## 3 Methodology
### Formulation of Optimal \(K_{cs}\) and MOGA-SVM Classifier
Searching the optimal weightings require a large computational power. As a result, there is a tradeoff between the accuracy and computational power. As it is difficult to find the optimal values of \(w_{c,m}\) and \(w_{cc,m}\) that are attributable to the complexity of the objective function, a good trial of \(w_{c,m}\) and \(w_{cc,m}\) are primarily important, which determines the accuracy. In this paper, an multi-objective optimization approach, MOGA, is employed to determine the weighting \(sw_{c,m}\) and \(w_{cc,m}\). However, exhaustive search algorithms may not be the appropriate choices for searching the solution of the optimization problem. The reason is that it deals with a huge range of combinations. Indeed, heuristic search algorithms efficiently and effectively perform the searching of the optimal solutions. In particular, the GA is a robust searching heuristic algorithm that imitates the process of the natural evolution for searching the solution of the optimization problem by the operations in selection, inheritance, crossover and mutation. Aforementioned, to be a proper kernel for SVM classification, the fulfillment of [PERSON]'s theorem is essential [30]. The evaluation of eigenvalues helps to determine the positive semi-definite of \(K_{c}\) and \(K_{cc}\). \[\left\{ \begin{array}{c}K_{c}V_{c}=D_{c}V_{c}\\ K_{cc}V_{cc}=D_{cc}V_{cc}\end{array} \right. \tag{7}\]  
where \(V_{c}\) and \(V_{cc}\) are non-zero eigenvectors for \(K_{c}\) and \(K_{cc}\), respectively. \(D_{c}\) and \(D_{cc}\) are the corresponding eigenvalues. All of the eigenvalues must be positive in order to ensure \(K_{c}\) and \(K_{cc}\) are positive semi-definite. Define \(K_{cs}\) as the sum of \(K_{c}\) and \(K_{cc}\). It takes the advantageous from \(K_{c}\) and \(K_{cc}\). It is proved below that the sum of Mercer's kernels is also a Mercer's kernel. If \(K_{c}\) and \(K_{cc}\) are positive semi-definite, then for any \(c\in\Re^{n}\), \(c^{T}K_{c}c\geq 0\) and \(c^{T}K_{cc}c\geq 0\). Hence,  
\[c^{T}K_{CS}c=c^{T}(K_{c}+K_{cc})c \tag{8}\]\[c^{T}K_{CS}c=c^{T}K_{c}c+c^{T}K_{cc}c\geq 0 \tag{9}\]  
Therefore, the \(K_{CS}\) is positive semi-definite. Thus, it is a Mercer's kernel. The multi objective optimization problem is formulated with two objective functions:  
\[ \begin{array}{ll}\begin{array}{c}\begin{array}{c}\begin{array}{c}Max\\ Max\\ \end{array}\end{array}&F_{1}=M(\alpha,w)\\ \begin{array}{c}\begin{array}{c}Max\\ \end{array}&F_{2}=OA=0.5(S_{e}+S_{p})\end{array}\end{array}\end{array}  \tag{10}\]  
\[\begin{array}{c}\begin{array}{c} \begin{array}{c}\begin{array}{c}\begin{array}{c} \alpha_{i}\geq 0,\sum\limits_{i=1}^{N}\alpha_{i}y_{i}=0,i=1,\ldots,N\\ \begin{array}{c}\begin{array}{c}2N-1\\ \sum\limits_{n=1}^{2N-1}w_{c,n}=1,\sum\limits_{n=1}^{2N-1}w_{cc,n}=1\\ D_{c,i}\geq 0,D_{cc,i}\geq 0,\forall i\end{array}\end{array}\end{array}\end{array}\end{array}  \tag{11}\]  
where \(M(\alpha,w)\) is the margin function of the classifier, \(\alpha_{i}\) is the Lagrange multiplier, \(S_{e}\) is the sensitivity, \(S_{p}\) is the specificity, \(y_{i}\in\{-1,+1\}\) is the output of the classifier, and \(D_{c,i}\) and \(D_{cc,i}\) are the entries of \(D_{c}\) and \(D_{cc}\), respectively. The margin function is defined as  
\[M(\alpha,w)=\sum\limits_{i=1}^{N}\alpha_{i}-\frac{1}{2}\sum\limits_{i=1}^{N} \sum\limits_{j=1}^{N}\alpha_{i}\alpha_{j}y_{i}y_{j}K_{CS} \tag{12}\]  
Figure 3 shows the MOGA-SVM for the optimal design of the \(K_{CS}\).