pp 1-9 | Cite as

A Bayesian Nonlinear Regression Model Based on t-Distributed Errors

  • Alexander DorndorfEmail author
  • Boris Kargoll
  • Jens-André Paffenholz
  • Hamza Alkhatib
Part of the International Association of Geodesy Symposia book series


In this contribution, a robust Bayesian approach to adjusting a nonlinear regression model with t-distributed errors is presented. In this approach the calculation of the posterior model parameters is feasible without linearisation of the functional model. Furthermore, the integration of prior model parameters in the form of any family of prior distributions is demonstrated. Since the posterior density is then generally non-conjugated, Monte Carlo methods are used to solve for the posterior numerically. The desired parameters are approximated by means of Markov chain Monte Carlo using Gibbs samplers and Metropolis-Hastings algorithms. The result of the presented approach is analysed by means of a closed-loop simulation and a real world application involving GNSS observations with synthetic outliers.


Bayesian nonlinear regression model Gibbs sampler Markov Chain Monte Carlo Metropolis-Hastings algorithm Scaled t-distribution 


  1. Alkhatib H, Schuh W-D (2006) Integration of the Monte Carlo covariance estimation strategy into tailored solution procedures for large-scale least squares problems. J Geod 81:53–66Google Scholar
  2. Alkhatib H, Kargoll B, Paffenholz J-A (2017) 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. Contributions to statistics. Springer, Cham, pp 25–38Google Scholar
  3. Bossler JD (1972) Bayesian inference in geodesy. Dissertation, The Ohio State University, Columbus, Ohio, USAGoogle Scholar
  4. Gamerman D, Lopes HF (2006) Markov chain Monte Carlo – stochastic simulation for bayesian inference, 2nd edn. Chapman and Hall/CRC, Boca RatonGoogle Scholar
  5. Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB (2014) Bayesian data analysis, 3rd edn. CRC Press, Taylor & Francis Group, Boca RatonGoogle Scholar
  6. Geweke J (1993) Bayesian treatment of the independent student-t linear model. J Appl Economet 8:19–40Google Scholar
  7. Gundlich B, Koch KR, Kusche J (2003) Gibbs sampler for computing and propagating large covariance matrices. J Geod 77:514–528Google Scholar
  8. Haario H, Laine M, Mira A, Saksman E (2006) DRAM: efficient adaptive MCMC. Stat Comput 16:339–354Google Scholar
  9. Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57:97–109Google Scholar
  10. Johnathan MB, Solonen A, Haario H, Laine M (2014) Randomize-then-optimize: a method for sampling from posterior distributions in nonlinear inverse problems. SIAM J Sci Comput 36:A1895–A1910Google Scholar
  11. Koch KR (1988) Bayesian statistics for variance components with informative and noninformative priors. Manuscr Geodaet 13:370–373Google Scholar
  12. Koch KR (1999) Parameter estimation and hypothesis testing in linear models, 2nd edn. Springer, BerlinGoogle Scholar
  13. Koch KR (2007) Introduction to Bayesian statistics, 2nd edn. Springer, BerlinGoogle Scholar
  14. Koch KR (2017) Monte Carlo methods. Int J Geomath 9:117–143Google Scholar
  15. Koch KR (2018) Bayesian statistics and Monte Carlo methods. J Geod Sci 8:18–29Google Scholar
  16. Koch KR, Kargoll B (2013) Expectation-maximization algorithm for the variance-inflation model by applying the t-distribution. J Appl Geod 7:217–225Google Scholar
  17. Kroese DP, Taimre T, Botev ZI (2011) Handbook of Monte Carlo methods. Wiley, HobokenGoogle Scholar
  18. Paffenholz J-A (2012) Direct geo-referencing of 3D point clouds with 3D positioning sensors. Deutsche Geodätische Kommission, Series C (Dissertation), No. 689, MunichGoogle Scholar
  19. Parzen E (1979) A density-quantile function perspective on robust estimation. In: Launer L, Wilkinson GN (eds) Robustness in statistics. Academic Press, New York, pp 237–258Google Scholar
  20. Riesmeier K (1984) Test von Ungleichungshypothesen in linearen Modellen mit Bayes-Verfahren. Deutsche Geodätische Kommission, Series C (Dissertation), No. 292, MunichGoogle Scholar
  21. Saltelli A, Ratto M, Andres T, Campolongo F, Cariboni J, Gatelli D, Saisana M, Tarantola S (2008) Global sensitivity analysis: the primer. Wiley, ChichesterGoogle Scholar
  22. Schaffrin B (1987) Approximating the Bayesian estimate of the standard deviation in a linear model. Bull Geod 61(3):276–280Google Scholar
  23. Yang Y (1991) Robust Bayesian estimation. Bull Geod 65(3):145–150Google Scholar
  24. Zhu J, Santerre R, Chang X-W (2005) A Bayesian method for linear, inequality-constrained adjustment and its application to GPS positioning. J Geod 78:528–534Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Alexander Dorndorf
    • 1
    Email author
  • Boris Kargoll
    • 1
  • Jens-André Paffenholz
    • 1
  • Hamza Alkhatib
    • 1
  1. 1.Geodetic InstituteLeibniz University HannoverHannoverGermany

Personalised recommendations