pp 1-9 | Cite as

Adjustment of Gauss-Helmert Models with Autoregressive and Student Errors

  • Boris KargollEmail author
  • Mohammad Omidalizarandi
  • Hamza Alkhatib
Part of the International Association of Geodesy Symposia book series


In this contribution, we extend the Gauss-Helmert model (GHM) with t-distributed errors (previously established by K.R. Koch) by including autoregressive (AR) random deviations. This model allows us to take into account unknown forms of colored noise as well as heavy-tailed white noise components within observed time series. We show that this GHM can be adjusted in principle through constrained maximum likelihood (ML) estimation, and also conveniently via an expectation maximization (EM) algorithm. The resulting estimator is self-tuning in the sense that the tuning constant, which occurs here as the degree of freedom of the underlying scaled t-distribution and which controls the thickness of the tails of that distribution’s probability distribution function, is adapted optimally to the actual data characteristics. We use this model and algorithm to adjust 2D measurements of a circle within a closed-loop Monte Carlo simulation and subsequently within an application involving GNSS measurements.


Autoregressive process Circle fitting Constrained maximum likelihood estimation Expectation maximization algorithm Gauss-Helmert model Scaled t-distribution Self-tuning robust estimator 



Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 386369985.


  1. Alkhatib H, Kargoll B, Paffenholz JA (2018) Further results on robust multivariate time series analysis in nonlinear models with autoregressive and t-distributed errors. In: Rojas I, Pomares H, Valenzuela O (eds) Time series analysis and forecasting. ITISE 2017, pp 25–38. Contributions to Statistics. Springer, Cham. Scholar
  2. Kargoll B, Omidalizarandi M, Loth I, Paffenholz JA, Alkhatib H (2018) An iteratively reweighted least-squares approach to adaptive robust adjustment of parameters in linear regression models with autoregressive and t-distributed deviations. J Geod 92(3):271–297. Scholar
  3. Koch KR (2012) Robust estimation by expectation maximization algorithm. J Geod 87:(2)107–116.
  4. Koch KR (2014) Robust estimations for the nonlinear Gauss Helmert model by the expectation maximization algorithm. J Geod 88(3):263–271. Scholar
  5. Koch KR (2014) Outlier detection for the nonlinear Gauss Helmert model with variance components by the expectation maximization algorithm. J Appl Geod 8(3):185–194. Scholar
  6. Koch KR, Kargoll B (2013) Expectation-maximization algorithm for the variance-inflation model by applying the t-distribution. J Appl Geod 7:217–225. Scholar
  7. Krasbutter I, Kargoll B, Schuh W-D (2015) Magic square of real spectral and time series analysis with an application to moving average processes. In: Kutterer H, Seitz F, Alkhatib H, Schmidt M (eds) The 1st international workshop on the quality of geodetic observation and monitoring systems (QuGOMS’11). International Association of Geodesy Symposia, vol 140. Springer International Publishing, Berlin, pp 9–14. Scholar
  8. Kuhlmann H (2003) Kalman-filtering with coloured measurement noise for deformation analysis. In: Proceedings of the 11th FIG International Symposium on Deformation Measurements, FIGGoogle Scholar
  9. Lange KL, Little RJA, Taylor JMG (1989) Robust statistical modeling using the t-distribution. J. Am. Stat. Assoc. 84:881–896. Scholar
  10. Lehmann R (2013) 3σ-rule for outlier detection from the viewpoint of geodetic adjustment. J. Surv. Eng. 139(4):157–165. Scholar
  11. Loth I, Schuh W-D, Kargoll B (2019) Non-recursive representation of an autoregressive process within the Magic Square, IAG Symposia (First Online), Springer.
  12. McLachlan GJ, Krishnan T (2008) The EM algorithm and extensions, 2nd edn. Wiley, HobokenGoogle Scholar
  13. Paffenholz JA (2012) Direct geo-referencing of 3D point clouds with 3D positioning sensors. Committee for Geodesy (DGK) of the Bavarian Academy of Sciences and Humanities, Series C: Dissertations, No. 689, MunichGoogle Scholar
  14. Parzen E (1979) A density-quantile function perspective on robust estimation. In: Launer L, Wilkinson GN (eds) Robustness in statistics, pp. 237–258. Academic Press, New York. Scholar
  15. Schuh WD (2003) The processing of band-limited measurements; filtering techniques in the least squares context and in the presence of data gaps. Space Sci Rev 108(1):67–78. Scholar
  16. Takai K (2012) Constrained EM algorithm with projection method. Comput Stat 27:701–714. Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Boris Kargoll
    • 1
    Email author
  • Mohammad Omidalizarandi
    • 1
  • Hamza Alkhatib
    • 1
  1. 1.Geodetic InstituteLeibniz University HannoverHannoverGermany

Personalised recommendations