## 2 Principle and Modeling  
### Principle of KELM  
The ELM is a practical and efficient feedforward neural network. The structure of the ELM network is shown in Figure 1.  
In the diagram, \(X,Y\) denote the input and output dataset, respectively. \(\mathbf{x_{k}}=\left[ \begin{array}{cccc}x_{k1}&x_{k2}&\cdots&x_{kn}\end{array}  \right]^{\mathsf{T}}\) is the input vector at dimension \(n\), and \(\mathbf{y_{k}}=\left[ \begin{array}{cccc}y_{k1}&y_{k2}&\cdots&y_{kl}\end{array}  \right]^{\mathsf{T}}\) is the output vector at dimension \(l\). \(\mathbf{w}_{j}=\left[ \begin{array}{cccc}w_{1j}&w_{2j}&\cdots&w_{nj}\end{array} \right]\) is the input weight of the node \(j\) in the hidden layer, and \(\mathbf{\beta}_{j}=\left[ \begin{array}{cccc}\beta_{j1}&\beta_{j2}&\cdots&\beta_{ jl}\end{array} \right]^{\mathsf{T}}\) is the output weight of the node\(j\) in the same hidden layer. \(\mathbf{b}=\left[ \begin{array}{cccc}b_{1}&b_{2}&\cdots&b_{m}\end{array} \right]^{ \mathsf{T}}\) is the threshold of the hidden  
Figure 1: ELM model structure diagram. layer. \(g(x)=\dfrac{1}{1+e^{-x}}\) is the excitation function of the hidden layer. \(f_{\text{ELM}}(\mathbf{x}_{k})\) is the network output of the ELM, which is expressed as follows:  
\[f_{\text{ELM}}(\mathbf{x}_{k})=\sum_{j=1}^{m}\mathbf{\beta}_{j}\cdot g(\mathbf{w}_{j}\cdot \mathbf{x}_{k}+b_{j}). \tag{1}\]  
The output matrix \(f_{\text{ELM}}(\mathbf{X})\) can be written as \(f_{\text{ELM}}(\mathbf{X})=\mathbf{H}\mathbf{\beta}\), where \(\mathbf{H}\) is the output matrix of the hidden layer:  
\[H= \begin{bmatrix}g(\mathbf{w}_{1}\cdot\mathbf{x}_{1}+b_{1})&\cdots&g(\mathbf{w}_{m}\cdot \mathbf{x}_{1}+b_{m})\\ g(\mathbf{w}_{1}\cdot\mathbf{x}_{2}+b_{1})&\cdots&g(\mathbf{w}_{m}\cdot\mathbf{x}_{2}+b_{m})\\ \vdots&\vdots&\vdots\\ g(\mathbf{w}_{1}\cdot\mathbf{x}_{N^{\prime}}+b_{1})&\cdots&g(\mathbf{w}_{m}\cdot\mathbf{x}_{N^ {\prime}}+b_{m})\end{bmatrix} _{N^{\prime}\times m}. \tag{2}\]  
\(\mathbf{X}\) and \(\mathbf{Y}\) are both known variables. The input weight matrix \(\mathbf{W}\) and the threshold \(b\) can be randomly given. \(\mathbf{H}\) can be calculated by Equation (2). The output weight matrix \(\mathbf{\beta}\) is calculated by the formula \(\mathbf{\beta}=\mathbf{H}^{+}\cdot\mathbf{Y}\), where \(\mathbf{H}^{+}\) is the pseudo-inverse matrix of \(\mathbf{H}\). The ELM overcomes the shortcomings of the traditional neural network, such as slow training speed, the overfitting that is prone to occur, and falling into local extremums, but there are still problems, such as difficulty in determining the number of nodes in the hidden layer and the occurrence of overfitting. To address the shortcomings of the ELM, inspired by the kernel function introduced into the support vector machines (SVM), the KELM is proposed in the literature [18], which denotes the mapping from the input to the output of the hidden layer as \(h(\mathbf{X})\). The kernel matrix is defined as  
\[\Omega_{KELM}(\mathbf{X})=h(\mathbf{X})\cdot\mathbf{H}^{T}. \tag{3}\]  
The elements at row \(i\) and column\(j\) of the matrix are shown by the following equation:  
\[\Omega_{KELM_{ij}}\big{(}x_{i},x_{j}\big{)}=h(x_{i})\cdot h\big{(}x_{j}\big{)} \triangleq K\big{(}x_{i}\cdot x_{j}\big{)} \tag{4}\]  
where \(K\big{(}x_{i}\cdot x_{j}\big{)}\) is the kernel function. \(f_{KELM}(\mathbf{X})\) can be estimated by Formula (5), which is described below. \[ \begin{split} f_{KELM}(\mathbf{X})=h(\mathbf{X})\cdot\mathbf{H}^{T}(\mathbf{ \Omega}_{KELM}(\mathbf{X}))^{-1}\cdot\mathbf{Y}=[K(\mathbf{X}\cdot\mathbf{x}_{1})\ \cdots K(\mathbf{X}\cdot\mathbf{x}_{N})]\times\\ (\mathbf{\Omega}_{KELM}(\mathbf{X}))^{-1}\cdot\mathbf{Y}.\end{split}