\begin{array}{l}IF\\ |Max(DCT\;coefficient)-Min(DCT\;coefficient|>Average(max-min)\times 1.1\\ AndSD(DCT\;coefficient)>Average(SD)\times 1.1\\ THEN\\ \text{remove window}\end{array} \tag{6}\]\[d_{x,y}=\sqrt{\left(x_{1}-y_{1}\right)^{2}+\left(x_{2}-y_{2}\right)^{2}} \tag{7}\]  
\[f(x)=w^{T}x+b \tag{8}\]  
With regards to the hyper plane of Equation (8), the random feature vector, \(x_{i}\) satisfies Equation (9).  
\[y_{i}=\left\{\begin{array}{c}1,\ w^{T}x_{i}+b\geq 0\\ -1,\ w^{T}x_{i}+b<0\end{array}\right. \tag{9}\]  
\[S_{B}=\sum_{i=1}^{c}N_{i}(\mu_{i}-\mu)(\mu_{i}-\mu)^{T} \tag{10}\]  
\[S_{W}=\sum_{i=1}^{c}\sum_{x_{k}\in X_{i}}(x_{k}-\mu_{i})(x_{k}-\mu_{i})^{T} \tag{11}\]  
where \(\mu_{i}\) indicates the average of class \(X_{i}\) and \(\mu\) indicates the global average. \(N_{i}\) indicates the number of data within class \(X_{i}\).  
\[J(W)=argmax\ w\ \frac{\left|W^{T}S_{B}W\right|}{W^{T}S_{W}W} \tag{12}\]  
Generally, in the case of multiple classes, one vs all (OAA) is used a lot but if the training set is unbalanced, classification is difficult in OAA. Therefore, this study uses the one vs one (OAO)  
Figure 2: Flow chart of window removal method. SD: standard deviation; DCT: discrete cosine transform.  
method. OAO, as a method of inspecting two classes by creating the classifier of \(M\)\((M-1)/2\), uses a hierarchical method of OAO to be used in individual identification. Hierarchical OAA is shown in Figure 3.