# Outdoor Node Localization Using Random Neural Networks for Large-Scale Urban IoT LoRa Networks +
## 5 Results and Analysis  
To analyze the performance and localization accuracy of our proposed RNN model, we used the MATLAB R2020b simulation-controlled environment. Figure 5 shows the system's average mean localization error values with the used learning rates of 0.001, 0.01, 0.1, and 1 at different epochs. The developed RNN-based localization model was evaluated using the average localization error (AE) defined by the following formula:  
\[AE=\sum_{i=1}^{n}{((X_{real}-X_{pred})^{2}+(Y_{real}-Y_{pred})^{2})^{0.5}} \tag{12}\]  
where \((X_{real},Y_{real})\) is the actual prerecorded position coordinates recorded using GPS; \((X_{pred},Y_{pred})\) is the estimated location of unknown location predicted by the developed localization system; and n represents the total number of data samples used in modeling.  
From Figure 6, generally, the highest accuracy of the proposed model was obtained with lower learning rates than higher learning rates at the expense of longer training time. Additionally, with higher learning rates, the localization system tends to be more unstable during training. We trained our model with multiple learning rates to offset these issues. Increasing the learning rate from 0.001 to 0.1 did not improve the developed system's accuracy by increasing the system's mean localization error by 0.03 m, which may be significant or not depending on the application. Meanwhile, increasing the learning rate further to 1 improved the accuracy of the localization system by minimizing the mean localization error by 0.02 m. Our system achieved a minimum mean localization error of 0.291 m while using a learning rate of 0.001. More details about obtained mean localization error values obtained for all the used learning rates are presented in Table 1. Additionally, a minimum mean square error (MSE) value of 0.09 m was obtained while training our model.  
Figure 5: Mean localization error for the used learning rates with various numbers of epochs.  
Next, we evaluated the impact of the different samples since we had a massive number of data points that took a very long training time, depending on the used learning rate. The mean localization error values of our system while using 1000, 3000, 5000, 10,000, and 15,000 data samples keeping the same RNN network architecture with the same used learning rates plus 0.0001 are shown in Figure 7. For 1000, 5000, and 10,000 data samples, increasing the learning rate from 0.0001 to 0.01 increased the mean localization error of the system, and when we increased further to 0.1 and 1, the system's error decreased. Furthermore, for 3000 and 15,000 data samples, increasing the learning rate from 0.0001 to 1 improved the system's performance by decreasing its mean localization error.  
The minimum mean localization error of 0.3 m was achieved with 15,000 samples while using learning rates of 0.1 and 1. Consequently, we investigated the total RNN training time elapsed (in sec) for the used number of samples and learning rates, as shown in Table 2. Hence, the highest system accuracy was achieved using the highest learning rate  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Learning Rate** & **Mean Localization Error (m)** \\ \hline
0.001 & 0.291 \\
0.01 & 0.294 \\
0.1 & 0.297 \\
1 & 0.295 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Average mean localization error (m) for used learning rates.  
Figure 6: Minimum mean localization error values for the used learning rates.  
Figure 7: Mean localization error for the used samples.  
of one with the shortest training time of 4970 s and a training error of MSE equal to 0.1 m. Mean localization values obtained for all the used samples are presented in Figure 8 with more details given in Table 2. The obtained results about data samples show that increasing data samples from 1000 to 15,000 and further, to 130,343 samples, did not significantly impact the system's accuracy as the difference is only 0.01 m at the expense of higher training times.