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Journal of Geodetic Science

Editor-in-Chief: Eshagh, Mehdi

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The Case of the Homogeneous Errors-In-Variables Model

B. Schaffrin
  • Corresponding author
  • Geodetic Science Program, School of Earth Sciences, The Ohio State University, Columbus, Ohio, USA
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ K. Snow
  • Corresponding author
  • Geodetic Science Program, School of Earth Sciences, The Ohio State University, Columbus, Ohio, USA
  • Topcon Positioning Systems, Inc., Columbus, Ohio, USA
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2014-12-10 | DOI: https://doi.org/10.2478/jogs-2014-0017


Recently, it has been claimed that the Homogeneous Errors-In-Variables (HEIV) Model, where the lefthand side (LHS) vector is allowed to be multiplied with an unknown scale factor, would represent a generalization of the regular EIV-Model for which a number of efficient algorithms already exist. Unfortunately, due to the forced rank deficiency in the case of the HEIV-Model, an additional constraint needs to be introduced to guarantee uniqueness of the TLS solution (“datum constraint”). If this constraint is linear, a simple manipulation will reduce the HEIV-Model with one constraint to the regular EIV-Model. But also in the case of a non-linear datum constraint, by introducing parameter ratios as unknowns, an EIV-Model may result that can be treated by standard TLS adjustment, followed by a solution of the datum constraint for the additional LHS scale parameter. This approach will be applied to an example where the datum constraint is chosen to be quadratic.

Keywords: Errors-In-Variables Model: Homogeneous vs. regular; Total Least-Squares adjustment: With and without datum constraint; model equivalency

  • Amiri-Simkooei, A. and Jazaeri, S., 2012, Weighted Total Least- Squares formulated by standard Least-Squares theory, Journal of Geodetic Science, 2(2):113–124.Google Scholar

  • Bôcher, M., 1915, Plane Analytic Geometry: with introductory chapters on the Differential Calculus, H. Holt and Co., New York.Google Scholar

  • Fang, X., 2011, Weighted Total Least-Squares Solutions for Applications in Geodesy, Publ. No. 294, Dept. of Geodesy and Geoinformatics, Leibniz University Hannover, Germany.Google Scholar

  • Inkilä, K., 2005, Homogeneous Least-Squares Problem, The Photogrammetric Journal of Finland, 19(2):34–42.Google Scholar

  • Linkwitz, K., 1976, On certain adjustment problems and their solutions by means of matrix eigenvalues (in German), In Ackermann, F., editor, Festschrift für E. Gotthardt, pages 111–126. German Geodetic Comm. B-216, Munich.Google Scholar

  • Mahboub, V., 2012, On weighted Total Least-Squares for geodetic transformations, Journal of Geodesy, 86(5):359–367.CrossrefWeb of ScienceGoogle Scholar

  • Matei, B. C. and Meer, P., 2006, Estimation of nonlinear Errors-In- Variables models for computer vision applications, IEEE Trans. Pattern Anal. and Machine Intell., 28(10):1537–1552.Google Scholar

  • Schaffrin, B., 2006, A note on Constrained Total Least-Squares estimation, Linear Algebra and its Applications, 417(1):245–258.Google Scholar

  • Schaffrin, B. and Felus, Y., 2005, On Total Least-Squares Adjustment with Constraints, In Sansò, F., editor, A Window on the Future of Geodesy, pages 417–421. IAG-Symp., Vol. 128, Springer: Berlin/Heidelberg/New York.Google Scholar

  • Schaffrin, B. and Wieser, A., 2008, On weighted total least-squares adjustment for linear regression, Journal of Geodesy, 82(7):415– 421.Web of ScienceCrossrefGoogle Scholar

  • Snow, K., 2012, Topics in Total Least-Squares Adjustment within the Errors-In-VariablesModel: Singular CofactorMatrices and Prior Information, Report No. 502, Div. of Geodetic Science, School of Earth Sciences, The Ohio State Univ., Columbus/OH, USA.Google Scholar

  • Xu, P., Liu, J., and Shi, C., 2012, Total Least-Squares adjustment in partial Errors-In-Variables models: Algorithm and statistical analysis, Journal of Geodesy, 86(8):661–675.CrossrefWeb of ScienceGoogle Scholar

  • Yeredor, A. and De Moor, B., 2004, On homogeneous Least-Squares problems and the inconsistency introduced by mis-constraining, Computational Statistics & Data Analysis, 47(3):455–465.Google Scholar

  • Zhou, Y., Kou, X., Zhu, J., and Deng, C., 2014, Quadratically constrained homogeneous Errors-In-Variables modeling and related Total Least-Squares adjustments, unpublished manuscript. Google Scholar

About the article

Received: 2014-10-10

Accepted: 2014-11-24

Published Online: 2014-12-10

Citation Information: Journal of Geodetic Science, Volume 4, Issue 1, ISSN (Online) 2081-9943, DOI: https://doi.org/10.2478/jogs-2014-0017.

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©2014 B. Schaffrin, K. Snow. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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