## 4 Prediction of Distribution of Drones
### Trajectory Prediction Results of the Drones  
To testify the performance of the QRDGRU model, several benchmark trajectory prediction models are constructed and compared, including the GRU, SVR, QRLSTM and QGRRU models. As a traditional classific model, the GRU model was first applied to justify whether to predict the three-dimensional coordinates at one time or to predict the three-dimensional coordinates separately. The prediction accuracies in the two ways are compared. The hyperparameters for the models are provided in Table 4 and the results are summarized as Model 1 and Model 2 in Table 5. The prediction performance indicators include RMSE, MAE, MAPE, and consumed time. The computer in this research is equipped with the R5-3600 CPU, with the frequency of 4.00 GHz and batch size of 100.  
It can be seen that all the models converge rapidly within several seconds. For the GRU model, the accuracy acquired by predicting the three-dimensional coordinates together tends to be higher than by predicting three-dimensional coordinates separately, indicating the fact that the drone's movements in the three axes are related. Therefore, in the following models, the three-dimensional coordinates were input into the model together.  
Comparing the prediction results from Model 1 to Model 6 in Table 6 and Figure 4, it can be found that the prediction performance indicators RMSE, MAE, and MAPE increase with the increase of timestep, due to the uncertainty of the drones' movement. In general, the QRDGRU model has the highest and most stable prediction accuracy, as compared with the SVR, GRU, QRLSTM, and QGRRU models for different timesteps. The proposed QRDGRU method improves the model structure by integrating two different prediction frameworks, including the D-GRU model proposed in our previous research as well as the quartile regression. The performance indicators confirm the superiority of the established model.  
According to the QRGRU method, the distribution of the probability for the drone with a specifically predicted timestep can be obtained. Figure 5 depicts the position distribution in the horizontal plane of the drone at a specific moment in the future for the 10 s, 20 s, 30 s, and 40 s timesteps, respectively. The darker the colour, the higher the position distribution probability is. It can be seen that the area of predicted location spreads with the increase of timestep, which further indicates the uncertainty of the drones' movement. The blue circle indicates the maximum uncertainty calculated using the maximum acceleration and speed for the given time-step, which is a circle with the initial position of the drone as its center and the furthest distance that can be arrived as its radius. The furthest distance is calculated based on the assumption that the drone accelerates with its initial speed to the maximum speed and then flies with the maximum constant speed to reach the largest distance within a given timestep. For the Mavic 2 drone, the maximum acceleration rate is 6 m/s\({}^{2}\) and the maximum speed is 20 m/s.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Timestep (s)**} & \multirow{2}{*}{**RMSE (m)**} & \multirow{2}{*}{**MAE (m)**} & \multirow{2}{*}{**MAPE (\%)**} & \multirow{2}{*}{**Time (s)**} & \multirow{2}{*}{**RMSE (m)**} & \multirow{2}{*}{**MAE (m)**} & \multirow{2}{*}{**MAPE (\%)**} & \multirow{2}{*}{**Time (s)**} \\  & & & & & & & & \\ \multicolumn{8}{c}{**Model 1: GRU (Predicting Three-Dimensional} & \multicolumn{4}{c}{**Model 2: GRU (Predicting Three-Dimensional} \\ \multicolumn{8}{c}{**Coordinates Together**} \\ \multicolumn{8}{c}{**Coordinates Separately**} \\ \multicolumn{8}{c}{**Coordinates Separately**} \\ \multicolumn{8}{c}{**} \\ \multicolumn