Exhaustive search procedure for multiple outlier detection

In the last decades many statistical tests based on the least squares solution have been proposed for multiple outlier detection. All of them suffer, however, from deficiencies that make them inefficient in their practical application. As recently demonstrated by the author, this situation is unavoidable in the framework of least squares theory. The present contribution elaborates on this impossibility of obtaining an unambiguous response for any statistical test based on the least squares solution and makes use of multiple least squares adjustments for statistically characterizing the equivalent sets of multiple gross error vectors. Several examples and a flexible Matlab implementation are provided.Baselga Moreno, S. (2011). Exhaustive search procedure for multiple outlier detection. Acta Geodaetica et Geophysica Hungarica. 46(4):401-416. doi:10.1556/AGeod.46.2011.4.3S401416464Baarda W 1968: A testing procedure for use in geodetic networks. Netherlands Geodetic Commission, Publications on Geodesy, New Series 2, No. 5, DelftBaselga, S. (2007). Global Optimization Solution of Robust Estimation. Journal of Surveying Engineering, 133(3), 123-128. doi:10.1061/(asce)0733-9453(2007)133:3(123)Baselga, S. (2011). Nonexistence of Rigorous Tests for Multiple Outlier Detection in Least-Squares Adjustment. Journal of Surveying Engineering, 137(3), 109-112. doi:10.1061/(asce)su.1943-5428.0000048Cen, M., Li, Z., Ding, X., & Zhuo, J. (2003). Gross error diagnostics before least squares adjustment of observations. Journal of Geodesy, 77(9), 503-513. doi:10.1007/s00190-003-0343-4Cross P A, Price D R 1984: A strategy for the detection of single and multiple gross errors in geodetic networks. In: Eighth UK Geophysical Assembly, Newcastle Upon TyneDing, X., & Coleman, R. (1996). Multiple outlier detection by evaluating redundancy contributions of observations. Journal of Geodesy, 70(8), 489-498. doi:10.1007/bf00863621Erenoglu, R., & Hekimoglu, S. (2010). Efficiency of robust methods and tests for outliers for geodetic adjustment models. Acta Geodaetica et Geophysica Hungarica, 45(4), 426-439. doi:10.1556/ageod.45.2010.4.3Eshagh, M., Sjöberg, L., & Kiamehr, R. (2007). Evaluation of robust techniques in suppressing the impact of outliers in a deformation monitoring network — a case study on the Tehran Milad tower network. Acta Geodaetica et Geophysica Hungarica, 42(4), 449-463. doi:10.1556/ageod.42.2007.4.6Gui, Q., Gong, Y., Li, G., & Li, B. (2007). A Bayesian approach to the detection of gross errors based on posterior probability. Journal of Geodesy, 81(10), 651-659. doi:10.1007/s00190-006-0132-yGuo, J.-F., Ou, J.-K., & Wang, H.-T. (2007). Quasi-Accurate Detection of Outliers for Correlated Observations. Journal of Surveying Engineering, 133(3), 129-133. doi:10.1061/(asce)0733-9453(2007)133:3(129)Hadi, A. S., & Simonoff, J. S. (1993). Procedures for the Identification of Multiple Outliers in Linear Models. Journal of the American Statistical Association, 88(424), 1264. doi:10.2307/2291266Huber, P. J. (1981). Robust Statistics. Wiley Series in Probability and Statistics. doi:10.1002/0471725250Knight, N. L., Wang, J., & Rizos, C. (2010). Generalised measures of reliability for multiple outliers. Journal of Geodesy, 84(10), 625-635. doi:10.1007/s00190-010-0392-