# An Expert Artificial Intelligence Model for Discriminating Microseismic Events and Mine Blasts
## 3 Methodology
### Extreme Learning Machine (ELM)  
ELM was presented in 2004 by [PERSON] et al., and belongs to the single-hidden layer feedforward neural network (SLFN) learning algorithms [52]. It can learn quickly with few training parameters, achieve good generalization, and has strong applicability [52]. ELM has the salient feature that the input weights and bias of hidden layer nodes (HLNs) are randomly generated using random algorithms and the only variable that needs to be calculated is the output weights, which is obtained through the least-square method.  
Therefore, compared with the conventional intelligent algorithms, it can learn faster while achieving better generalization [53].  
Generally, an ELM network includes three parts: an input layer, a hidden layer and an output layer (see Figure 5). For \(N\) arbitrary different samples \((x_{i},t_{i})\), where \(\mathbf{x}_{i}=[x_{i1},x_{i2},\ldots,x_{in}]^{T}\in\mathbf{R}^{n}\), \(t_{i}=[t_{i1},t_{i2},\ldots,t_{im}]^{T}\in\mathbf{R}^{m}\), a standard SLFN, of which the number of HLNs (NumHLNs) is \(l\) and the activation function is \(g(x)\), can be expressed mathematically as follows [52; 54]:  
\[\sum_{j=1}^{l}\mathbf{\beta}_{j}g(\mathbf{w}_{j}\cdot\mathbf{x}_{i}+\mathbf{b}_{j})=\mathbf{o}_{i} \tag{1}\]  
where \(i=1,\ldots,N\) and \(j=1,\ldots,l\). \(\mathbf{w}_{j}=[w_{j1},w_{j2},\ldots,w_{jn}]^{T}\) is the input weight which connects the \(j\)th hidden node with the input nodes, \(\mathbf{\beta}_{j}=[\beta_{j1},\beta_{j2},\ldots,\beta_{jm}]^{T}\) is the output weight which connects the \(j\)th hidden node with the output nodes, \(\mathbf{b}_{j}\) is the bias of the \(j\)th hidden node, and \(\mathbf{w}_{j}\cdot\mathbf{x}_{i}\) is the inner product of \(\mathbf{w}_{j}\) and \(\mathbf{x}_{i}\).  
When the above standard SLFN infinitely approximates the \(N\) samples and the error is equal to 0, i.e., \(\sum\limits_{i=1}^{N}\|\mathbf{o}_{i}-\mathbf{t}_{i}\|=0\), there exist \(\mathbf{\beta}_{j}\mathbf{w}_{j}\mathbf{b}_{j}\), so that:  
\[\sum_{j=1}^{l}\mathbf{\beta}_{j}g(\mathbf{w}_{j}\cdot\mathbf{x}_{i}+\mathbf{b}_{j})=\mathbf{t}_{i} \tag{2}\]  
The simplified form of Equation (2) is  
\[\mathbf{H}\mathbf{\beta}=\mathbf{T}\]  
Figure 5: ELM network structure.  
where  
\[\mathbf{H}(\mathbf{w}_{1},\ldots,\mathbf{w}_{l},\mathbf{b}_{1},\ldots,\mathbf{b}_{l},\mathbf{x}_{1}, \ldots\mathbf{x}_{N})=\left[\begin{array}{ccc}\mathcal{g}(\mathbf{w}_{1}\cdot\mathbf{x}_{1 }+\mathbf{b}_{1})&\ldots&g(\mathbf{w}_{l}\cdot\mathbf{x}_{1}+\mathbf{b}_{l})\\ \vdots&\cdots&\vdots\\ g(\mathbf{w}_{1}\cdot\mathbf{x}_{N}+\mathbf{b}_{1})&\cdots&g(\mathbf{w}_{l}\cdot\mathbf{x}_{N}+\bm {b}_{l})\end{array}\right]_{N\times l} \tag{3}\]  
\[\mathbf{\beta}=\left[\begin{array}{c}\mathbf{\beta}_{1}^{T}\\ \vdots\\ \mathbf{\beta}_{1}^{T}\end{array}\right]_{l\times m} \tag{4}\]  
\[\mathbf{T}=\left[\begin{array}{c}\mathbf{t}_{1}^{T}\\ \vdots\\ \mathbf{t}_{N}^{T}\end{array}\right]_{N\times m} \tag{5}\]  
where \(\mathbf{H}\) is the output matrix of the hidden layer in ELM.  
The difference between the ELM algorithm and the general SLFN algorithm is that the input weight \(\mathbf{w}\) and hidden layer bias \(\mathbf{b}\) of the former are given randomly at the beginning of the algorithm and accordingly calculating the output matrix \(\mathbf{H}\). Then, what needs to be done is to determine the parameter \(\mathbf{\beta}\). Training feedforward neural network can be regarded as seeking out the solution of \(\mathbf{H}\mathbf{\beta}=T\) by the least-square method. The output weight matrix \(\mathbf{\beta}\) can be obtained through solving this equation:  
\[\mathbf{\beta}=\mathbf{H}^{+}\mathbf{T} \tag{6}\]  
where \(\mathbf{H}^{+}\) is the Moore-Penrose generalized inverse of \(\mathbf{H}\).