# Analysis of 'Pre-Fit' Datasets of gLAB by Robust Statistical Techniques
## 2 Structure of the Dataset and Preliminary Analysis
** & **Satellite** & **Pre-Fit** & **Measured Range** & **Modeled Range** & **Elevation** \\ \hline
2018 & 203 & 30.00 & GPS & 30 & 1.1285 & 20,774,695.8820 & 20,774,694.7535 & 62.830 \\
2018 & 203 & 30.00 & GPS & 7 & 0.5062 & 20,902,305.0420 & 20,902,304.5358 & 62.107 \\
2018 & 203 & 30.00 & GPS & 5 & 0.6373 & 21,449,496.2580 & 21,449,495.6207 & 48.474 \\
2018 & 203 & 30.00 & GPS & 13 & 0.0678 & 22,580,115.9500 & 22,580,115.8822 & 32.131 \\
2018 & 203 & 30.00 & GPS & 27 & \(-\)0.7367 & 23,106,933.9460 & 23,106,934.6827 & 23.969 \\
2018 & 203 & 30.00 & GPS & 21 & 1.5017 & 24,496,859.0780 & 24,496,857.5763 & 19.849 \\
2018 & 203 & 30.00 & GPS & 9 & \(-\)0.6624 & 23,870,542.7020 & 23,870,543.3644 & 16.482 \\
2018 & 203 & 30.00 & GPS & 8 & \(-\)0.3127 & 24,198,683.3000 & 24,198,683.6127 & 14.795 \\
2018 & 203 & 30.00 & GPS & 28 & \(-\)1.0963 & 24,119,609.3400 & 24,119,610.4363 & 13.659 \\
2018 & 203 & 30.00 & GPS & 16 & 0.3602 & 24,836,613.8360 & 24,836,613.4758 & 9.066 \\
2018 & 203 & 77,520.00 & GPS & 9 & 0.6399 & 20,249,609.7060 & 20,249,609.0661 & 72.297 \\
2018 & 203 & 77,520.00 & GPS & 23 & \(-\)0.2187 & 21,521,386.7380 & 21,521,386.9567 & 50.623 \\
2018 & 203 & 77,520.00 & GPS & 2 & 2.2097 & 22,539,967.2700 & 22,539,965.0603 & 37.848 \\
2018 & 203 & 77,520.00 & GPS & 26 & \(-\)0.6910 & 22,878,625.4080 & 22,878,626.0990 & 29.222 \\
2018 & 203 & 77,520.00 & GPS & 16 & \(-\)0.4619 & 22,622,817.8300 & 22,622,818.2919 & 29.130 \\
2018 & 203 & 77,520.00 corresponding to hours 21\(h\)32 in Figure 1, satellite 3 shows a pre-fit of about 16 m, probably caused by its low elevation. It is remarked that it is general practice not to consider in the solution of the positioning problem the satellites with 'low' elevation. gLAB, while operating in SPP mode, excludes by default satellites with an elevation below five degrees already in the PREPROCESSING module. As a first essay with the dataset, we apply a LMS regression. The least median of squares estimator [16], described in more detail below, is historically among the first improvements of OLS and is implemented in FSDA by the LXS function. For LXS, we used a very conservative estimate, which was a _Bonferroni-corrected_[17] confidence level of \(1-0.01/N\simeq 0.9999997\) where \(N\) is the number of observations--32,000 in the present case. The quantity \(0.01/N\simeq 3\times 10^{-7}\) can be considered as the probability to falsely declare a dataset as contaminated by outliers, when one is testing a large number of genuine datasets. Figure 2 represents the outlier detection against LMS regression on 24 h of samples. The result with respect to other estimators, compared to LMS below in this section, will also be referred to in this Figure. With reference to Figure 2, the median pre-fit value for the whole dataset is \(\approx 0.2183\) m. This rather small value indicates that the receiver is steering its clock to the GPS time, hence the receiver clock offset value \(c\delta t_{rec}\) is close to 0 in Equation (4). The confidence interval found corresponds to about \(\pm 5\) m. The two main groups of outliers were already evident by visual inspection in Figure 1. The \"V-shaped\" subset between hours 4 and 8 and the smaller group at about hour 22 are correctly identified by LMS. It is remarked that three distinct points at about hours 9, 12 and 17, very close to the limits of the confidence interval, are marked as outliers as well. After this first essay on the data by LMS regression, three remarks can be made:  
Figure 2: Dataset from 4 th November 2018 (day 203 of year 2018). A single application to all data of LMS, re-weighted LTS and FS produces the outliers in red. More precisely, the red ‘+’ symbols are detected by all methods, while LMS also detects the red ‘*’ associated to an absolute pre-fit of about 5 m. FS is applied in the standard form, calibrated to simultaneous 1% confidence level while, to ensure same significance, we corrected the LMS and LTS with Bonferroni.