## 3 Proposed Algorithm
### ELM Algorithm  
In this section, we first introduce the principle of a single hidden-layer feed-forward neural network algorithm. After that, we describe the principle of ELM algorithm in details. Figure 2 above is a single hidden layer neural network containing only one hidden layer. Assuming there are \(N\) groups of samples \((x_{i},t_{i})\), so there are \(L\) hidden layer in \(x_{i}=[x_{i1},x_{i2},\ldots,x_{in}]^{T}\in\Re^{n}\), \(t_{i}=[t_{i1},t_{i2},\ldots,t_{in}]^{T}\epsilon\)\(\Re^{n}\). If \(g(x)\) is an activation function, the model is shown. Figure 2: Single hidden layer neural network framework. \[\sum_{i=1}^{L}\beta_{i}g_{i}(x_{j})=\sum_{i=1}^{L}\beta_{i}g(w_{i}.x_{i}+b_{i})=o _{j},\ j=1,\ 2,\ldots,N \tag{1}\]  
In the above formula, \(w_{i}=[w_{i1},w_{i2},\ldots,w_{in}]^{T}\) is the weight value between the hidden layer and the input layer. \(\beta_{1}=[\beta_{i1},\beta_{i2},\ldots,\beta_{in}]^{T}\) is the weight value between the hidden layer and the output layer. \(b_{i}\) is the bias of the \(Z\)th hidden point. The canonical model with \(L\) hidden layer points and \(g(x)\) can accurately approximate \(N\) data, expressing \(\sum_{j=1}^{L}||o_{j}-t_{j}||=0\). According to \(\beta_{i}\),\(w_{i}\), and \(b_{i}\), we can get  
\[\sum_{i=1}^{L}\beta_{i}g_{i}(w_{j}.x_{j}+b_{i})=t_{j},j=1,2,\ldots,N \tag{2}\]  
The above equations are summarized as  
\[H\beta=T \tag{3}\]  
Among  
\[H(w_{1},\ldots,w_{L},b_{1},\ldots,b_{n},x_{1},\ldots,x_{N})=\left( \begin{array} []{c}G(w_{1},b_{1},x_{1})\cdots G(w_{L},b_{L},x_{L})\\ \vdots\ddots\vdots\\ G(w_{1},b_{1},x_{N})\cdots G(w_{L},b_{L},x_{N})\end{array} \right) \tag{4}\]  
\[\beta=\left[ \begin{array}{c}\beta_{1}^{T}\\ \vdots\\ \beta_{L}^{T}\end{array} \right]_{L\times m},T=\left[ \begin{array}{c}t_{1}^{T }\\ \vdots\\ t_{N}^{T}\end{array} \right]_{N\times m} \tag{5}\]  
\(H\) is the corresponding output matrix of the hidden layer. The column of \(H\) is the output of the first hidden point. Generally speaking, in the process of training a hidden layer neural network, \(\hat{w}_{i},\hat{b}_{i},\ \hat{\beta}_{i}(i=1,\ \ldots,\ L)\) need meet  
\[\left||H\Big{(}\hat{w}_{1},\ldots,\hat{w}_{L},\hat{b}_{i},\ldots,\hat{b}_{L} \Big{)}\hat{\beta}_{1}\hat{b}_{i}-T\right||=\min_{w_{i}b_{i}\beta_{i}}\|H \Big{(}\Big{(}\hat{w}_{1},\ldots,\hat{w}_{L},\hat{b}_{i},\ldots,\hat{b}_{L} \Big{)}\hat{\beta}_{1}\hat{b}_{i}-T\Big{)}\| \tag{6}\]  
The equation above is equivalent to the equation below minimizing the following cost function. \[E_{0}=\sum_{j=1}^{N}\left(\sum_{i=1}^{L}\beta_{i}g\big{(}w_{i}.w_{j}+b_{i} \big{)}-t_{j}\right)^{2} \tag{7}\]  
The solution of \(H\) is obtained by gradient descent learning algorithm for getting the minimum to \(||H\beta-T||\). The joint vector of \((w_{i},\beta_{i})\) and \((b_{i})\) is \(W\). After finishing the iteration, we get the following formula:  
\[W_{k}=W_{k-1}-\eta\frac{\partial E(W)}{\partial W} \tag{8}\]  
In the formula above, \(\eta\) is learning rate. So, in view of the above single hidden-layer feed-forward neural network algorithm, it is easy to cause over-fitting in the actual operation process, the experimental real-time data acquisition rate being low and so on. Therefore, this type of method is called a problem that restricts the development of the neural network. The ELM algorithm is distinct from the neural network algorithm in the above sense. It sets different hidden layer numbers and obtains different analytical results for the output weights [16]. The ELM algorithm will be described and proved in detail as follows. Figure 3 displays the structure of the ELM algorithm, including three levels: input, output layer, and hidden layer [17]. Assuming that the activation function g is infinitely different, so the number of hidden layers set by the ELM training is much smaller than that of the sample, namely \(L\leq N\). So, get the following theorem:  
**Theorem 3**. : _Assuming that there are \(N\) implicit nodes in the ELM corresponding to \(g:\Re\rightarrow\Re\) is infinitely different.