# Influence of Noise-Limited Censored Path Loss on Model Fitting and Path Loss-Based Positioning +
## 1 Introduction
### Contributions  
The censored PL data (may) exist in both the training phase and the positioning phase. Therefore, in terms of taking or not taking into account the censored samples, there are four options. In [10], fitting without the censored data is called ordinary least squares (OLS) fitting, as opposed to the Tobit MLE. Similarly, for the positioning there is the ordinary positioning, using only the distance estimates from the contacted BSs, which we now call ordinary trilateration positioning (OTP). Then, there is the option to include the censored data with Tobit MLE positioning. The four options are [13]:  
Figure 1: Likelihood function illustration example in 1D: measured path loss (PL) of 130 dB to BS1, censored PL \(>\) 140 dB to BS2, and the combined likelihood function. Note that one measured and one censored PL are sufficient to get a unique positioning solution and the same is true in 2D and 3D.  
* OLS-OPT, with ordinary least squares fitting to the training data and the ordinary trilateration positioning.
* MLE-OPT, with Tobit MLE fitting to the training data and the ordinary trilateration positioning.
* OLS-MLE, with ordinary least squares fitting to the training data and the Tobit MLE positioning.
* MLE-MLE, with Tobit MLE fitting to the training data and the Tobit MLE positioning.  
Traditionally, distance estimates between two sources in one dimension (e.g., corridor), three sources in two dimensions (e.g., antennas at about the same height), or four sources in the typical three dimensions, are needed to give a unique position estimate. When the censored PLs are included, this requirement is loosened significantly, as only one contacted and one censored source are needed to give the unique position estimate. The one contacted BS provides one distance estimate (two points in 1D, circle in 2D, or sphere in 3D) and the censored BS selects the furthest point as the unique position estimate. An example in 1D is presented in Figure 1.  
To the best of the authors' knowledge, this is the first time (1) the effect of the noise-limited censored path loss data has been studied in both phases of the PL model-based positioning and both phases are explicitly written as Tobit MLE; (2) a selection criterion accounting for BSs with censored PL in the positioning phase is presented; and (3) the reduced amount of information needed for a unique positioning solution is noted. We examine the influence of including the censored data through simple simulations. The simulations use a typical log-distance PL model and illustrate the potential of the proposed approach for considerable improvements in positioning accuracy. Three different realistic PL distributions are considered, including examples in which line-of-sight (LOS) and non-line-of-sight (NLOS) follow different distributions according to [14]. Additionally, the dependence on the noise threshold level and LOS detection probability is examined. Usually, in positioning the antennas are, or are assumed to be, omnidirectional [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23]. In this paper, the positioning simulations compare omnidirectional and directional BS antennas assuming a simple antenna model.