# Analysis of 'Pre-Fit' Datasets of gLAB by Robust Statistical Techniques
## 4 Results, Discussion and Proposal for Further Work
### Discussion  
Figure 8 is a detail of Figure 7 for small position errors; Figure 9 is a zoom of Figure 8 for the time period between 18 and 23 h. As can be seen in the upper part of the plot, most of the time, robust outlier detection methods improve the 3D position errors of the original SPP obtained from pre-fit residuals with outliers. However, there are four epochs between 21\(h\)/00 and 21\(h\)30, in which the solution, especially when the outliers found with FS are removed, is deteriorated compared to  
Figure 8: Zoom of Figure 7 for positioning errors smaller than 10 m. 3D position errors of HUEG using the SPP. In red, the original PVT obtained without detecting outliers. The PVT obtained after removing the outliers detected by LMS (blue), LTS (green) and FS (black). Figure 7: 3D position errors of HUEG using the SPP. In red, the original PVT obtained without detecting outliers. The PVT obtained after removing the outliers detected by LMS (blue), LTS (green) and FS (black). the original one. In order to explain such a result, the bottom plot depicts the geometric dilution of precision (GDOP), which is a measure of how the measurement noise impacts (i.e., dilutes) the positioning accuracy (see, for instance [4, 7]). The GDOP metric evaluates the geometric distribution of the satellites over the sky, only being computed with the elevation and azimuth of the satellites whose measurements determine the position in the FILTER module. The GDOP depends not only on the number of satellites used to compute the coordinates, but also on its distribution in the sky. In this regard, discarding satellites typically increases the GDOP and the positioning error. At \(21h15\), it can be seen that the GDOP values for the FS reach four times the GDOP values of the original solution, which uses nine satellites in view with an elevation greater than 5 degrees. The reason is that the FS discards two low-elevation satellites that greatly contribute to the positioning estimation. Fortunately, these high GDOP values are an indication to the user that the coordinates are being estimated with a weak geometry and consequently should be used with caution. There are a number of measures to mitigate the deterioration in the positioning error. First, one should use a more accurate modeling of the measurements than the one used in SPP, as in [21]. This would, in principle, reduce the dispersion of the pre-fit residuals, easing outlier detection. Second, one should consider the pre-fit residuals obtained from distributions other than the normal one for detecting outliers. Third, the information of the satellite geometry (elevation and azimuth) should be included in the robust methods, since the present exercise has done a blind examination taking into consideration only pre-fit residuals, but not, for instance, their possible correlations. Fourth and finally, the robustness of the Kalman filter itself should be increased, coordinating the effort of the outlier detection in the pre-fit residuals domain and in the final filtering module domain. Figure 10 shows in detail the behavior of FS during one of the computations described in Section 3 on a time window of 5 min, starting at \(20h34\). The horizontal axis represents the size \(m\) of the subset used at a given step of the FS process to fit the regression model. The FS process is such that outliers enter the subset 'towards the end'. The vertical purple lines show the moment when the monitored statistic (minimum deletion residual--\(r_{min}(m,n)\)--among the observations not yet included in the subset) exceeds the confidence bands at different levels of significance. The FS algorithm checks whether the exceedance from the bands is not spurious (that is, it remains _persistently_ out of the confidence limits in the sense formalized by [22]), identifies its precise position, in addition to the units forming a homogeneous subset, and ultimately those to be declared outliers.