# Influence of Noise-Limited Censored Path Loss on Model Fitting and Path Loss-Based Positioning +
## 7 Positioning Simulation Results
### Example 1  
In this section, we analyze the positioning results assuming the true PL follows the Example 1 distribution. The OLS and MLE model fitting results are presented in Section 5.1 and positioning is based on (4) with parameters from Table 1 and \(PL^{*}=140\) dB. Three BS grid densities are considered, leading to an average of 2, 3, or 5 contacted BSs. Each average number of contacted BSs, \(\overline{N}_{\text{BS}}\), corresponds to a constant distance between neighbours \(d_{\text{BS}}\). 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 3 and three of the error CDFs are presented in Figure 7.  
The considered BSs with censored PL are limited to an area around the OTP result with a radius equal to the link distance where the PL model predicts \(\overline{PL}(d)-1\cdot\sigma=PL^{*}\). This radius depends on the PL model. With the MLE fitted model, this is 150 m and with the OLS it is 300 m. The OLS fitted model overestimates the large link distances, and therefore, more censored BSs are taken into account with OLS-MLE than with the MLE-MLE. For example, in the case of \(\overline{N}_{\text{BS}}=5\) and \(N=8\), OLS-MLE includes an average of 13 censored BSs and MLE-MLE includes only an average of 2.9. Note that in this example the positioning error statistics do not change significantly even if more censored BSs are included. When comparing OLS-MLE and MLE-MLE there are two effects. First, the OTP positioning based on OLS fitted model creates typically larger error than the OTP positioning based on MLE fitted model. Second, a larger number of censored BSs accounted for in case of OLS-MLE may correct the error caused by positioning with OLS fitted model, whereas in case of MLE-MLE a smaller number of censored BSs being included may be sufficient. The BS grid density affects the positioning accuracy. In a regular hexagonal grid, the distance between BSs is constant. The positioning error with the directive antennas is smaller than with the omnidirectional antennas, and similarly, with the denser BS grid  
Figure 7: CDF of positioning error of Example 1. 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). (a) \(N=1\), \(\overline{N}_{\text{BS}}=2\); (b) \(N=8\), \(\overline{N}_{\text{BS}}=2\); (c) \(N=8\), \(\overline{N}_{\text{BS}}=5\). the errors are smaller. These trends apply to all the presented cases and also for the other examples of the true PL distribution. In all of the simulated cases, OLS-OTP is the worst. It is the worst-case scenario with the PL model that does not fit the reality and has positioning that does not use all the available information. For example, with omnidirectional antennas and \(\overline{N}_{\text{BS}}=2\), the median errors are 60, 43, 43, and 42 m for OLS-OTP, MLE-OTP, OLS-MLE, and MLE-MLE, respectively. In this case, there is a 29% improvement from OLS-OTP to MLE-MLE. Using MLE in either phase improves accuracy. The best example of this is a dense BS grid with \(\overline{N}_{\text{BS}}=5\) and the directive antennas resulting in the 90 th percentile error of 74 m with OLS-OTP and a 42-55% improvement to about 43-33 m with MLE in either or both phases (see Table 3). When using the noise-limited censored PL with MLE in both phases, MLE-MLE is in general the most accurate option. The improvement from OLS-OTP to MLE-MLE ranged from 29% (from 60 to 42 m with \(N=1\) and \(\overline{N}_{\text{BS}}=2\)) to 55% (from 53 to 24 m with \(N=8\) and \(\overline{N}_{\text{BS}}=2\)) for the median error and from 24% (from 149 to 114 m with \(N=1\) and \(\overline{N}_{\text{BS}}=2\)) to 56% (from 101 to 45 m with \(N=8\) and \(\overline{N}_{\text{BS}}=3\)) for the 90 th percentile. Lastly, let us compare MLE-OTP to OLS-MLE. The median and the 90 th percentiles are close to each other.