## 2 Basic Theory
### [PERSON] et al. put forward the theory of extreme learning machine in 2006. On account of this theory, many scholars have innovated in addition to a variety of models, such as online sequential extreme learning machine and KELM [31; 32]. KELM is a single-layer feedforward neural network algorithm, which is more accurate than the extreme learning machine (ELM) algorithm. Compared with BPNN and SVM, the kernel function extreme learning machine has shorter time to calculate results, faster calculation speed, and greatly improves the adaptability of the model to the samples [33]. KELM algorithm has been effectively applied in many fields, which proves its effectiveness in prediction. The following introduces the construction principle of the general ELM model, and the specific neural network function is as follows:  
\[g(x)=h_{i}(x)\cdot\beta_{i} \tag{12}\]  
In the formula: \(g(x)\) denotes the output value of the network, \(h_{i}\) (\(x\)) represents the output of the \(i\) hidden layer neurons corresponding to the input \(x\); \(\beta_{i}\) represents the connection weights between the \(i\) hidden layer neurons and the output neurons. The regression accuracy of ELM is measured by the error. The smaller the error is, the greater the accuracy is. Therefore, the minimum output error is calculated to obtain the optimal result. The formula is as follows:  
\[\lim_{L\rightarrow\infty}\left|\left|g(x)-g_{0}(x)\right|\right|=\lim_{L \rightarrow\infty}\left|\left|\sum_{i=1}^{L}\beta_{i}h_{i}(x)-g_{0}(x) \right|\right|=0 \tag{13}\]  
In the formula, \(L\) represents the quantity of hidden layer neurons; \(g_{0}(x)\) represents the function to be forecasted composed of the target value. The generalization ability of neural networks is measured by minimizing the output weight \(\beta\). Usually \(\beta\) takes its least squares solution, the calculation method is as follows:  
\[\beta=H^{+}O=H^{T}\Big{(}HH^{T}\Big{)}^{-1}O=H^{T}\bigg{(}\frac{1}{C}+HH^{T} \bigg{)}^{-1}O \tag{14}\]  
In the formula: \(H\) represents the hidden layer matrix of neural network; \(H^{+}\) represents the generalized inverse matrix of \(H\) matrix; \(O\) represents the prediction target value vector. On the basis of ridge regression theory, by improving the normal number \(1/C\), the results will be more stable with better generalization. The introduction of kernel function to KELM algorithm can obtain better regression prediction accuracy. This paper uses Mercer' s condition to define the kernel matrix, and the calculation formula is as follows:  
\[\left\{ \begin{array}{c}\Omega_{ELM}=HH^{T}\\ \Omega_{i,j}=h(x_{i})\cdot h\left(x_{j}\right)=K\left(x_{i},x_{j}\right)\end{array} \right. \tag{15}\]  
Kernel matrix \(\Omega\) replaces random matrix HHT in ELM and uses kernel function to map all input samples from n-dimensional space to high-dimensional hidden feature space. After setting the kernel parameters, the mapping value of the kernel matrix \(\Omega\) is fixed [33]. The kernel function includes RBF kernel function, linear kernel function and polynomial kernel function, which is usually set as RBF kernel function:  
\[K(\mu,v)=\exp\ \left[-(\mu-\frac{v^{2}}{\sigma})\right] \tag{16}\]  
The parameter 1/C is added to the main diagonal of the unit diagonal matrix HHT, so that the characteristic root will not be equal to zero, and then the weight vector \(\beta^{*}\) is obtained. A more stable and better generalization ELM model is obtained. At this time the output weights of ELM method become [34]:  
\[\beta^{*}=H^{T}(\frac{I}{C}+HH^{T})^{-1}O \tag{17}\]  
In the formula, \(I\) denotes diagonal matrix; \(C\) denotes penalty coefficient for weighing the proportion between structural risk and empirical risk; \(HHT\) denotes generated by mapping input samples from kernel functions. From the above formulas, the output of the KELM model is described as follows:  
\[f(x)=h(x)H^{T}(\frac{I}{C}+HH^{T})^{-1}O=\left[ \begin{array}{c}K(x,x_{i})\\ K(x,x_{N})\end{array}