# Influence of Noise-Limited Censored Path Loss on Model Fitting and Path Loss-Based Positioning +
## 4 Censored Path Loss and Model Fitting  
The first phase of the two-phase PL model-based positioning is the measurement of training data and fitting the model to the data. Training data consist of \(N_{s}\) samples of PL and the corresponding linked distances. In OLS fitting, only these measurable PL samples are used. In the case of noise-limited censored PL, the link distance is known and PL is more than the threshold limit \(PL^{*}\). Using Tobit MLE, the censored PL samples can be taken into account [10; 11; 13].  
The likelihood of measuring \(PL\) is [10]  
\[l(PL)=(1/\sigma)\phi((PL-\overline{PL})/\sigma), \tag{6}\]  
where \(\sigma\) is the standard deviation (std) of the shadow fading, and \(\overline{PL}\) is the expected path loss model. Here, \(\phi(\cdot)\) is the standard normal probability density function (PDF). The log-likelihood function for known \(PL\) samples at distances \(d_{i}\) is  
\[LLF=\sum_{i=1}^{N_{s}}\biggl{(}-\ln\sigma+\ln\phi\biggl{(}\frac{PL(d_{i})- \overline{PL}(d_{i})}{\sigma}\biggr{)}\biggr{)}, \tag{7}\]  
where \(N_{s}\) is the number of uncensored data samples. The likelihood of measuring \(PL>PL^{*}\) is [10]  
\[l(PL>PL^{*})=1-\Phi((PL^{*}-\overline{PL})/\sigma), \tag{8}\]  
where \(\Phi(\cdot)\) is the cumulative distribution function (CDF) of the standard normal distribution. The log-likelihood function for censored samples at distances \(d_{i}\) is  
\[LLF^{*}=\sum_{i=1}^{N_{s}^{*}}\ln\biggl{(}1-\Phi\biggl{(}\frac{PL^{*}- \overline{PL}(d_{i})}{\sigma}\biggr{)}\biggr{)}, \tag{9}\]  
where \({}^{*}\) refers to censored data; i.e., \(N_{s}^{*}\) is the number of censored data points.  
The path loss parameters are then estimated as the minimum of the negative of the log-likelihood function. For the OLS fitting, the censored samples are not used; i.e.,  
\[[\hat{\alpha},\hat{\beta},\hat{\sigma}]=\operatorname*{argmin}_{\alpha,\hat{ \beta},\sigma}\{-LLF\}, \tag{10}\]  
where parameter estimates are marked with \(\hat{\gamma}\); i.e., \(\hat{\alpha}\), \(\hat{\beta}\), and \(\hat{\sigma}\). The MLE fitting uses both (7) and (9) as  
\[[\hat{\alpha},\hat{\beta},\hat{\sigma}]=\operatorname*{argmin}_{\alpha,\hat{ \beta},\sigma}\{-(LLF+LLF^{*})\}. \tag{11}\]  
Therefore, the only difference is whether the censored data are used or not. 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.  
It should be noted that, e.g., in the case of a very small ratio of \(\sigma/\alpha\) or if only very short link distances exist in the training data, then the OLS and MLE fitted models can be practically identical. In practice, in many typical radio channel conditions, there is a wide range of link distances where there is a relatively large probability of censored PL.  
Path loss-based positioning can be done also without the training phase. In that case, the PL model parameters can be taken, e.g., from a standard channel model. The OLS fitting result can be seen as serving double duty, both as the OLS fitting result and as a (rather poor) example of an educated guess in the absence of training data.