## 3 Methods
### Extreme Learning Machine Optimized by the Crow Search Algorithm  
(1) Extreme learning machine  
The extreme learning machine (ELM) is a single hidden-layer forward neural network (SLFM) learning method proposed by [PERSON] et al. [34]. In ELM, the connection weights between the input and hidden layer and the threshold value of the hidden layer are generated randomly. It is not necessary to adjust during the training, which could avoid the cumbersome adjustment of neural network parameters in the iterative process. Therefore, the ELM method has a fast learning speed and good generalization performance. Compared with traditional learning algorithms based on gradient descent theory, such as the BP algorithm, its learning speed is faster and it can obtain better generalization performance. Compared with support vector machines, the model of ELM is simpler. In this paper, it is supposed that there are \(N\) arbitrary samples \((X_{i},t_{i})\). Among them, the physiological characteristic parameter is the input variable, \(X_{i}=(x_{i1},x_{i2}\cdots,x_{in})^{T}\in R^{n}\). Intention is expressed as an output variable, \(t_{i}=(t_{i1},t_{i2}\cdots,t_{in})^{T}\in R^{n}\). \(n\) is the dimension of the input layer and \(m\) is the dimension of the output layer. For a single hidden-layer feedforward neural network with \(L\) hidden layer nodes, the output can be expressed by Equation (5):  
\[\sum_{j=1}^{L}\beta_{j}\beta(X_{i})=\sum_{j=1}^{L}\beta_{j}\beta(W_{j}X_{i}+q_ {j})=o_{i} \tag{5}\]  
where \(i=1,2,\cdots,N\), \(W_{j}=(\omega_{j1},\omega_{j2},\cdots,\omega_{jn})^{T}\) represents the input weight, \(\beta=(\beta_{j1},\beta_{j2},\cdots,\beta_{jn})^{T}\) is the output weight, \(q_{j}\) is the bias of the \(j\)th hidden-layer elements, \(W_{j}\cdot X_{i}\) expresses the inner product of \(W_{j}\) and \(X_{i}\), and \(g(x)\) is the activation function. Figure 4: Diagram of the positioning process. The learning goal of SLFM is to approximate \(N\) samples under the minimum error, which can be defined by Equation (6):  
\[\sum_{i=1}^{N}\beta_{j}\|o_{i}-t_{i}\|=0,\ i=1,2,\cdots,N \tag{6}\]  
That is, there are \(W_{j}\), \(\beta_{j}\), \(q_{j}\) such that:  
\[\sum_{j=1}^{L}\beta_{j}g(W_{j}\cdot X_{i}+q_{j})=t_{i},\ i=1,2,\cdots,N \tag{7}\]  
It can be simplified as \(H\beta=T\), where \(H\), \(\beta\), and \(T\) are the hidden-layer node output matrix, the output weight, and the desired output, respectively. \[ \begin{split}& H(W_{1},W_{2},\cdots,W_{L},q_{1},q_{2},\cdots q _{L},X_{1},X_{2},\cdots,X_{L})=\\ &\left[\begin{array}{cccc}g(W_{1}\cdot X_{1}+b_{1})&g(W_{2} \cdot X_{1}+b_{2})&\cdots&g(W_{L}\cdot X_{1}+q_{L})\\ g(W_{1}\cdot X_{2}+b_{1})&g(W_{2}\cdot X_{2}+b_{2})&\cdots&g(W_{L}\cdot X_{2}+q _{L})\\ \vdots&\vdots&\ddots&\vdots\\ g(W_{1}\cdot X_{N}+b_{1})&g(W_{2}\cdot X_{N}+b_{2})&\cdots&g(W_{L}\cdot X_{N}+q _{L})\\ \end{array}\right]_{N\times L}\end{split}  \tag{8}\]  
\[\text{where }\beta=\left[ \begin{array}{c}\beta_{1}^{T}\\ \beta_{2}^{T}\\ \vdots\\ \beta_{L}^{T}\end{array} \right]_{L\times m},\ T=\left[ \begin{array}{c}T_{1} ^{T}\\ T_{2}^{T}\\ \vdots\\ T_{L}^{T}\end{array} \right]_{N\times m}.\]  
To train a single hidden-layer neural network, a set of parameters \(\hat{W}_{j}\), \(\hat{\beta}_{j}\) and \(\hat{q}_{j}\) are obtained such that:  
\[\|H(\hat{W}_{j},\hat{q}_{j})\hat{\beta}_{j}-T\|=\min_{W,q,\beta}\|H(W_{j},q_{ j})\beta_{j}-T\|,\ j=1,2,\cdots,L \tag{9}\]  
The output matrix of hidden layer \(H\) will be obtained when the single hidden-layer feedforward neural network input weight \(W_{i}\) and hidden-layer bias \(q_{i}\) are determined randomly. The solution process of SLFM is transformed into a linear system \(H\beta=T\). The output weight \(\beta\) could be determined by Equation (10):  
\[\hat{\beta}=H+T \tag{10}\]  
In the extreme learning machine, the weights from the input layer to the hidden layer and the threshold of the hidden-layer nodes are random.