## 2 Kernel Principal Component Analysis (KPCA)  
Kernel Principal Components Analysis is a nonlinear PCA extension that comprises observations to a more enormous (potentially infinite) dimensional feature structure using a nonlinear array and then measuring dot products in a feature space. As an approximation is used in future space, the KPCA is retrieved. KPCA is a kind of kernel-based computing system. KPCA's central premise is, therefore, logical and popular. The fundamental premise of KPCA is to allocate the source values x to the function feature space, first employing a nonlinear projection and afterward implementing a regular PCA in future space. A dataset matrix was identified by \(n\) measurements representing m variables, so that the matrix \(X\) is defined as:  
\[X=(x_{1},x_{2},x_{3}\ldots x_{n})=\left( \begin{array}{ccc}x_{11}&\cdots&x_{1n }\\ \vdots&\ddots&\vdots\\ x_{m1}&\cdots&x_{mn}\end{array} \right)\]  
The covariance matrix in the domain of a function is then determined as:  
\[C_{F}=1/n\underline{\sum_{i=1}^{n}(\Phi(x_{i}),\Phi(x_{i}))} \tag{1}\]  
where \(\Phi\) is the nonlinear mapping due to which the the measured inputs are extended into hyper dimensional feature space and \(\Phi(x_{i})\) is anticipated to take unit variance and zero mean. To diagonalize the covariance matrix, we illustrate the eigenvalue hitch in the feature space as:  
\[\lambda_{A}=C_{F}A \tag{2}\]  
where \(\lambda\) is an eigenvalue of \(C_{F}\) and \(A\) is the consequent eigenvector (\(\lambda\geq 0,\ A\
eq 0\)), where \(A=\sum_{i=1}^{n}\alpha_{i}\langle\Phi(x_{j})\rangle\). The eigenvectors of \(C_{F}\) are specified by:  
\[\frac{1}{n}\underline{\sum_{j=1}^{n}A_{i}\langle\Phi(x_{k}),\underline{\sum_{ i=1}^{n}\Phi(x_{i})}\rangle\langle\Phi(x_{i}),\Phi(x_{j})\rangle} \tag{3}\]  
Rather than conducting eigenvalue reduction explicitly on \(C_{F}\) in Equation (1) and figuring eigenvalue and principal components (PCs), we employ the kernel trick by describing the \(n\times n\) (kernel) matrix as follows:  
\[[\mathbf{K}]_{ij}=K_{ij}=(\Phi(x_{i}),\Phi(x_{i}))=k\big{(}x_{i}x_{j}\big{)} \tag{4}\]  
For all \(i,j=1,\ 2,\ \ldots,n\), presenting the kernel function of the system \(k(x,y)=\langle\phi(x),\phi(y)\rangle\) in Equation (4) allows the computation of the inner products \(\big{\langle}\phi(x_{i}),\phi(x_{j})\big{\rangle}\) in the feature space as a function of the response statistics. This prohibits the necessity to conduct the nonlinear mappings and the explicit computation of inner products in the feature space ([PERSON] et al., 1998) [35]. Employing the kernel matrix, we revise Equation (4) as \(\lambda m\mathbf{K}\alpha=\mathbf{K}^{2}\alpha\) for \(k=1,\ 2,\ \ldots,n\), where it is comparable to the eigenvalue problem as \(\lambda m\alpha=\mathbf{K}\alpha\). The kernel matrix can be mean centered as follows:  
\[\mathbf{K}_{ct}=\mathbf{K}-U\mathbf{K}-\mathbf{K}U+U\mathbf{K}U \tag{5}\]  
where \(U\) is an \(n\times n\) matrix in which each integral is equivalent to \(\frac{1}{n}\), and \(\mathbf{K}\) is the kernel trick. Eigen breakdown of \(\mathbf{K}_{ct}\) is identical to the output of PCA in \(R^{F}\). This totals to determining the eigenvalue problem, which produces eigenvectors \(\alpha_{1}\,\alpha_{2}\,\ \ldots\ \alpha_{n}\) and the consequent eigenvalues \(\lambda_{1}\geq\lambda_{2}\ldots\lambda_{n}\). Meanwhile, the kernel matrix \(\mathbf{K}_{ct}\) is symmetric, and the resulting principal components are orthonormal, such as:  
\[\big{\langle}\alpha_{i},\alpha_{j}\big{\rangle}=\delta_{ij} \tag{6}\]  
where \(\delta_{ij}=(i,j=1,2\ldots m)\) signifies the Dirac delta function. The score vectors of the nonlinear mapping of mean centered training observations \(x_{j}\,\ i=1\,\ 2\,\ldots,n\), can then be obtained by projecting \(\phi(x_{j})\) onto the principal component space spanned by the eigenvectors \(\ a_{k}\,\ k=1,\ \ldots,\ m\), and applying the kernel trick, so we can obtain the below equation, and by using this equation we can obtain the nonlinear components.