## 3 Methodology
### Formulation of Optimal \(K_{cs}\) and MOGA-SVM Classifier  
Kernel is essential in SVM classification and it has to obey [PERSON]'s theorem. That is, the kernel is positive semi-definite. A common interpretation of kernel is that it captures the correlation between pairs of data. Thus, the proposed \(K_{CS}\) is optimally designed using convolution and cross-correlation. The \(K_{CS}\) is formulated as multi-objective optimization problem and is solved by MOGA [29]. Let \(X_{i,j}(n)\) of length 120 (zero padding for length\(<\)120) be the WPS sample. The subscript \(i\) refers to the class label from 0 to 4 and that of \(j\) refers to the sample number. Therefore, the sets in Class 0 to Class 4 are \(\{X_{0,1}(n),\,\ldots\,X_{0,1800}(n)\}\), \(\{X_{1,1}(n),\,\ldots\,X_{1,630}(n)\}\),\(\{X_{2,1}(n),\,\ldots\,X_{2,970}(n)\}\),\(\{X_{3,1}(n),\,\ldots\,X_{3,1380}(n)\}\),and \(\{X_{4,1}(n),\,\ldots\,X_{4,820}(n)\}\), respectively. The convolution between two WPSs \(X_{a,b}(n)\) and \(X_{c,d}(n)\) is given by  
\[C^{a,b}_{c,d}(n)=X_{a,b}(n)*X_{c,d}(n)=\sum_{k=0}^{N-1}X_{a,b}(k)X_{c,d}(n-k) \tag{1}\]  
where \(N=120\) is the length of the WPS sample. The cross-correlation between two WPSs \(X_{a,b}(n)\) and \(X_{c,d}(n)\) can be expressed as  
\[R^{a,b}_{c,d}(k)=\left\{ \begin{array}{ccc}\sum\limits_{n=k}^{N-1}X_{a,b}(n )X_{c,d}(n-k),&k\geq 0\\ \sum\limits_{n=0}^{N-|k|-1}X_{a,b}(n)X_{c,d}(n-k),&k<0\end{array} \right. \tag{2}\]  
The customized similarity kernel \(K_{CS}\) is formulated by customized convolution kernel \(K_{c}\) and customized cross-correlation kernel \(K_{cc}\). \(K_{c}\), and \(K_{cc}\) are defined as  
\[K_{c}=\left[ \begin{array}{ccc}X_{c,1,1}&\cdots&X_{c,1,N_{t}}\\ \vdots&\ddots&\vdots\\ X_{c,N_{t},1}&\cdots&X_{c,N_{t},N_{t}}\end{array} \right] \tag{3}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Class**} & \multirow{2}{*}{**Name**} & \multicolumn{5}{c}{**Age**} & \multirow{2}{*}{**Total**} \\ \cline{3-4} \cline{6-7}  & & **[0,20]** & **[20,40]** & **[40,60]** & **[60,100]** \\ \hline
0 & Healthy & 8 & 26 & 30 & 16 & 100 \\
1 & Appendicitis & 0 & 22 & 0 & 0 & 22 \\
2 & Acute Appendicitis & 20 & 8 & 10 & 0 & 38 \\
3 & Duodenitis & 4 & 26 & 6 & 6 & 42 \\
4 & Pancreatitis & 16 & 26 & 4 & 0 & 46 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Sample distribution of the datasets. \[K_{cc}=\left[ \begin{array}{ccc}X_{cc,1,1}&\cdots&X_{cc,1,N_{t}}\\ \vdots&\ddots&\vdots\\ X_{cc,N_{t},1}&\cdots&X_{cc,N_{t},N_{t}}\end{array} \right] \tag{4}\]  
where \(N_{t}\) = 5040 is the 90% of the training samples in Class 0 to Class 4. Here, \(X_{c,i,j}\) refers to the weighting sum of convolution coefficients between \(i\)th and \(j\)th WPS sample. The 1 st to 1620 th samples come from Class 0. The 1621 th to 2187 th samples come from Class 1. The 2188 th to 3060 th samples come from Class 2. The 3061 th to 4302 th samples come from Class 3. The 4303 th to 5040 th samples come from Class 4. \(X_{c,i,j}\) and \(X_{cc,i,j}\) are computed by  
\[X_{c,i,j}=\sum_{m=1}^{2N-1}w_{c,m}C_{c,d}^{a,b}(m) \tag{5}\]  
\[X_{cc,i,j}=\sum_{m=1}^{2N-1}w_{cc,m}R_{c,d}^{a,b}(m) \tag{6}\]  
where \(w_{c,m}\) and \(w_{cc,m}\) are the weightings of convolution coefficients and cross-correlation coefficients, respectively. These weightings are optimally designed while using MOGA. The kernels \(K_{c}\) and \(K_{cc}\) predominantly affect the maximum margin and the accuracy (Acc) of the organ inflammations classifier. From (3) and (4), the maximum margin is directly related to both \(X_{c,i,j}\) and \(X_{cc,i,j}\). Based on (5) and (6), an optimal design of both \(X_{x,i,j}\) and \(X_{cc,i,j}\) can be obtained by an optimal design of the weighting \(sw_{c,m}\) and \(w_{cc,m}\) form \(=1,\,\ldots,2N-1\), for the given sequences of \(C_{c,d}^{a,b}(m)\) and \(R_{c,d}^{a,b}(m)\), respectively. In general, an optimally designed kernel will speed up the convergence of the training algorithm MOGA-SVM. However, varying the combinations of \(w_{c,m}\) and \(w_{cc,m}\) will deduce the different kernels.