## 2 Improved Skewness Decision Tree SVM (ISDTSVM) Algorithm
### The Classification Order Determination Method of the ISDTSVM Algorithm
#### 2.2.2 The Realization Method of the ISDTSVM Algorithm  
For a _k_-class classification problem, the classification order is determined by the sum of the Euclidean distances between the multi-class sample centers. The larger the sum of Euclidean distances, the earlier separation. The related calculation expression are presented as follows:  
(1) The center of the class  
If class \(i\) has \(n\) samples, the center of the class \(\overline{x_{i}}\) is given by Equation (6). \[\overline{x_{i}}=\sum_{j=1}^{n}x_{ij}/n \tag{6}\]  
where, \(x_{ij}\) is the _j_th simple of class \(i\), \(x_{ij}\in R^{d}\). (2) The Euclidean distances between the class centers  
The Euclidean distances between class \(i\) and class \(j\) is given by Equation (7). \[d_{ij}=\sqrt{(\overline{x_{i}}-\overline{x_{j}})^{2}},i,j=1,2,\cdots,k \tag{7}\]  
(3) The sum of the Euclidean distances between the class centers  
Figure 3: Illustration of the conventional Skewness Decision Tree Support Vector Machine (SVM). The sum of Euclidean distances between the class centers is given by Equation (8). \[Sumd(i)=\sum_{j=1}^{k}d_{ij},\ i=1,2,\cdots,k \tag{8}\]  
The ISDTSVM algorithm can overcome the influence of the interference points by calculating the Euclidean distances between the classes centers, compared to Euclidean distance between the nearest sample vectors between classes. It is presented as follows:  
Assuming there are three types of sample signals, one of which has two interference points for points A and B. The two-dimensional spatial distribution of the three types of sample signals is presented in Figure 4. In Figure 4, \(dc_{12}\) is the Euclidean distance between the center of class 1 and class 2, \(dc_{13}\) is the Euclidean distance between the center of class 1 and class 3, \(dc_{23}\) is the Euclidean distance between the center of class 2 and class 3, \(dn_{11}\) is the Euclidean distance between the nearest sample vector of class 1 and class 2, \(dn_{13}\) is the Euclidean distance between the nearest sample vector of class 1 and class 3 and \(dn_{23}\) is the Euclidean distances between the nearest sample vector of class 2 and class 3. As can be seen in Figure 4, \(dc_{12}>dc_{13}>dc_{23}\) and \(dn_{13}>dn_{23}>dn_{12}\) can be obtained and class 1 is the easiest to be separated. Due to \(dc_{12}>dc_{13}>dc_{23}\) and \(dn_{13}>dn_{23}>dn_{12}\) presented in Figure 4, it can be stated that \(dc_{12}+dc_{13}>dc_{12}+dc_{23}>dc_{13}+dc_{23}\) and \(dn_{13}+dn_{23}>dn_{13}+dn_{12}>dn_{12}+dn_{23}\). Therefore, class 1 is the easiest to be separated by using the sum of the Euclidean distances between the class centers. Whereas, it can be concluded that class 3 is the easiest to be separated by using the sum of the Euclidean distances between the nearest sample vectors between the classes. The result obtained by the sum of the Euclidean distances between the class centers is consistent with what can be seen in reality. It shows that the ISDTSVM algorithm can overcome the influence of the interference points by calculating the Euclidean distances between the class centers. The procedure of performing the ISDTSVM algorithm is described as follows:  
1. According to Equation 6, the center of the classes is obtained, _labeli_ = 0. 2. Calculating the Euclidean distances between the class centers. \(d_{ij}\) (\(i,j=1,2,\cdots,k\)) are obtained by Equation (7) and a symmetric matrix \(D_{k\times k}\) is obtained as well. 3. According to Equation (8), \(Sumd(i)\), the sums of the Euclidean distances between classes are obtained by calculating the sum of each row in a symmetric matrix \(D_{k\times k}\). 4. Sorting the \(Sumd(i)\), _labeli_ = _labeli_ + 1. The class label with maximum \(Sumd(i)\) is for the _labeli_. Elements of line \(i\) and column \(i\) of \(D_{k\times k}\) is assigned to be zero. \(k=k-1\), repeating Step 3 to Step 4 until _labeli_ = \(k\). 5. Multi-class classification is conducted using the conventional Skewness Decision Tree SVM. Figure 4: Two-dimensional spatial distribution of the three types of sample signals.