# Robust Estimators in Geodetic Networks Based on a New Metaheuristic: Independent Vortices Search
## 5 Results and Discussion
### Comparing LTS/LTS-RC Trim Limit and LMS Median Position  
We performed tests with the two values for the limits of LTS and LTS-RC and for the median position of LMS. The lower limit, \(h=\left(n+1\right)/2\), will be shown as \(h_{n2}\) and the upper limit, \(h=\left(n+p+1\right)/2\), as \(h_{n2}\).  
Figure 9 shows that the adjustments with the limit of \(h_{n2}\) had worse results in the classification for LMS and LTS. Classifications of type \"A\" were reduced by adopting the expected threshold, as compared to more robust, increased \"B\" classifications. For these estimators, the lower limit maintained outliers identification, but pointed out more false positives. The results for the LTS-RC present a slight improvement to the limit of \(h_{n2}\).  
Comparing the results of LTS and LTS-RC, we notice that the results presented almost the same classifications, whereas the variations with \(h_{n2}\) presented a significant difference. This is because the larger limit (\(h_{n2}\)) removed fewer residuals, hardly breaking the redundancy. Furthermore, since the network had several observations among its three-dimensional points (although not repeated), it was hard for any parameter to remain without at least two residuals (not breaking redundancy). Using the limit of \(h_{n2}\), the classic LTS more easily broke the redundancy in the residuals trim, whereas the LTS-RC achieved more satisfactory results, retaining the redundancy of the network.  
Figure 9: Occurrence of each classification in the 900 scenarios for the three estimators and their variation in the trim/median position. The suffix “-N2” shows the limit of \(h_{n2}\), for the respective estimator. Non-sufficed REs show the limit of \(h_{n2}\).  
Figure 10 confirms a gain of false positives with the limit of \(h_{n2}\) for LMS and LTS, and stability with a slight improvement for LTS-RC. There were no significant differences in outlier identification for the different limits, while, for undetected errors, only the LTS showed a 6.6% worsening.  
We also compared the distance of the co-ordinates from their true values. Figure 11 displays the mean distance difference of the co-ordinates in each scenario package, between the limits \(h_{n2}\) and \(h_{np2}\), for each RE. The mean differences were negligible in most situations, being below 1 mm with either \(h_{n2}\) or \(h_{np2}\). LTS produced inferior results with \(h_{n2}\), whereas the LTS-RC reduced the variation in two cases, with one and four simultaneous outliers.  
Figure 11: Mean distance difference of the co-ordinates from their official values. The limits \(h_{n2}\) and \(h_{np2}\) divided by scenario packages from 0 to 4 simultaneous outliers. Negative values mean smaller deviations with the limit of \(h_{n2}\).  
Figure 10: Percent difference for each estimator divided into identified errors (false and undetected) for the limits of \(h_{n2}\) and \(h_{np2}\). A positive percentage value means higher occurrence with \(h_{n2}\).  
The redundancy constraint has proven to be interesting for any problems where the number of observations per parameter is more limited and/or the trim limit is smaller. To the best of our knowledge, this is the first time the \(h\) value has been analyzed for a geodetic application. As the measurements of these networks do not contain data related to all the parameters, the redundancy constraint also played an important role. Besides this specific treatment, the \(h\) value can be considered in any studies that wish to apply LMS, LTS, or LTS-RC.