Studia Geophysica et Geodaetica

, Volume 61, Issue 1, pp 35–52 | Cite as

Bayesian inference for the Errors-In-Variables model

  • Xing Fang
  • Bofeng Li
  • Hamza Alkhatib
  • Wenxian ZengEmail author
  • Yibin Yao


We discuss the Bayesian inference based on the Errors-In-Variables (EIV) model. The proposed estimators are developed not only for the unknown parameters but also for the variance factor with or without prior information. The proposed Total Least-Squares (TLS) estimators of the unknown parameter are deemed as the quasi Least-Squares (LS) and quasi maximum a posterior (MAP) solution. In addition, the variance factor of the EIV model is proven to be always smaller than the variance factor of the traditional linear model. A numerical example demonstrates the performance of the proposed solutions.


Errors-In-Variables Total Least-Squares Bayesian inference quasi solution Maximum Likelihood noninformative prior informative prior 


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  1. Akyilmaz O., 2007. Total least squares solution of coordinate transformation. Surv. Rev., 39, 68–80.CrossRefGoogle Scholar
  2. Amiri-Simkooei A., 2013. Application of least squares variance component estimation to errors-invariables models. J. Geodesy, 87, 935–944.CrossRefGoogle Scholar
  3. Amiri-Simkooei A. and Jazaeri S., 2012. Weighted total least squares formulated by standard least squares theory. J. Geod. Sci., 2, 113–124.Google Scholar
  4. Amiri-Simkooei A. and Jazaeri S., 2013. Data-snooping procedure applied to errors-in-variables models. Stud. Geophys. Geod., 57, 426–441.CrossRefGoogle Scholar
  5. Bauwens L., Lubrano M. and Richard J.-F., 1999. Bayesian Inference in Dynamic Econometric Models. Oxford University Press, Oxford, U.K.Google Scholar
  6. Bolfarine H. and Rodrigues J., 2007. Bayesian inference for an extended simple regression measurement error model using skewed priors. Bayesian Anal., 2, 349–364.CrossRefGoogle Scholar
  7. Dellaportas P. and Stephens D.A., 1995. Bayesian analysis of errors-in-variables regression models. Biometrics, 51, 1085–1095.CrossRefGoogle Scholar
  8. Fang X., 2011. Weighted Total Least Squares Solutions for Applications in Geodesy. PhD Thesis. Leibniz University, Hannover, Germany.Google Scholar
  9. Fang X., 2013. Weighted Total Least Squares: necessary and sufficient conditions, fixed and random parameters. J. Geodesy, 87, 733–749.CrossRefGoogle Scholar
  10. Fang X., 2014a. A structured and constrained Total Least-Squares solution with cross-covariances. Stud. Geophys. Geod., 58, 1–16.CrossRefGoogle Scholar
  11. Fang X., 2014b. On non-combinatorial weighted Total Least Squares with inequality constraints. J. Geodesy, 88, 805–816.CrossRefGoogle Scholar
  12. Fang X., 2014c. A total least squares solution for geodetic datum transformations. Acta Geod. Geophys., 49, 189–207.CrossRefGoogle Scholar
  13. Fang X., 2015. Weighted total least-squares with constraints: a universal formula for geodetic symmetrical transformations. J. Geodesy, 89, 459–469.CrossRefGoogle Scholar
  14. Felus F., 2004. Application of Total Least Squares for Spatial Point Process Analysis. J. Surv. Eng., 130, 126–133.CrossRefGoogle Scholar
  15. Felus F. and Burtch R., 2009. On symmetrical three-dimensional datum conversion. GPS Solut., 13, 65–74.CrossRefGoogle Scholar
  16. Florens J.-P., Mouchart M. and Richard J.-F., 1974. Bayesian inference in error-in-variables models. J. Multivar. Anal., 4, 419–452.CrossRefGoogle Scholar
  17. Golub G. and Van Loan C., 1980. An analysis of the Total least-squares problem. SIAM J. Numer. Anal., 17, 883–893.CrossRefGoogle Scholar
  18. Grafarend E. and Awange J.L., 2012. Applications of Linear and Nonlinear Models. Fixed Effects, Random Effects, and Total Least Squares. Springer-Verlag, Berlin, Germany.Google Scholar
  19. Grafarend E.W. and Schaffrin B., 1993. Ausgleichungsrechnung in linearen Modellen. BIWissenschaftsverlag, Mannheim, Germany (in German).Google Scholar
  20. Huang H.-J., 2010. Bayesian Analysis of Errors-in-Variables Growth Curve Models. PhD Thesis. University of California, Riverside, CA.Google Scholar
  21. Koch K., 1986. Maximum likelihood estimate of variance components. Bull. Geod., 60, 329–338.CrossRefGoogle Scholar
  22. Koch K., 1999. Parameter Estimation and Hypothesis Testing in Linear Models. 2nd Edition. Springer-Verlag, Berlin, Germany.CrossRefGoogle Scholar
