# Robust Estimators in Geodetic Networks Based on a New Metaheuristic: Independent Vortices Search
## 6 Conclusions and Future Works  
Geodetic networks are the basis for mapping, geoinformation, land cadastre, and other location-based services. They play an important role in society in infrastructural works that depend on accurate control points. This work tested the strategy of using Robust Estimators (REs) in geodetic network adjustment and for detection of outliers. Several REs were tested. A metaheuristic optimization was conducted for the LTS, LTS-RC, LMS, SRLS, and LTA estimators, whereas, for the IRLS, an iterative process was handled. The two main contributions of the research were successfully demonstrated: (1) the Independent Vortices Search (IVS) overcame the VS, MVS, HVS, and ABC in all aspects; (2) we performed a deep investigation of several REs, separating the analysis scenarios into no outliers, small outliers, and outliers of great magnitude. This led to the classifications in Table 4, something not explored in the literature before. In addition, other minor contributions were also presented. One of them was the search space proposed for applying IVS using Equation (27), which is valid not only for geodetic networks, but for any problem based on the Gauss-Markov model. Furthermore, a more detailed analysis of the results, obtained with Equations (7) and (18), was given in Section 5.2, showing which equation is more appropriate in each case. The experiments with the geodetic network were all performed by applying the IVS, built on the Modified Vortex Search. IVS works with the vortices independently, which can also facilitate the parallelization of procedures which require high performance. This includes exploring more complex problems or even larger geodetic networks. In the experiments of quality control in the geodetic networks, in situations with small outliers or seeking outlier identification, we do not recommend the LS method. In these cases, according to the experiments, it is better to work with IRLS, or, for a better identification at a cost of more false positives, LMS, LTS, or LTS-RC-N2 can achieve better results. Even though LS detects fewer outliers in these scenarios, it remains as the best estimator for the parameters, as the co-ordinates remain closer to their true values. For estimating the co-ordinates in scenarios with large-scale outliers, REs present a more satisfactory estimate than LS. This showed the sensitivity of LS to outliers of high magnitude and the strength of the robust methods. Although the computational cost of REs that use MHs is greater than the classical techniques, it is not possible to establish a cost-benefit at the moment. For this, it is first necessary to consider a minimum amount of FE to get good results, based on the RE and the network dimension. However, a disadvantage of this technique is that is it not possible to estimate the precision of the points. To achieve that, we need an LS estimation after removing the outliers from the observation vector. Future works can focus on a scalability study and the parallelization of the IVS to achieve better performance. In robust estimation, other strategies can also be studied, such as testing new constraints to REs or checking other REs not contemplated by this work. Data curation, I.K. ; methodology, I.E.K. ; software, I.E.K. ; validation, I.E.K. and I.K. ; formal analysis, I.E.K. and I.K. ; investigation, I.E.K. ; resources, L.G.J. and M.R.V. ; writing--original draft, I.E.K., writing--review and editing, I.K., L.G., M.T.M., V.F.R., and M.R.V. ; supervision, L.G. and M.R.V. This research was funded in part by the Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior--Brasil (CAPES)--Finance Code 001. \begin{table}
\begin{tabular}{l l l} \hline  & **Detection** & **Estimation** \\ \hline No outlier & - & LS \\ Small magnitude outliers & IRLS, LMS *, LTS *, LTS-RC-N2 * & LS \\ Large magnitude outliers & LS, IRLS, LMS, LTS-RC-N2 & IRLS, LMS, LTS, LTS-RC, LTS-RC-N2 \\ \hline \end{tabular}
\end{table} 
Table 4: Estimators with best results, according to the demand and scenario.