The maximum likelihood estimation for multivariate EIV model

  • Qisheng Wang
  • Youjian Hu
  • Bin WangEmail author
Original Study


In this paper, a new method of parameter estimation for multivariate errors-in-variables (MEIV) model was proposed. The formulae of parameter solution for the MEIV model were deduced based on the principle of maximum likelihood estimation, and two iterative algorithms were presented. Since the iterative process is similar to the classical least square, both of the proposed algorithms are easy to program and understand. Finally, real and simulation datasets of affine coordinate transformation were employed to verify the applicability of the proposed algorithms. The results show that both of the proposed algorithms can achieve identical parameter estimators as those obtained by Lagrange algorithm and Newton algorithm. Additionally, the proposed Algorithm 2 can solve the MEIV model with higher convergence efficiency than Algorithm 1.


Total least squares Multivariate errors-in-variables model Parameter estimation Iterative algorithm 



This research was supported by the Natural Science Foundation of Hunan Province (No.2017JJ5035) and the Natural Science Foundation of Jiangsu Province (No. BK20180720).

Compliance with ethical standards

Conflicts of interest

The authors declare no conflict of interest.


  1. Akyilmaz O (2007) Total least squares solution of coordinate transformation. Surv Rev 39:68–80CrossRefGoogle Scholar
  2. Amiri-Simkooei AR (2013) Application of least squares variance component estimation to errorsin-variables models. J Geod 87:935–944CrossRefGoogle Scholar
  3. Amiri-Simkooei A, Jazaeri S (2013) Data-snooping procedure applied to errors-in-variables models. Stud Geophys Geod 57(3):426–441CrossRefGoogle Scholar
  4. Amiri-Simkooel AR, Jazaeri S (2012) Weighted total least squares formulated by standard least squares theory. J Geod Sci 2(2):113–124Google Scholar
  5. Fang X (2011) Weighted total least squares solutions for applications in geodesy. Ph.D. Thesis. Leibniz University, Hannover, GermanyGoogle Scholar
  6. Fang X (2013) Weighted total least squares: necessary and sufficient conditions, fixed and random parameters. J Geod 87:733–749CrossRefGoogle Scholar
  7. Fang X (2014a) A structured and constrained total least-squares solution with cross-covariances. Stud Geophys Geod 58(1):1–16CrossRefGoogle Scholar
  8. Fang X (2014b) A total least squares solution for geodetic datum transformations. Acta Geod Geoph 49:189–207CrossRefGoogle Scholar
  9. Fang X (2014c) On non-combinatorial weighted total least squares with inequality constraints. J Geod 88(8):805–816CrossRefGoogle Scholar
  10. Fang X (2015) Weighted total least-squares with constraints: a universal formula for geodetic symmetrical transformations. J Geod 89(5):459–469CrossRefGoogle Scholar
  11. Fang X, Li B, Alkhatib H, Zeng WX, Yao YB (2017) Bayesian inference for the errors-in-variables model. Stud Geophys Geod 61:35–52CrossRefGoogle Scholar
  12. Golub G, Van Loan C (1980) An analysis of the total least squares problem. SIAM J Numer Anal 17(6):883–893CrossRefGoogle Scholar
  13. Jazaeri S, Amiri-Simkooei AR, Sharifi MA (2014) Iterative algorithm for weighted total least squares adjustment. Surv Rev 46(334):19–27CrossRefGoogle Scholar
  14. Li B et al (2013) Seamless multivariate affine error-in-variables transformation and its application to map rectification. Int J Geogr Inf Sci 27(8):1572–1592CrossRefGoogle Scholar
  15. Mahboub V (2012) On weighted total least-squares for geodetic transformations. J Geod 86:359–367CrossRefGoogle Scholar
  16. Neitzel F (2010) Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation. J Geod 84:751–762CrossRefGoogle Scholar
  17. Schaffrin B, Felus YA (2008a) On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms. J Geod 82:372–383Google Scholar
  18. Schaffrin B, Felus Y (2008b) Multivariate total least-squares adjustment for empirical affine transformations. In: Xu P, Liu J, Dermanis A (eds) VI Hotine-Marussi symposium on theoretical and computational geodesy international association of geodesy symposia, 29 May–2 June, Wuhan, China, vol 132, pp 238–242. Springer, BerlinGoogle Scholar
  19. Schaffrin B, Wieser A (2008) On weighted total least-squares adjustment for linear regression. J Geod 82:415–421CrossRefGoogle Scholar
  20. Schaffrin B, Wieser A (2009) Empirical affine reference frame transformations by weighted multivariate TLS adjustment. Geodetic reference frames. Springer, Berlin, pp 213–218Google Scholar
  21. Shen YZ, Li BF, Chen Y (2011) An iterative solution of weighted total least-squares adjustment. J Geod 85:229–238CrossRefGoogle Scholar
  22. Wang B, Li JC, Liu C (2016a) A robust weighted total least squares algorithm and its geodetic applications. Stud Geophys Geod 60(2):177–194CrossRefGoogle Scholar
  23. Wang L, Zhao YW, Chen XY, Zang DY (2016b) A Newton algorithm for multivariate total least squares problems. Acta Geodaetica Cartogr Sin 45:411–417 (in Chinese with English Abstract) Google Scholar
  24. Wang B, Li JC, Liu C, Yu J (2017) Generalized total least squares prediction algorithm for universal 3D similarity transformation. Adv Space Res 59(3):815–823CrossRefGoogle Scholar
  25. Wang B, Yu J, Liu C, Li MF, By Zhu (2018) Data snooping algorithm for universal 3D similarity transformation based on generalized EIV model. Measurement 119:56–62CrossRefGoogle Scholar
  26. Xu PL, Liu JN, Shi C (2012) Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis. J Geod 86:661–675CrossRefGoogle Scholar
  27. Zeng WX, Liu JN, Yao YB (2015) On partial errors-in-variables models with inequality constraints of parameters and variables. J Geod 89(2):111–119CrossRefGoogle Scholar
  28. Zhang SL, Tong XH, Zhang K (2013) A solution to EIV model with inequality constraints and its geodetic applications. J Geod 87:23–28CrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó 2019

Authors and Affiliations

  1. 1.Faculty of Information EngineeringChina University of GeosciencesWuhanChina
  2. 2.School of Geomatics Science and TechnologyNanjing Tech UniversityNanjingChina
  3. 3.Hunan Software Vocational InstituteXiangtanChina

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