  23. Koch K., 2007. Introduction to Bayesian Statistics. 2nd Edition. Springer-Verlag, Berlin, Germany.Google Scholar
  24. Markovsky I. and Van Huffel S., 2007. Overview of total least-squares methods. Signal Process., 87, 2283–2302.CrossRefGoogle Scholar
  25. Kwon J.H., Lee J.K., Schaffrin B., Yun S.C. and Lee I., 2009. New affine transformation parameters for the horizontal network of Seoul/Korea by multivariat TLS-adjustment. Surv. Rev., 41, 279–291CrossRefGoogle Scholar
  26. Li B., Shen Y., Zhang X., Li C. and Lou L., 2013. Seamless multivariate affine error-in-variables transformation and its application to map rectification. Int. J. Geogr. Inf. Sci., 27, 1572–1592.CrossRefGoogle Scholar
  27. Lindley D.V. and El Sayyad G.M., 1968. The Bayesian estimation of a linear functional relationship. J. R. Stat. Soc. Ser. B-Stat. Methodol., 30, 190–202.Google Scholar
  28. Neitzel F., 2010. Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation. J. Geodesy, 84, 751–762.CrossRefGoogle Scholar
  29. Neri F., Saitta G. and Chiofalo S., 1989. An accurate and straightforward approach to line regression analysis of error-affected experimental data. J. Phys. E-Sci. Instr., 22, 215–217.CrossRefGoogle Scholar
  30. Pope A., 1972. Some pitfalls to be avoided in the iterative adjustment of nonlinear problems. In: Proceedings of the 38th Annual Meeting, American Society of Photogrammetry, Washington, D.C., Mar. 12-17, 1972. American Society of Photogrammetry, Bethesda, MD, 449–477.Google Scholar
  31. Polasek W., 1995. Bayesian generalized errors in variables (GEIV) models for censored regrssions. In: Mammitzsch V. and Schneeweiss H. (Eds), Symposia Gaussiana. Conference B: Statistical Sciences. De Gruyter, Berlin, Germany, 261–279.Google Scholar
  32. Reilly P.M. and Patino-Lea H., 1981. A Bayesian study of the error-in-variables model. Technometrics, 23, 221–231.CrossRefGoogle Scholar
  33. Schaffrin B., 1997. Reliability measures for correlated observations. J. Surv. Eng.-ASCE, 123, 126–137.CrossRefGoogle Scholar
  34. Schaffrin B., 2006. A note on Constrained Total Least-Squares estimation. Linear Alg. Appl., 417, 245–258.CrossRefGoogle Scholar
  35. Schaffrin B. and Felus Y., 2008. On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms. J. Geodesy, 82, 373–383.CrossRefGoogle Scholar
  36. Schaffrin B. and Wieser A., 2008. On weighted total least-squares adjustment for linear regression. J. Geodesy, 82, 415–421.CrossRefGoogle Scholar
  37. Shan J., 1989. A fast recursive method for repeated computation of the reliability matrix QvvP. Photogrammetria, 43, 337–346.CrossRefGoogle Scholar
  38. Shen Y., Li B.F. and Chen Y., 2010. An iterative solution of weighted total least-squares adjustment. J. Geodesy, 85, 229–238.CrossRefGoogle Scholar
  39. Snow K., 2012. Topics in Total Least-Squares Adjustment within the Errors-In-Variables Model: Singular Cofactor Matrices and Priori Information. PhD Thesis. Report 502. Geodetic Science Program, School of Earth Sciences, The Ohio State University, Columbus, OH.Google Scholar
  40. Stigler S.M., 1986. The History of Statistics: The Measurement of Uncertainty before 1900. Belknap Press of Harvard University Press, Cambridge, MA, ISBN 0-674-40340-1.Google Scholar
  41. Teunissen P.J.G., 1988. The nonlinear 2D symmetric Helmert transformation: an exact nonlinear least squares solution, Bull. Geod., 62, 1–15.CrossRefGoogle Scholar
  42. Xu P., Liu Y., Shen Y. and Fukuda Y., 2007. Estimability analysis of variance and covariance components. J. Geodesy, 81, 593–602.CrossRefGoogle Scholar
  43. Xu P., Liu J. and Shi C., 2012. Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis. J. Geodesy, 86, 661–675.CrossRefGoogle Scholar
  44. Xu P., 2016. The effect of errors-in-variables on variance component estimation. J. Geodesy, DOI: 10.1007/s00190-016-0902-0Google Scholar
  45. Zellner A., 1971. An Introduction to Bayesian Inference in Econometrics. John Wiley & Sons.Google Scholar

Copyright information

© Institute of Geophysics of the ASCR, v.v.i 2017

Authors and Affiliations

  • Xing Fang
    • 1
    • 2
  • Bofeng Li
    • 3
  • Hamza Alkhatib
    • 4
  • Wenxian Zeng
    • 1
    Email author
  • Yibin Yao
    • 1
  1. 1.School of Geodesy and GeomaticsWuhan UniversityWuhanChina
  2. 2.School of Earth ScienceOhio State UniversityColumbusUSA
  3. 3.College of Surveying and Geo-InformaticsTongji UniversityShanghaiChina
  4. 4.Geodetic InstituteLeibniz UniversityHannoverGermany

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