# Influence of Noise-Limited Censored Path Loss on Model Fitting and Path Loss-Based Positioning +
## 6 Censored Path Loss and Positioning  
Path loss model-based positioning is based on getting distance estimates from measured PL to BSs with known locations. For a given distance the PL model describes a probability distribution for PL, or inversely a PL value gives a distribution for the link distance. The width of the distribution is proportional to shadow fading. In ordinary trilateration positioning (OTP), distance estimates to contacted BSs are used to trilaterate the positioning estimate. When fitting the model to training data, the censored PL is a measurement result at a known link distance and PL known to be more than the noise threshold level. In positioning, censored PL is an uncontacted BS due to PL larger than the threshold. Therefore, the minimum link distance has a probability distribution associated with the noise threshold level. Thus, failing to contact a given BS has position information as it means that the true position is unlikely to be close to that particular BS. Figure 1 illustrates examples of likelihood functions associated with measured PL and censored PL. Next, we will frame the positioning problem as a Tobit maximum likelihood estimation (MLE) in a similar manner as in the case of fitting the PL model to training data. Let us first write the log-distance as \(q=10\cdot\log_{10}(d/d_{0})\), then the expected distance is  
\[\overline{q}(PL_{i},\theta_{i})=(PL_{i}-\beta+G(\theta_{i}))/\alpha, \tag{19}\]  
\[q^{*}(\theta_{i})=(PL^{*}-\beta+G(\theta_{i}))/\alpha, \tag{20}\]  
where \(PL_{i}\) is the measured PL to \(i\)th base station (or beam), \(PL^{*}\) is the noise threshold, \(q^{*}\) is the corresponding log-distance, and the standard deviation corresponding to shadow fading is \(\sigma/\alpha\). \(q_{i}\) is the log-distance from point \((x,y)\) to BS\({}_{i}\) (or beam) and \(\theta_{i}\) is the beam offset angle. The log-likelihood function for known measured \(PL\) at point \((x,y)\) is  
\[LLF=\sum_{i=1}^{N_{BS}}\biggl{(}-\ln\frac{\sigma}{\alpha}+\ln\phi\biggl{(} \frac{q_{i}-\overline{q}(PL_{i},\theta_{i})}{\sigma/\alpha}\biggr{)}\biggr{)}, \tag{21}\]  
where \(N_{BS}\) is the number of BSs (or directive beams) with measured PL under the noise threshold. The log-likelihood function for censored PL at point \((x,y)\) is  
\[LLF^{*}=\sum_{i=1}^{N_{BS}}\ln\biggl{(}1-\Phi\biggl{(}\frac{q^{*}(\theta_{i}) -q_{i}}{\sigma/\alpha}\biggr{)}\biggr{)}, \tag{22}\]  
where \(N_{BS}^{*}\) is the number of BSs (or directive beams) with censored PL. The position estimate \((\hat{x},\hat{y})\) is derived as the minimum of the negative of the log-likelihood function. For the OTP positioning, the censored samples are not used; i.e.,  
\[[\hat{x},\hat{y}]=\operatorname*{argmin}_{x,y}\{-LLF\}. \tag{23}\]  
The MLE positioning uses both (21) and (22) as  
\[[\hat{x},\hat{y}]=\operatorname*{argmin}_{x,y}\{-(LLF+LLF^{*})\}. \tag{24}\]Thus, the only difference is whether or not the location information from the noise-limited censored PL data is used. In Section 7, the influence of noise-limited PL in model fitting is then examined by comparing positioning error statistics using the OLS fitted model and MLE fitted model. Far away, uncontacted BSs have no effect on the positioning result, and therefore, it is useful to limit the BSs that are considered. In principle, the considered BSs should be chosen based on the properties of the likelihood functions, i.e., based on the PL model. In practice, of course, the database of known BS locations is also limiting which BSs can be considered. In this work, we propose to limit the considered censored BSs 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 limit was chosen due to a small performance difference observed between selection criteria \(\overline{PL}(d)-1\cdot\sigma=PL^{*}\) and \(\overline{PL}(d)-2\cdot\sigma=PL^{*}\). Importantly, the OLS and MLE fitted models give different radii for the area.