# Influence of Noise-Limited Censored Path Loss on Model Fitting and Path Loss-Based Positioning +
## 7 Positioning Simulation Results
### Example 2  
In this section, we analyze the positioning results while assuming the true PL follows Example 2's distribution. The OLS and MLE model fitting results are presented in Section 5.2 and positioning is based on (4) with parameters from Table 1 and \(PL^{*}=95\) dB. Two BS grid densities are considered, leading to an average of five or ten contacted BSs. Two BS antennas are considered, omni-directional and directional beams with \(N=8\) and \(\theta_{\text{3 dB}}=45^{\circ}\). Positioning error 50 th and 90 th percentiles are listed in Table 4 and three of the error CDFs presented in Figure 8. The BSs with censored PL around the OTP results, OLS-OTP and MLE-OTP, are included if they lie withing 85 and 220 m in case of OLS-MLE and MLE-MLE, respectively. For example, in the case of \(\overline{N}_{\text{BS}}=5\) and \(N=8\), OLS-MLE includes an average of 21 censored BSs and MLE-MLE includes only an average of 1.2. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(\overline{N}_{\text{BS}}\) & 2 & 3 & 5 & 2 & 3 & 5 \\ \cline{2-11} \(N\) & 1 & 1 & 1 & 8 & 8 & 8 & 8 \\ \cline{2-11} \(d_{\text{BS}}\) & 159 & 129 & 100 & 245 & 199 & 154 \\ \cline{2-11}  & 50\% & 90\% & 50\% & 90\% & 50\% & 90\% & 50\% & 90\% & 50\% & 90\% \\ \hline OLS-OTP & 60 & 149 & 43 & 108 & 29 & 77 & 53 & 137 & 34 & 101 & 20 & 74 \\ \hline MLE-OTP & 43 & 117 & 29 & 72 & 20 & 48 & 27 & 88 & 19 & 59 & 14 & 43 \\ \hline OLS-MLE & 43 & 116 & 28 & 72 & 20 & 44 & 25 & 64 & 19 & 44 & 14 & 33 \\ \hline MLE-MLE & 42 & 114 & 28 & 69 & 19 & 43 & 24 & 69 & 18 & 45 & 13 & 34 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Positioning error, in meters, 50 th and 90 th percentiles of Example 1. Base stations have omnidirectional (\(N=1\)) or directional antennas (\(N=8\)) and the average number of contacted BSs is \(\overline{N}_{\text{BS}}=2\), 3, or 5. The relative performances of OLS-OTP, MLE-OTP, OLS-MLE, and MLE-MLE are the same as in Example 1. Using MLE in either case reduces the positioning errors; OLS-OTP is the worst and MLE-MLE is the best method. For example, in the case of \(\overline{N}_{\text{BS}}=5\) and omnidirectional BS antennas, the improvement in median error is 35% or 28% for Examples 1 and 2, respectively. The errors are larger in Example 2 than in Example 1 due to larger \(\sigma/\alpha\), i.e., larger standard deviation of the expected log-distance (19). The improvement from OLS-OTP to MLE-MLE ranged from 28% (from 96 to 69 m with \(N=1\) and \(\overline{N}_{\text{BS}}=5\)) to 57% (from 64 to 28 m with \(N=8\) and \(\overline{N}_{\text{BS}}=10\)) for the median error and from 24% (from 251 to 190 m with \(N=1\) and \(\overline{N}_{\text{BS}}=5\)) to 47% (from 146 to 77 m with \(N=8\) and \(\overline{N}_{\text{BS}}=10\)) for the 90 th percentile. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\overline{N}_{\text{BS}}\) & \multicolumn{2}{c}{5} & \multicolumn{2}{c}{10} & \multicolumn{2}{c}{5} & \multicolumn{2}{c}{10} \\ \cline{2-9} \(N\) & \multicolumn{2}{c}{1} & \multicolumn{2}{c}{1} & \multicolumn{2}{c}{8} & \multicolumn{2}{c}{8} \\ \cline{2-9} \(d_{\text{BS}}\) & \multicolumn{2}{c}{459} & \multicolumn{2}{c}{325} & \multicolumn{2}{c}{515} & \multicolumn{2}{c}{420} \\ \cline{2-9}  & 50\% & 90\% & 50\% & 90\% & 50\% & 90\% & 50\% & 90\% \\ \hline OLS-OTP & 96 & 251 & 66 & 164 & 101 & 228 & 64 & 146 \\ \hline MLE-OTP & 71 & 182 & 46 & 105 & 61 & 168 & 34 & 90 \\ \hline OLS-MLE & 82 & 352 & 49 & 259 & 49 & 236 & 28 & 78 \\ \hline MLE-MLE & 69 & 190 & 45 & 106 & 47 & 149 & 28 & 77 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Positioning error, in meters, 50 th and 90 th percentiles of Example 2. Base stations have omnidirectional (\(N=1\)) or directional antennas (\(N=8\)) and the average number of contacted BSs is \(\overline{N}_{\text{BS}}=5\) or 10. Figure 8: CDF of positioning error of Example 2. Model fitting to training data with ordinary least squares (OLS-) or Tobit MLE fitting (MLE-) and positioning with either ordinary trilateration (-OTP) of Tobit MLE positioning (-MLE).