## 2 Proposed Method
### Obtaining the Adjusting Factor for RUL Prediction
#### 2.2.1 The Construction of the ELM  
In this paper, an extreme learning machine (ELM) is used to construct the relationship between the degradation rate data and capacity data in Section 2.2 for the first time, as well as the relationship between the capacity data and the RUL in Section 2.3 for the second time. In the experiment section, the hyper-parameter for the ELM is 10 nodes at the input layer, 7 nodes at the hidden layer, and 1 node at the output layer. The values of the hyper-parameter are determined according to stability and efficiency examined in pre-experiments and historical experiences.  
Construct a single hidden-layer extreme learning machine (ELM) network. Suppose the capacity of the battery sample can be expressed as (\(X_{i}\), \(t_{i}\)) and \(X_{i}\) = [\(x_{i,m}\)] (\(m\) = 1, 2, 3...), \(Y_{i}\) = [\(y_{i,m}\)] (\(m\) = 1, 2, 3...), where \(i\) represents the \(i\)th battery, \(X_{i}\)  is the input to the network, and \(Y_{i}\)  is the output of the network:  
\[\sum_{i=1}^{length}\beta_{i}g(W_{i}\cdot X_{j}+b_{i})=O_{j},\ j=1,2,3,\ldots \text{N} \tag{2}\]  
where \(W_{i}\) is the input weight matrix, \(\beta_{i}\) is the output weight matrix, \(b_{i}\) is the bias matrix, \(g(x)\) is the activate function where the sigmoid function is used, \(O_{j}\) is the output, and N is the total quantity.  
To minimize the error, there should be:  
\[\sum_{j=1}^{N}|O_{j}-y_{j}|=0 \tag{3}\]  
which means there exist \(\beta_{i}\), \(W_{i}\), and \(b_{i}\), such that:  
\[\sum_{i=1}^{length}\beta_{i}g(W_{i}\cdot X_{j}+b_{i})=y_{j} \tag{4}\]The complex form according to the formula is:  
\[H=\begin{bmatrix}g\Big{(}w_{1,1}^{T}X_{1}-\theta_{1}\Big{)}&\cdots&g\Big{(}w_{N,1 }^{T}X_{1}-\theta_{N}\Big{)}\\ \vdots&\ddots&\vdots\\ g\Big{(}w_{1,D}^{T}X_{D}-\theta_{1}\Big{)}&\cdots&g\Big{(}w_{N,D}^{T}X_{D}- \theta_{N}\Big{)}\end{bmatrix}_{D\times N}\]  
where the other two parameters are the following, respectively:  
\[\beta=\begin{bmatrix}\beta_{1}^{T}\\ \vdots\\ \beta_{N}^{T}\end{bmatrix}_{N\times n}Y=\begin{bmatrix}Y_{1}^{T}\\ \vdots\\ Y_{D}^{T}\end{bmatrix}_{D\times n}\]  
where \(H\) is the output at the hidden layer nodes and \(Y\) is the expected output.  
In order to minimize the error, the mission can be turned into minimizing:  
\[|(W_{i},\;\;b_{i})\;\beta_{i}-Y|\;i=1,\;2,\;3,\;\ldots \tag{5}\]  
which is identical to minimizing the loss function:  
\[\text{L}=\sum_{j=1}^{N}\left[\sum_{i=1}^{length}\beta_{i}g(W_{i}\!:\!X_{j}+b_ {i})\!-\!y_{j}\right]^{2} \tag{6}\]  
Compared with other traditional gradient-degradation methods, such as backpropagation (BP), the advantage of the ELM is that once \(W_{i}\) and \(b_{i}\) are chosen arbitrarily, training the neural network can be turned into solving a first-order linear task, which is more efficient.  
Suppose the step width of the historical data trained is l.  
The output of the network can be expressed as:  
\[H(W_{1},\ldots W_{L},b_{1},\ldots,b_{L},X_{1},\ldots,X_{N})=\begin{bmatrix}g(W_ {1}\!\cdot\!X_{1}+b_{1})&\cdots&g(W_{L}\!\cdot\!X_{1}+b_{L})\\ \vdots&\ddots&\vdots\\ g(W_{1}\!\cdot\!X_{N}+b_{1})&\cdots&g(W_{L}\!\cdot\!X_{N}+b_{L})\end{bmatrix} \tag{7}\]  
Thus, the purpose is to minimize:  
\[\text{E}=\sum_{j=1}^{N}\left[\sum_{i=1}^{L}\beta_{i}\;g(W_{i}\!\cdot\!X_{j}+b _{i})\!-\!t_{j}\right]^{2} \tag{8}\]