## 5 Results
### Experimental Results and Analysis
#### 5.2.4 Discussion on Running Time  
Table 5 shows the average running time obtained after 100 Monte Carlo experiments of each algorithm. Due to the absence of sampling and propagation of particles, UKF took the shortest running  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Overall RMSE (m)**} \\ \cline{2-4}  & \(\sigma_{\text{R}}\) = 50 & \(\sigma_{\text{R}}\) = 100 & \(\sigma_{\text{R}}\) = 150 \\ \hline PF & 143.3577 & 256.4032 & 376.2221 \\ PS-PF & 125.8578 & 239.6098 & 359.9481 \\ UKF & 127.4355 & 252.4281 & 374.7265 \\ UPF & 123.8671 & 238.5487 & 356.6859 \\ MMNF & 122.0391 & 236.6279 & 354.3415 \\ PS-UPF & 120.0796 & 230.2823 & 349.8173 \\ PS-AUPF & 119.1287 & 227.8623 & 347.2862 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Tracking RMSE of each algorithm with Rayleigh noise.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Overall RMSE (m)**} \\ \cline{2-4}  & \(\sigma_{\text{G}}\) = 50 & \(\sigma_{\text{G}}\) = 100 & \(\sigma_{\text{G}}\) = 150 \\ \hline PF & 115.7570 & 182.6600 & 247.5047 \\ PS-PF & 88.1420 & 156.7254 & 227.3936 \\ UKF & 89.9564 & 167.2354 & 264.3609 \\ UPF & 81.8241 & 150.7658 & 215.7149 \\ MMNF & 80.4735 & 148.3845 & 214.0359 \\ PS-UPF & 77.8543 & 142.9482 & 202.3238 \\ PS-AUPF & 75.9206 & 139.1967 & 199.8989 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Tracking RMSE of each algorithm with Gaussian noise.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Overall RMSE (m)**} \\ \cline{2-4}  & \(\pm\)**50 m Range of Noises** & \(\pm\)**100 m Range of Noises** & \(\pm\)**150 m Range of Noises** \\ \hline PF & 89.8381 & 121.3732 & 156.4902 \\ PS-PF & 56.8432 & 94.8275 & 135.7030 \\ UKF & 59.2855 & 101.2696 & 154.4144 \\ UPF & 51.7421 & 89.4955 & 129.6116 \\ MMNF & 50.5371 & 88.3553 & 128.9389 \\ PS-UPF & 48.4721 & 86.1945 & 124.6514 \\ PS-AUPF & 46.4825 & 84.4288 & 123.3828 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Tracking RMSE of each algorithm with Uniform noise.  
time. The MMNF also has no particle propagation, but unlike UKF, as a multiple-model method based on GRNN, MMNF needs to train multiple models simultaneously to adapt to different motion patterns of the target. Although those models can be implemented in parallel, it still takes a lot of time in the process of parameter training. For the traditional PF, it needs to generate 3000 particles during operation, which means that it spent the longest time. The sample size of PS-PF is reduced to 1000 by partitioned sampling, and its running time is reduced to 11.35% of PF with higher accuracy. For UPF, its accuracy is already greatly improved by using UKF to establish the importance density. Therefore, only 30 particles needed could allow UPF to achieve lower tracking errors with less running time than PF, even though the extra computation of UKF increased the computational complexity. After introducing partitioned sampling, the running time of PS-UPF is reduced to 24.66% of UPF with the same sample size. In addition, by adaptively adjusting our sample size, with the same initial sample size, the time complexity of PS-AUPF is further reduced by nearly half of PS-UPF. Figure 9 shows changes of the total number of particles in PS-AUPF during a tracking process with \(N_{\text{min}}=9\).