## 2 Materials and Methods
### RNN Autoencoder-Based Method for Extracting Capability Features
#### 2.2.2 Capability Feature Extraction Method
\begin{bmatrix}\left(w_{h^{\prime}h^{\prime}}\right) _{1,1}&\cdots&\left(w_{h^{\prime}h^{\prime}}\right)_{1,m}\\ \vdots&\ddots&\vdots\\ \left(w_{h^{\prime}h^{\prime}}\right)_{m,1}&\cdots&\left(w_{h^{\prime}h^{ \prime}}\right)_{m,m}\end{bmatrix} _{m\times m} \tag{17}\]  
\(W_{h^{\prime}x}\), as shown in Equation (15), represents the joint weight matrix of the hidden state \(h^{\prime}\) reconstruction feature of the Decoder section. In the process of feature extraction ability, through the study of the matrix, we can obtain the Decoder part hidden state and reconstruction feature \(\hat{x}\) mapping relation. 1. Feature extraction steps 1. Feature coding \[W_{h^{\prime}x}= \begin{bmatrix}\left(w_{h^{\prime}x}\right)_{1,1}&\cdots& \left(w_{h^{\prime}x}\right)_{1,m}\\ \vdots&\ddots&\vdots\\ \left(w_{h^{\prime}x}\right)_{M,1}&\cdots&\left(w_{h^{\prime}x}\right)_{M,m} \end{bmatrix} _{M\times m}\] (18)The capability extraction process begins with the raw capability data being mapped by \(f\)\(x_{i}\) mapping to the final state \(a_{i}\), as the feature representation of this sequence, and the mapping function is  
\[a_{i}=f(x_{i}) \tag{19}\]  
If the dimension of the capability feature is \(M\), then \(x_{i}^{(t)}=\left(x_{i}^{(t)1},x_{i}^{(t)2},\cdots,x_{i}^{(t)M}\right)\), and the \(t\in\{1,2,\cdots T_{i}\}\). \(a_{i}\) is the coded value of the \(i\)th sample, and if the \(m\)th sequence of dimensional features is to be extracted, then \(a_{i}=\left(a_{1}^{1},a_{2}^{2},\cdots,a_{i}^{m}\right)\). (b) Feature decoding  
By \(d\) mapping the extracted \(a_{i}\) to the output \(\hat{x}_{i}\):  
\[\hat{x}_{i}=d(a_{i}) \tag{20}\]  
where \(\hat{x}_{i}\) is the reconstruction result of the \(i\)th capability feature sequence; see Equation (21). \[\hat{x}_{i}=\left(\hat{x}_{i}^{(1)},\hat{x}_{i}^{(2)},\cdots,\hat{x}_{i}^{(T_{ i})}\right) \tag{21}\]  
\(i\in\{1,2,\cdots N\}\), the set of reconstructed capability feature sequences is  
\[\hat{X}=\{\hat{x}_{1},\hat{x}_{2},\cdots,\hat{x}_{N}\} \tag{22}\]  
(c) Loss function calculation  
In order to extract the optimal features, the set of reconstructed features of the RNN autoencoder requires \(\hat{X}\) to be as close as possible to the real set of capability feature sequences \(X\), so its learning process is to minimize the \(L_{2}\) loss function is as follows. \[L\left(\hat{X},\ X\right)=\sum_{t=1}^{T}L^{t}=\sum_{i=1}^{N}(\hat{x}_{i}-x_{i} )^{2} \tag{23}\]  
(d) Parameter learning  
The time-based backpropagation algorithm Backpropagation Through Time (BPTT) [23] is used to learn the parameters of the encoder-decoder architecture. The actual solution process can be combined with certain optimization strategies, such as stochastic gradient descent (SGD), to obtain the final encoder after several iterations of training and stopping when the termination condition or the predefined number of training rounds is reached \(f^{*}\). Given a sample set \(S=(X,X)\), the number of samples is N, and the set of ability feature sequences is \(X=\{x_{1},x_{2},\cdots,x_{N}\}\); then, the ability feature of the \(i\)th feature sequence after dimensionality reduction is  
\[a_{i}=f^{*}(x_{i}) \tag{24}\]  
where \(x_{i}\in X\), \(x_{i}=\left(x_{i}^{(1)},x_{i}^{(2)},\cdots,x_{i}^{(T_{i})}\right)\), \(x_{i}^{(t)}=\left(x_{i}^{(t)1},x_{i}^{(t)2},\cdots,x_{i}^{(t)M}\right)\), and \(a_{i}=\left(a_{i}^{1},a_{i}^{2},\cdots,a_{i}^{m}\right)\). The set of samples after feature extraction \(S^{\prime}=(X,A)\), \(A=\{a_{1},a_{2},\cdots,a_{N}\}\) is the \(A\in\mathbb{R}^{m\times N}\). According to the principle of feature extraction, the final state can be set \(a\) with dimensionality less than the input feature dimensionality, i.e., \(m<M\), which can effectively reduce the risk of overfitting of the encoder-decoder architecture, while forcing it to capture the most significant features in the capability sequence for the purpose of dimensionality reduction.