Robust estimations for the nonlinear Gauss Helmert model by the expectation maximization algorithm

• Published in: Journal of geodesy Vol. 88; no. 3; pp. 263 - 271
• Main Author: Koch, Karl-Rudolf
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2014; Springer Nature B.V
• Subjects: Algorithms; Earth and Environmental Science; Earth Sciences; Geodetics; Geophysics/Geodesy; Multivariate analysis; Normal distribution; Original Article; Variance-inflation model; Errors-in-variables model; Laser scanner; distribution; Mean-shift model; Multivariate; Mixture of normal distributions
• ISSN: 0949-7714; 1432-1394
• DOI: 10.1007/s00190-013-0681-9

For deriving the robust estimation by the EM (expectation maximization) algorithm for a model, which is more general than the linear model, the nonlinear Gauss Helmert (GH) model is chosen. It contains the errors-in-variables model as a special case. The nonlinear GH model is difficult to handle because of the linearization and the Gauss Newton iterations. Approximate values for the observations have to be introduced for the linearization. Robust estimates by the EM algorithm based on the variance-inflation model and the mean-shift model have been derived for the linear model in case of homoscedasticity. To derive these two EM algorithms for the GH model, different variances are introduced for the observations and the expectations of the measurements defined by the linear model are replaced by the ones of the GH model. The two robust methods are applied to fit by the GH model a polynomial surface of second degree to the measured three-dimensional coordinates of a laser scanner. This results in detecting more outliers than by the linear model.