## 4 Data Partition and Modeling
### Modeling
#### 4.2.2 RNN  
RNN has provided excellent performance in various sequential data applications, such as NLP, voice, and time-series signals [33]. In this study, we used a basic LSTM cell for the authentication model because its performance is the same as that of the other variants of the RNN cell [34]. The LSTM cell is illustrated in Figure 8, and Equations (5)-(10) provide the computation of each component in the figure. The line on the top denotes the cell state, which is the internal memory, whereas that on the bottom indicates the hidden state. We denote the cell and the hidden state by \(c\) and \(h\), respectively. The input gate, \(i\), forget gate, \(f\), output gate, \(o\), and internal hidden state, \(g\), are the distinctive features of LSTM, and they are designed to prevent the vanishing gradient problem [35]. During training, the weights and bias in each gate are trained. The forget gate, \(f_{t}\), can control how much the previous hidden state, \(h_{(t-1)}\), should be preserved in the current state, whereas the input gate, \(i_{t}\), can control how much the current input, \(x_{t}\), should be reflected in the current state. The output gate, \(o_{t}\), regulates the amount of hidden state, \(h_{(t-1)}\), in the next sequence and the internal hidden state, \(g_{t}\), is calculated from the input, \(x_{t}\), and the previous hidden state \(h_{(t-1)}\).  
Figure 8: LSTM cell.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **Coordinate-A** & **Coordinate-B** & **Full** & **Angular-A** & **Angular-B** \\ \hline \(nI\) & 12 & 9 & 12 & 12 & 10 \\ \(nO\) & 2 & 2 & 5 & 3 & 3 \\ Total & 16,674 & 16,314 & 16,869 & 16,739 & 16,499 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Number of TCN parameters according to five combinations of variable sets.  
\[f_{t} =\sigma(W_{f}*[h_{t-1},x_{t}]+b_{f}), \tag{5}\] \[i_{t} =\sigma(W_{i}*[h_{t-1},x_{t}]+b_{i}),\] (6) \[o_{t} =\sigma(W_{o}*[h_{t-1},x_{t}]+b_{o}),\] (7) \[g_{t} =tanh(W_{g}*[h_{t-1},x_{t}]+b_{g}),\] (8) \[c_{t} =(c_{t-1}\otimes f_{t})\oplus(g_{t}\otimes i_{t}),\] (9) \[h_{t} =tanh(c_{t})\otimes o_{t}. \tag{10}\]  
The RNN-based DDP model used in this study is illustrated in Figure 9, and its parameter details are provided in Table 10. The number of hidden nodes of LSTM and FC are set to 33 and 64, respectively. The number of RNN parameters for the used five combinations of the variable set is presented in Table 11, which confirms that most networks have 17k parameters, which is similar to the case of TCN models.  
\begin{table}
\begin{tabular}{c c} \hline
**Forward** & **Backward** \\ \hline LSTM-33 & LSTM-33 \\  & Concatenate \\  & FC-64 \\  & FC-\(nO\) \\ \hline \end{tabular}
\end{table}
Table 10: Network parameters for the proposed RNN-based DDP model.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Method** & **Coordinate-A** & **Coordinate-B** & **Full** & **Angular-A** & **Angular-B** \\ \hline \(nI\) & 12 & 9 & 12 & 12 & 10 \\ \(nO\) & 2 & 2 & 5 & 3 & 3 \\ Total & 16,434 & 15,642 & 16,629 & 16,499 & 15,971 \\ \hline \end{tabular}
\end{table}
Table 11: Number of RNN parameters according to five combinations of variable sets.  
Figure 9: Structure of RNN-based DDP model.