# Analysis of 'Pre-Fit' Datasets of gLAB by Robust Statistical Techniques
## 4 Results, Discussion and Proposal for Further Work
### Discussion
Note that the progression of \(r_{min}(m,n)\) (\(m=45,\ldots,90\) in the example) re-enters in the middle of the confidence bands at the end of the FS: this is the so-called'masking effect' occurring in presence of outliers if all the data fit, such as the case of OLS. In this case, outliers mask each other and one cannot detect them. In a nutshell: OLS provides only \(r_{min}(n,n)\), which is excellent if all the \(n\) data points are good; LTS/LMS only provide the very robust fit for \(r_{min}(\lceil n/2\rceil,n)\), which is excellent if there are many outliers (up to \(\lceil n/2\rceil\)); FS mediates between the two extremes by providing a sequence of fits with \(r_{min}(m,n)\), \(m=m_{0},\ldots,n\) and by identifying the optimal subset with \(m^{*}\) good units. Figure 9: Zoom of Figure 8 for the time period comprised between 18 and 23 h. The top plot depicts the 3D position errors, whereas the bottom plot the GDOP. Figure 11 gives three views of the intercepts estimated as the moving window of 10 epochs progresses with time. The top panels represent the distribution of the deviations of the estimated values from the desired one, which should be zero. The histogram suggests that the fluctuations observed in the time representation of the bottom panel are rather symmetrical. On the other hand, the view provided by the normal probability plot on the top right, which compares the empirical distribution of the intercept estimates to the normal, indicates the extent to which the left and right tails deviating from the normal differ. This suggests the presence of a small systematic bias in the way positions are received. This possibility is corroborated by the experiments performed with the robust estimators in Section 3 above, which raise some questions. While considering the plots of the mean and median in Figure 4, we already remarked that the mean looks 'biased' by the negative outliers. These problematic data points correspond to pre-fit residuals of low  
Figure 10: The FS outlier detection mechanism. The top panel enlarges Figure 4 (third row, first column) in an interesting zone between hours 19 and 22, where three somewhat deviating satellites were detected as outliers. The zone refers to the window of 10 epochs starting at iteration 2470 of 2879, that is at time \(20\lambda 34m\). The bottom-left panel shows the \(n=90\) observations (9 satellites for 10 epochs) monitored in that zone; here, the x axis tick marks are unique observation identifiers (index-numbers) not necessarily ordered in time: they refer to the random order the satellites enter in the sliding window. The red crosses were detected as outliers by the FS, whose progress is shown on the right panel; the x axis ticks of the right panel refer in fact to the FS steps. elevation satellites, when the receiver is rising or setting above the local horizon. In such circumstances, the SPP cannot accurately model propagation delays (e.g., at troposphere and at ionosphere). Moreover, the noise of the measurements increases due to multipath (i.e., reflections of the GNSS signal on the ground or metallic structures). A more precise modeling of the measurements to the one used in SPP (see for instance, [21]) would, in principle, reduce the dispersion of the pre-fit residuals. This would probably ease the detection of outliers, although new borderline cases (i.e., on the limit to be considered as outliers) would appear.