## 2 Principle and Modeling
### Hybrid Kernel Extreme Learning Machine  
By using the kernel function to map the low-dimensional space to the high-dimensional space, the KELM greatly reduces the complexity of the network, and makes the prediction and the generalization ability better [19]. However, even within the same sample, the performance of various kernel functions for prediction is extremely diverse [20], indicating that it is difficult for a single kernel function in the standard KELM algorithm to accommodate diverse logging sample data. Therefore, a hybrid kernel extreme learning machine (HKELM) is proposed. By adding a hybrid kernel, the defect of ELM with a single core can be overcome, and the problem of low prediction accuracy and insufficient generalization ability can be solved.  
Based on the synthesis of the characteristics of common kernels and the trade-off between model accuracy and computational complexity, two kinds of kernel functions, polynomial and Gaussian radial basis, are selected to carry out a weighted combination, and the equivalent kernel function, combining the two kinds of kernels, is constructed. The expressions of the two basic kernel functions are shown in Equations (8) and (9), respectively.  
Polynomial kernel function can be expressed as  
\[K_{poly}(\mathbf{X},\mathbf{x}_{k})=\left[\left(\mathbf{x}_{k}^{T}\mathbf{X}\right)+C_{0} \right]^{d},C_{0}>0 \tag{8}\]  
where \(C_{0}\) and \(d\) are the parameters of the polynomial kernel function. This kernel function can generally represent the nonlinear mapping of the system.  
Gaussian radial basis kernel function can be expressed as  
\[K_{RBF}(\mathbf{X},\mathbf{x}_{k})=exp\bigg{(}\frac{-\parallel\mathbf{X}-\mathbf{x}_{k} \parallel^{2}}{2\sigma^{2}}\bigg{)} \tag{9}\]  
where \(\sigma\) is the kernel width. This kernel function can generally represent the nonlinearity of the system.  
Therefore, the equivalent kernel function used in HKELM algorithm can be expressed as the following equation:  
\[\Omega_{MKELM}(\mathbf{X},\mathbf{x}_{k})=C_{1}K_{poly}(\mathbf{X},\mathbf{x}_{k})+C_{2}K_{ RBF}(\mathbf{X},\mathbf{x}_{k}) \tag{10}\]  
where \(C_{1}\) and \(C_{2}\) are the weighted coefficients of the kernel function, ranging between (0, 1) and \(C_{1}+C_{2}=1\). After cancelling out \(C_{2}\),  
\[\Omega_{MKELM}(\mathbf{X},\mathbf{x}_{k})=C_{1}\cdot K_{poly}(\mathbf{X},\mathbf{x}_{k})+(1-C_ {1})\cdot K_{RBF}(\mathbf{X},\mathbf{x}_{k}). \tag{11}\]  
After substituting Equation (11) into Equation (7), the output \(f_{MKELM}(\mathbf{x}_{k})\) of the proposed HKELM algorithm that integrates the two kernel functions is expressed as  
\[\begin{split} f_{MKELM}(\mathbf{x}_{k})=\\ \sum_{k=1}^{N}\Omega_{MKELM}(\mathbf{X}\cdot\mathbf{x}_{k})\Big{\{}C_{1} \big{[}\big{(}\mathbf{x}_{k}^{T}\mathbf{X}\big{)}+C_{0}\big{]}^{d}+(1-C_{1})\,exp\Big{(} \frac{-\parallel\mathbf{X}-\mathbf{x}_{k}\parallel^{2}}{2\sigma^{2}}\Big{)}+\Big{(} \frac{\lambda}{\lambda}\Big{)}_{k}\Big{\}}^{-1}\cdot\mathbf{Y}_{k}.\end{split} \tag{12}\]  
The benefits of the global kernel and the local kernel can be combined with the hybrid kernel function of HKELM. So, the HKELM not only has good local search ability but also strengthens the global search ability. Due to the inefficiency in manually determining the hyper-parameter, the HKELM model should be optimized by the optimization algorithm.