# Enconv1d Model Based on Pseudolite System for Long-Tunnel Positioning
## 4 Experiment and Result Analysis
### Analysis of Experimental Result  
To accurately assess the positioning accuracy, three performance evaluation metrics are used in this study: Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Distance Error. Let there be N positioning points, with \((x_{0},\ y_{0})\) representing the true coordinates and (x, y) representing the predicted coordinates. The formulas for the Mean and Distance Error are as follows:  
\[\mathit{distance\_error}=(\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}) \tag{9}\]  
\[\mathit{Mean}=(\sum_{i}^{N}\mathit{distance\_error})/N \tag{10}\]  
We demonstrate the localization performance of our proposed model by comparing it with three other models: 1D CNN (Conv1d), LSTM, and encoder. The 1D CNN model can learn spatial information from one-dimensional data, while the LSTM model constrains localization errors using temporal characteristics. The encoder model, composed of multi-head attention mechanisms and feed-forward neural networks, aims to capture the input matrix's global features and temporal correlations. The detailed parameter settings are shown in Table 1.  
For the four test paths, the number of localization points for each path is 13,622, 10,678,  
Table 2, the average Mean Distance Error of Enconv1d across the four test paths is 0.975 m, which represents improvements of 49.15%, 58.77%, and 32.29% over LSTM, Conv1d, and encoder, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Enconv1d** & **Encoder** & **Conv1d** & **LSTM** \\ \hline Epochs & 500 & 500 & 500 & 500 \\ Leaning rate & 0.0002 & 0.0002 & 0.00015 & 0.00015 \\ Batch size & 64 & 64 & 64 & 64 \\ Optimizer & Adam & Adam & Adam & Adam \\ CNN layers & 3, 3, 3 & / & 3 & / \\ Multi-head & 2 & 2 & / & / \\ Multi-head layers & 2, 2 & 4 & / & / \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter settings for four models. Figure 7: Comparison of the errors of the four localization models in the four test paths. (**a**) Comparison of Mean Distance Error, (**b**) comparison of MAE, (**c**) comparison of RMSE, (**d**) CDF comparison for distance error. Figure 7b,c illustrate the MAE and RMSE of the four models, where the curves in different colors highlight the advantages of our model. From the fluctuations in the curves, it can be observed that all models show an increase in error on the third test path. This is due to multiple lateral lane changes within the tunnel during the third test path. From the perspective of the RMSE, the metric most sensitive to error, it can be seen that relying solely on LSTM to learn long-term dependencies for constraining localization results without other auxiliary modules makes the model highly susceptible to influences from previous time steps. This over-constraining prevents the model from updating in real-time during lane changes, resulting in a maximum error of up to 7.42 m. The Conv1d model, composed of 1D convolutions, is significantly affected by local feature interference when learning spatial information and is not robust to outliers, with a minimum RMSE of 2.74 m.  
In contrast, the encoder structure, composed of multi-head attention mechanisms, learns data features from a global perspective, achieving a minimum RMSE of 1.49 m, significantly better than the previous two models. The Enconv1d model, which incorporates multi-scale convolutional kernels, exhibits lower sensitivity to outliers and effectively constrains the range of abnormal values. The minimum RMSE value achieved is 0.98 m.  
From the model testing data, Table 3 shows that the average MAE of Enconv1d across the four test paths is 1.148 m, representing reductions of 48.08%, 57.14%, and 30.77% compared to the LSTM, Conv1d, and encoder models, respectively. Table 4 presents the RMSE of the localization results for the four models across different test paths. The data indicate that the average RMSE of Enconv1d is 1.105 m, showing reductions of 71.11%, 65.14%, and 35.1% compared to the LSTM, Conv1d, and encoder models, respectively.