## 4 Method
### Semi-Supervised Kernel Extreme Learning Machine Improvement Scheme
#### 4.2.1 Improved Design of Kernel Function  
To increase the applicability of a single kernel function, combining multiple kernel functions is a good solution. In the article [29], [PERSON] introduced various ways of combining kernels and showed that using multiple kernels is generally more effective than using a single kernel. After many experiments, the weighted kernel function calculated by this calculation method is applied to this semi-supervised learning task. It also shows a better effect than single-kernel learning. The following is a brief list of four common basic kernel functions:  
(1) Linear Kernel function, whose form is shown in Equation (6):  
\[\mathrm{K}\left(\mathbf{x}_{\mathrm{i}},\mathbf{x}_{\mathrm{j}}\right)=x_{ \mathrm{i}}^{\mathrm{T}}x_{\mathrm{j}} \tag{6}\]  
(2) Polynomial Kernel function, \(\mathrm{c},\ \mathrm{d}\) are hyperparameters whose form is shown in Equation (7):  
\[\mathrm{K}\left(\mathbf{x}_{\mathrm{i}},\mathbf{x}_{\mathrm{j}}\right)=\left( x_{\mathrm{i}}^{\mathrm{T}}\mathbf{x}_{\mathrm{j}}+\mathrm{c}\right)^{ \mathrm{d}} \tag{7}\]  
(3) Radial basis function kernel, \(\sigma\) is a hyperparameter whose form is shown in Equation (8):  
\[\mathrm{K}\left(\mathbf{x}_{\mathrm{i}},\mathbf{x}_{\mathrm{j}}\right)=\exp \left(-\frac{\|\mathbf{x}_{\mathrm{i}}-\mathbf{x}_{\mathrm{j}}\|^{2}}{2\sigma^ {2}}\right) \tag{8}\]  
(4) Sigmoid kernel function, where \(\mathrm{a},\ \mathrm{c}\) are hyperparameters, and its form is shown in Equation (9):  
\[\mathrm{K}\left(\mathbf{x}_{\mathrm{i}},\mathbf{x}_{\mathrm{j}}\right)=\tanh \left(\mathbf{ax}_{\mathrm{i}}^{\mathrm{T}}\mathbf{x}_{\mathrm{j}}+\mathrm{c}\right) \tag{9}\]A linear kernel function is a linear map and a special case of the RBF kernel. RBF can not only non-linearly map data to high-dimensional space but also has fewer parameters than the poly kernel function, so it has less complexity. In addition, when the parameter \(d\) in the poly kernel function is high, the element value of the kernel matrix tends to be infinitely large or infinitely small, which greatly increases the difficulty of calculation. Sigmoid as the basic core will not conform to [PERSON]'s theorem. Therefore, in this study, the RBF kernel is selected as the basic kernel function to obtain the weighted kernel function through a weighted combination. The calculation process is as follows:  
First, define a weighted nonlinear mapping function \(\Phi(x)\), where \(\theta\) is a weighting parameter, \(\Phi_{1}(x)\) and \(\Phi_{2}(x)\) are two different nonlinear mapping functions, the form is shown in Equation (10):  
\[\Phi(x)=\theta\Phi_{1}(x)+(1-\theta)\Phi_{2}(x) \tag{10}\]  
Then define a kernel function \(K\big{(}x_{i},x_{j}\big{)}=<\Phi(x_{i}),\Phi(x_{j})>\), and bring Equation (10) into it to get Equation (11):  
\[K\big{(}x_{i},x_{j}\big{)}=\theta^{2}K_{1}\big{(}x_{i},x_{j}\big{)}+(1-\theta) ^{2}K_{2}\big{(}x_{i},x_{j}\big{)}+2\theta(1-\theta)K_{1,2}\big{(}x_{i},x_{j} \big{)} \tag{11}\]  
where \(K_{1}\big{(}x_{i},x_{j}\big{)}=<\[PERSON]_{1}(x_{i}),\[PERSON]_{1}(x_{j})>,K_{2}\big{(} x_{i},x_{j}\big{)}=<\Phi_{2}(x_{i}),\Phi_{2}\big{(}x_{j}\big{)}>\) is a common basic kernel function, \(K_{1,2}(x_{i},x_{j})\) is the cross-kernel function.  
When RBF is selected as the basic kernel function, their forms can be expressed as Equations (12)-(14):  
\[K_{1}\big{(}x_{i},x_{j}\big{)}=\exp\!\left(-\frac{\left\|x_{i}-x_{j}\right\|^{ 2}}{2\sigma_{1}{}^{2}}\right) \tag{12}\]  
\[K_{2}\big{(}x_{i},x_{j}\big{)}=\exp\!\left(-\frac{\left\|x_{i}-x_{j}\right\|^{ 2}}{2\sigma_{2}{}^{2}}\right) \tag{13}\]  
\[K_{1,2}\big{(}x_{i},x_{j}\big{)}=\left(\frac{2\sigma_{1}\sigma_{2}}{\sigma_{1 }{}^{2}+\sigma_{2}{}^{2}}\right)^{\frac{N}{2}}\exp\!\left(-\frac{\left\|x_{i }-x_{j}\right\|^{2}}{\sigma_{1}{}^{2}+\sigma_{2}{}^{2}}\right) \tag{14}\]  
where \(\sigma_{1}\) and \(\sigma_{2}\) are the parameters of the RBF kernel function, and N is the dimension of the input data \(x\).