## 3 Proposed Algorithm
### ELM Algorithm
Then, all least-squares solutions of \(x_{0}=A^{+}y\) and \(Ax=y\) have the smallest modulus: \[E_{0}=\sum_{j=1}^{N}\left(\sum_{i=1}^{L}\beta_{i}g\big{(}w_{i}.w_{j}+b_{i} \big{)}-t_{j}\right)^{2} \tag{7}\]0 (15) 3. \(Ax=y\) with the only smallest modulus and the least-squares solution is \(x=A^{+}y\mathsf{x}=A^{+}y\). The Learning Rules of ELM: Known training samples \(\aleph=\Big{\{}\frac{x_{i},t_{i}}{x_{i}cR^{n}}\Big{\}},t_{i}eR,\ i=1,\ldots,N\), and transfer function \(g\). \(L\) is the hidden layer node [24]:  
1. Get random hidden node parameters \((a_{i},b_{i}),i=1,\ldots,L\);
2. Getting of Hidden layer output matrix \(H\);
3. The calculation of output weight \(\beta:\beta=H^{+}T\). \(H^{+}\) is the pseudo-inverse MP of \(H\). \(T=[t_{1},\ldots,t_{N}]\). There are many calculation methods for the pseudo-inverse MP of \(H\): the orthographic projection method, orthogonal method, iterative method, and singular value decomposition (SVD), etc. [25]. To reduce search and iteration in ELM algorithms, when \(H^{T}H\) is non-singular, or the hogonal projection method is used with \(H^{-1}=\big{(}H^{T}H\big{)}^{-1}H^{T}\). However, \(H^{T}H\) cannot be known to be non-singular, and in some cases it is singular, so the orthogonal projection methods do not achieve the desired effect [26]. However, SVD is often used to obtain the MP pseudo-inverse method of \(H\). According to the research above, the following conclusions can be drawn:  
1. Training error term can reach the minimum. Especially, solving the least-squares solution of \(\hat{\beta}=H^{+}T\) being \(H\beta=T\), shows that this solution can be used to obtain the smallest error term: \[E_{0}=\sum_{j=1}^{N}\left(\sum_{i=1}^{L}\beta_{i}g\big{(}w_{i}.w_{j}+b_{i} \big{)}-t_{j}\right)^{2} \tag{7}\]1 (16) In the actual operation of the algorithm, all the learning algorithms are aimed at obtaining the minimum error. However, the application process has limitations on the number of iterations,trapping into the local minimum, and other conditions, resulting in failure to obtain the desired effect, thereby reducing the performance of the algorithm. 2. Minimum modulus of weight. All the solutions of \(\hat{\beta}=H^{+}T\) and \(H\beta=T\) can find the minimal solution: \[E_{0}=\sum_{j=1}^{N}\left(\sum_{i=1}^{L}\beta_{i}g\big{(}w_{i}.w_{j}+b_{i} \big{)}-t_{j}\right)^{2} \tag{7}\]2 (17)
3. The solution of \(H\beta=T\) has the uniqueness, \(\hat{\beta}=H^{+}T\). In this paper, a new single-layer feed-forward neural network algorithm-extreme learning machine algorithm is introduced. After random estimation of the initial parameters is determined, the output values are analyzed and acquired. Any parameters in the ELM algorithm can be achieved by analysis instead of a simple iterative acquisition. Therefore, this paper also proves in theory that the ELM algorithm can guarantee the learning speed and has strong ability for generalization of the data. The ELM algorithm can be well applied, and the speed is improved by many times compared with the ordinary learning algorithm [18]. Compared with the commonly used BP neural network and SVM algorithm [27], the introduction of the ELM algorithm in this study has the following main advantages:  
1. Ease of use. The model does not have to set a large number of parameters, and the training speed does not need to intervene and also can obtain better experimental results;
2. Fast learning. The learning time of the model is quite short, which is more conducive to the user's real-time information acquisition;
3. Generalization performance is high, and the ELM algorithm has a better generalization performance for different data types;
4. Activation function selection is flexible. Alternative types of activation function in ELM algorithms are adequate.