## 3 Materials and Methods
### Extreme Learning Machine  
Figure 6 shows the ELM network structure, which is formed by three layers: the input layer with \(n\) nodes, the hidden layer with \(L\) nodes, and the output layer with \(m\) nodes. The neurons in adjacent layers are fully linked. The input weight values and hidden layer biases in ELM are set randomly, and the activation function in the hidden layer maps the input space onto a new space, then a linear combination is performed on the new space. Thus, the parameter which has to be solved is the weight matrix that links the hidden and output neurons, which can be determined through the least square approach.  
Given \(N\) training samples \((\mathbf{x}_{i},\mathbf{t}_{i})\), \(i=1,2,\ldots,N\), in which \(\mathbf{x}_{i}=\left[x_{1}^{i},\ x_{2}^{i},\ldots,\ x_{n}^{i}\right]^{T}\) is an \(n\)-dimensional input vector of the \(i\)th sample and \(\mathbf{t}_{i}\)=\(\left[\begin{matrix}i_{1}^{i},\ t_{2}^{i},\ldots,\ i_{m}^{i}\end{matrix}\right]^{T}\) is an \(m\)-dimensional target vector associated with \(\mathbf{x}_{i}\), the mathematical model can be written as  
\[\mathbf{o}_{i}=\sum_{j=1}^{L}\mathbf{\beta}_{j}\mathbf{\beta}_{h}(\mathbf{w}_ {j}\cdot\mathbf{x}_{i}+b_{j})\ \ i=\ 1,\ 2,\ 3,\ \ldots,\ N. \tag{24}\]  
where \(\mathbf{o}_{i}=\left[\begin{matrix}o_{1}^{i},\ o_{2}^{i},\ldots,\ o_{m}^{i} \end{matrix}\right]^{T}\) is the \(m\)-dimensional output vector of the network with input vector \(\mathbf{x}_{i}\), \(\mathbf{\beta}_{j}=\left[\begin{matrix}\mathbf{\beta}_{j1},\ \mathbf{\beta}_{j2},\ldots,\mathbf{\beta}_{jm} \end{matrix}\right]^{T}\) is the weight vector between the \(j\)th hidden neuron and the output layer, \(\mathbf{w}_{j}=\left[\begin{matrix}w_{j1},\ w_{2},\ldots,w_{jn}\end{matrix} \right]^{T}\) is the weight vector between the \(j\)th hidden neuron and the input layer, \(b_{j}\) is the bias parameter of the \(j\)th hidden neuron, \(\mathbf{w}_{j}\cdot\mathbf{x}_{i}\) denotes the inner product of \(\mathbf{w}_{j}\) and \(\mathbf{x}_{i}\), and \(\mathbf{\beta}_{h}(x)\) is the activation function in the hidden layer.  
Figure 6: Structure of the extreme learning machine.  
According to the theory proposed by [PERSON] [21], the ELM can approximate all given training samples without error, which implies that there are \(\mathbf{w}_{j}\), \(b_{j}\), and \(\mathbf{\beta}_{j}\) such that  
\[\sum_{j=1}^{L}\mathbf{\beta}_{j}g_{h}(\mathbf{w}_{j}\cdot\mathbf{x}_{i}+b_{j})= \mathbf{t}_{i}\;\;i\;=\;1,\;2,\;3,\;\ldots,\;N. \tag{25}\]  
Equation (25) can be written in a linear system as  
\[\mathbf{H}\mathbf{\beta}=\mathbf{T}. \tag{26}\]  
where \(\mathbf{H}\) is the output matrix of the hidden layer and is defined as  
\[\mathbf{H}=\left[\begin{array}{cccc}\mathbf{\beta}_{1}^{T}\\ \mathbf{\beta}_{2}^{T}\\ \vdots\\ \mathbf{\beta}_{L}^{T}\end{array}\right]_{\text{L}\times m}\text{ and }\mathbf{T}=\left[\begin{array}{c}\mathbf{t}_{1}^{T}\\ \mathbf{t}_{2}^{T}\\ \vdots\\ \mathbf{t}_{N}^{T}\end{array}\right]_{\text{N}\times m},\text{ where }\mathbf{\beta}\text{ is the weight matrix connecting the hidden layer}\]  
and the output layer, and \(\mathbf{T}\) is the target output matrix.  
After assigning the coefficient matrices for random parameters, the matrix \(\mathbf{\beta}\) can be estimated analytically by finding the minimum norm least squares solution:  
\[\mathbf{\hat{\beta}}=\underset{\mathbf{\beta}}{\text{argmin}}\|\mathbf{H} \mathbf{\beta}-\mathbf{T}\|=\mathbf{H}^{+}\mathbf{T}. \tag{28}\]  
where \(\mathbf{H}^{+}\) is the Moore-Penrose generalized inverse of \(\mathbf{H}\).