# A Robust Tracking Algorithm Based on a Probability Data Association for a Wireless Sensor Network
## 4 Experiment and Result Analysis
### Exponential Distribution  
It was assumed that the measurements' noise obeyed a Gaussian distribution \(N\big{(}0,\sigma_{L}^{2}\big{)}\) and the NLOS errors obeyed an exponential distribution \(E(\lambda)\). The default parameters of the exponential distribution are shown in the Table 3.  
We investigated how the positioning accuracy of the EKF, IMM-EKF, MPDA, and the proposed algorithm varied with the number of beacon nodes when the NLOS errors were exponential distribution (Figure 5(a)). As the number of beacon nodes increased, the RMSEs of the EKF, IMM-EKF, MPDA, and the proposed algorithm all decreased, but the difference was that the RMSEs of the EKF and IMM-EKF decreased slowly and the improvement of the positioning accuracy was not high, while the RMSEs of the MPDA and the proposed algorithms decreased rapidly and the positioning accuracy improved significantly. In the case of a few beacon nodes, the positioning accuracy of the proposed algorithm was higher than that of the EKF, IMM-EKF, and MPDA algorithms. In the case of more beacon nodes, the positioning accuracy of the proposed algorithm was slightly higher than that of the MPDA, and far higher than that of the EKF and IMM-EKF. Compared with the EKF, IMM-EKF, and MPDA algorithms, the proposed algorithm improved the positioning accuracy by 63.60%, 59.57%, and 20.27%, respectively, on average. We also explored the influence of different NLOS errors probabilities on the positioning results of the EKF, IMM-EKF, MPDA, and the proposed algorithm when NLOS errors were given as an exponential distribution. From Figure 5(b), we can see that the positioning accuracy of the EKF, IMM-EKF, MPDA, and the proposed algorithm declined with the increasing probability of NLOS errors. With the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Default Values** \\ \hline The number of beacon nodes & \(M\) & 6 \\ \hline The NLOS error probability & \(P_{NLOS}\) & 0.5 \\ \hline The measurements noise & \(N\big{(}0,\sigma_{L}^{2}\big{)}\) & \(N\big{(}0,1^{2}\big{)}\) \\ \hline The NLOS errors & \(E(\lambda)\) & \(E(8)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Exponential distribution parameter. increase of the NLOS error probability, the RMSEs of the EKF and IMM-EKF increased almost linearly, and the RMSE growth rate of the MPDA and the proposed algorithm gradually accelerated. When the NLOS error was less than 0.3, the positioning accuracy of the proposed algorithm was slightly higher than that of the MPDA, but significantly higher than that of the EKF and IMM-EKF. When the probability of the NLOS errors was relatively large, the proposed algorithm had a better localization performance than the EKF, IMM-EKF, and MPDA. The proposed algorithm had a higher positioning accuracy than the EKF, IMM-EKF, and MPDA with about 65.83%, 53.01%, and 18.56%, respectively, on average. In Figure 6c, the influence of changing the parameter \(\lambda\) of exponential distribution on the localization accuracy is shown. We can see that when the parameter \(\lambda\) changed from 3 to 10, the RMSEs of the EKF and IMM-EKF algorithm increased rapidly, while the RMSE of the MPDA algorithm grew slowly. The change of the parameter \(\lambda\) from 3 to 10 had no significant impact on the localization accuracy of the proposed algorithm. Compared with the EKF, IMM-EKF, and MPDA algorithms, the proposed algorithm improved the positioning accuracy by at least 36.16%, 33.35%, and 11.43%, respectively. The error cumulative distribution function of the localization errors is depicted in Figure 6d. It shows that the 90 th percentile of the proposed algorithm was less than 3.882 m. In contrast, the 90 th percentile of the EKF, IMM-EKF, and MPDA were achieved at 12.22 m, 10.54 m, and 5.065 m, respectively. Figure 6: (**a**) Performance comparison between the EKF, IMM-EKF, MPDA, and the proposed algorithm under different numbers of beacon nodes \(M\), where \(P_{NLOS}=\ 0.5\), \(N\big{(}0,1^{2}\big{)}\), and \(E\)(8).