# Influence of Noise-Limited Censored Path Loss on Model Fitting and Path Loss-Based Positioning +
## 7 Positioning Simulation Results  
The four options, with and without the censored data, were compared by conducting simulations in a regular BS grid, following a hexagonal layout. The true PL values were calculated based on the three examples, and positioning simulation results are presented and analyzed in Sections 7.1-7.3. The positioning is based on PL model derived, in Sections 5.1-5.3, either by OLS or by MLE. Positioning is done either with OTP or with Tobit MLE, as presented in Section 6. Therefore, the four options are OLS-OTP, MLE-OTP, OLS-MLE, and MLE-MLE. Positioning error is the distance between the true position and the estimated position. These statistics are based on 10,000 samples with random true locations and different realizations of SF. In all examples, two BS antennas are considered, omni-directional (\(N=1\)) and directional beams with eighth beams (\(N=8\) and \(\theta_{\text{3 dB}}=45^{\circ}\)). A few BS grid densities are considered where the distance between BSs, \(d_{\text{BS}}\), is selected such that an average number of contacted BSs, \(\overline{N}_{\text{BS}}\), is, e.g., 5.0. It is assumed that the signals from the BSs do not interfere and the BS locations and beam directions are known.  
Before we analyze the positioning error statistics, let us look at the likelihood function illustrations presented in Figures 5 and 6. These figures were calculated assuming that the true PL follows the distribution in Example 1 and positioning using OLS and MLE fitted models presented in Section 5.1. In Figure 5a is the \(LLF_{i}\) of a single contacted BS with an omnidirectional antenna where the highest likelihood is found on a ring around the BS. In Figure 5b is the \(LLF_{i}^{*}\) of an uncontacted BS showing a low likelihood close to the BS. Figure 5c,d shows the corresponding examples with single directive antennae where the likelihood functions are stretched by the antenna gain pattern. Figure 6 shows an example of the sum of the likelihood functions in one location using the four options. In this example, the distance between BSs is 245 m, BSs antennas have eight beams (\(N=8\), \(\theta_{\text{3 dB}}=45^{\circ}\)), and three BSs are contacted with a total of four beams (\(PL<PL^{*}\)). The positioning errors are 81, 72, 9, and 14 m for OLS-OTP, MLE-OTP, OLS-MLE, and MLE-MLE, respectively. Using MLE, positioning results in smaller and narrower likelihood function optimum. Let us compare OLS-OTP, Figure 6a, and OLS-MLE, Figure 6c. The OTP result is to the right-hand side to the true location, and the the closest uncontacted BSs are on the right side of the OTP result at \((x_{\text{BS}},y_{\text{BS}})=(385,521)\) and \((430,123)\). In MLE positioning these BSs are taken into account, which corrects the positioning away from the uncontacted BSs and thus reduces the error. Thus, MLE positioning can, at least sometimes, compensate for the error resulting from the OLS fitted model, which overestimates the link distances.  
Figure 5: Likelihood function illustrations: (**a**) measured PL and omnidirectional antenna, (**b**) censored PL and omnidirectional antenna, (**c**) measured PL and directive beam (\(N=8\)), (**d**) censored PL and directive beam (\(N=8\)). White is likely, gray is possible, and black is an unlikely location.  
Figure 6: Likelihood function illustrations: true location (+), position estimate (\(\times\)), base stations (BSs; stars), contact with three BSs with directive antennas (\(N=8\)) at \((x_{\rm BS},y_{\rm BS})\) = (302,324), (547,363), and (146,516). White is likely, gray is possible, and black is an unlikely location.(**a**) OLS-OPT: ordinary least squares fitting and trilateration positioning. (**b**) MLE-OPT: Tobit MLE fitting and the ordinary trilateration positioning. (**c**) OLS-MLE: ordinary least squares fitting and the Tobit MLE positioning. (**d**) MLE-MLE: Tobit MLE fitting and the Tobit MLE positioning.