# Influence of Noise-Limited Censored Path Loss on Model Fitting and Path Loss-Based Positioning +
## 5 Model Fitting Results
### Example 1  
In this example the true PL distribution is the log-distance PL model (1) and (2) with the parameters \(\alpha=4\), \(\beta=60\) dB, \(\sigma=7\) dB, and noise threshold 140 dB [13]. The PL distribution, threshold level, and the fitted models are illustrated in Figure 3. Note that the figure shows only samples of measurable PL data points but the link distances and the number of noise-limited censored PL data points are also known.  
Training data were created with a uniform distribution of distances between 20 and 500 m. The PL model was fitted to the data using OLS or Tobit MLE. With OLS the noise-limited censored data are ignored and with MLE all data are taken into account. The fitted PL models are illustrated in Figure 3 and the parameter estimates are listed in Table 1. The OLS fitting underestimates PL distribution for large link distances because there are more censored samples above the true mean, and as they were omitted the expected value was lowered. The OLS fitting gives parameter estimates \(\hat{\alpha}\approx 2.5\), \(\hat{\beta}\approx 83\) dB, and \(\hat{\sigma}\approx 5.7\) dB. The MLE gives parameter estimates \(\hat{\alpha}\approx 4\), \(\hat{\beta}\approx 60\) dB, and \(\hat{\sigma}\approx 7\) dB. The MLE estimates are very close to the true values \(\alpha=4\), \(\beta=60\) dB, and \(\sigma=7\) dB. As can be seen, ignoring the censored samples can lead to significant errors in the PL model distribution. The same conclusion was made in [10; 11], and in this paper, we study the effects on positioning accuracy.  
The noise threshold level \(PL^{*}\) limits the maximum path loss that can be measured. The expected PL reaches the 140-dB level at a link distance of 100 m. Therefore, PL at that distance has a 50% probability of being immeasurable, i.e., censored. Due to the large shadow fading, \(\sigma=7\) dB, there is significant probability of censored PL between 40 and 220 m, i.e., where the dash lines \(\overline{PL}(d)\pm 1.96\sigma\) cross the noise threshold of 140 dB in  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & \(\hat{\mathbf{\delta}}\) & \(\hat{\mathbf{\beta}}\) & \(\hat{\mathbf{\phi}}\) \\ \hline \multirow{2}{*}{Example 1} & OLS & 2.5 & 83 & 5.8 \\ \cline{2-5}  & MLE & 4.0 & 60 & 7.0 \\ \hline \multirow{2}{*}{Example 2} & OLS & 1.3 & 63 & 5.6 \\ \cline{2-5}  & MLE & 2.2 & 51 & 7.6 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Fitted ordinary least squares (OLS) and maximum likelihood estimation (MLE) parameter estimates for Example 1 (\(\alpha=4\), \(\beta=60\), \(\sigma=7\)) and Example 2 (\(\alpha=2.2\), \(\beta=51\), \(\sigma=7.6\)).  
Figure 3: Fitting the path loss model to the training data with OLS (green) and MLE (red)—Example 1 (\(\alpha=4\), \(\beta=60\), \(\sigma=7\)) and Example 2 (\(\alpha=2.2\), \(\beta=51\)). Training data with \(PL\) under the noise threshold are shown with black dots, solid lines show \(\overline{PL}(d)\), and the dash lines are \(\overline{PL}(d)\pm 1.96\sigma\).  
Figure 3. The OLS and MLE fitted models give the same expected PL at a link distance of about 37 m. Therefore, for most reasonable link distances the OLS model underestimates the expected PL, i.e., overestimates the link distance for a given PL. With MLE, the link distance corresponding to \(\overline{PL}=PL^{*}\) is 100 m and with the OLS fitted model, it is about 170 m. The OLS fitted model predicts a significant probability of censored PL between 63 and 490 m (\(\overline{PL}(d)\pm 1.96\sigma\)).