Sensitivity of robust estimators applied in strategy for testing stability of reference points. EIF approach
In deformation analyses, it is important to find a stable reference frame and therefore the stability of the possible reference points must be controlled. There are several methods to test such stability. The paper's objective is to examine one of such methods, namely the method based on application of R-estimation, for its sensitivity to gross errors. The method in question applies three robust estimators, however, it is not robust itself. The robustness of the method depends on the number of unstable points (the fewer unstable points there are, the more robust is the proposed method). Such property makes it important to know how the estimates applied and the strategy itself respond to a gross error. The empirical influence functions (EIF) can provide necessary information and help to understand the response of the strategy for a gross error. The paper presents examples of EIFs of the estimates, their application in the strategy and describes how important and useful is such knowledge in practice.
Baarda W., (1968): A testing procedure for use in geodetic networks, Publication on Geodesy. New series 2, No 5, Netherlands Geodetic Commission, Delft.
Denli H.H., (2008): Stable Point Research on Deformation Networks, Survey Review, Vol. 40, pp. 74-82.
Ding X., Coleman R., (1996): Multiple outlier detection by evaluating redundancy contributions of observations, Journal of Geodesy, Vol. 70, pp. 489-498.
Duchnowski R., (2008): R-estimation and its application to the LS adjustment, Bollettino di Geodesia e Scienze Affini, Vol. 67, No 1, pp. 17-32.
Duchnowski R., (2009): Geodetic Application of R-estimation - Levelling Network Examples, Technical Sciences, Vol. 12, pp. 135-144.
Duchnowski R., (2010): Median-based estimates and their application in controlling reference mark stability, Journal of Surveying Engineering, Vol. 136, No 2, pp. 47-52.
Duchnowski R., (2011): Robustness of strategy for testing levelling mark stability based on rank tests, Survey Review, Vol. 43, pp. 687-699.
Gui Q., Gong Y., Li G., Li B., (2007): A Bayesian approach to the detection of gross errors based on posterior probability, Journal of Geodesy, Vol. 81, pp. 651-659.
Gui Q., Gong Y., Li G., Li B., (2011): A Bayesian unmasking method for locating multiple gross errors based on posterior probabilities of classification variables, Journal of Geodesy, Vol. 85, pp. 191-203.
Hampel F.R., Ronchetti E.M., Rousseeuw P.J., Stahel W.A., (1986): Robust Statistics. The Approach Based on Influence Functions, Wiley, New York.
Hekimoglu S., Erenoglu R.C., (2007): Effect of heteroscedasticity and heterogeneousness on outlier detection for geodetic networks, Journal of Geodesy, Vol. 81, pp. 137-148.
Hekimoglu S., Erdogan B., Butterworth S., (2010): Increasing the efficacy of the conventional deformation analysis methods: alternative strategy, Journal of Surveying Engineering, Vol. 136, No 2, pp. 53-62.
Huber P.J., (1981): Robust Statistics, Wiley, New York.
Prószyński W., Kwaśniak M., (2006): Basis of geodetic calculations of displacements. Notions and methodology elements (in Polish), Oficyna Wydawnicza Politechniki Warszawskiej, Warszawa.
Prószynski W., (2010): Another approach to reliability measures for systems with correlated observations, Journal of Geodesy, Vol. 84, pp. 547-556.
Rousseeuw P.J., Croux C., (1993): Alternative to the Median Absolute Deviation, J. Am. Stat. Assoc., 88, pp. 1273-1283.
Rousseeuw P.J., Verboven S., (2002): Robust estimation in very small samples, Computation Statistics Data Analysis, Vol. 40, No 4, pp. 741-758.
Shaorong Z., (1990): On separability for deformations and gross errors, Journal of Geodesy, Vol. 64, pp. 383-396.
Wiśniewski Z., (2009): Estimation of parameters in a split functional model of geodetic observations (Msplit estimation), Journal of Geodesy, Vol. 83, pp. 105-120.
Xu P., (2005): Sign-constrained robust least squares, subjective breakdown point and effect of weights of observations on robustness, Journal of Geodesy, Vol. 79, pp. 146-159.