Skip to main content

On weighted total least-squares for geodetic transformations

Abstract

In this contribution, it is proved that the weighted total least-squares (WTLS) approach preserves the structure of the coefficient matrix in errors-in-variables (EIV) model when based on the perfect description of the dispersion matrix. To achieve this goal, first a proper algorithm for WTLS is developed since the quite recent analytical solution for WTLS by Schaffrin and Wieser is restricted to the condition \({{P}_{\rm A} =\left({P_0 \otimes P_x}\right)}\) (where \({\otimes}\) is used to denote the Kronecker product) for the weight matrix of the coefficient matrix in the EIV model. This situation can be seen in the case of an affine transformation where the univariate approach can be an appropriate alternative to the multivariate WTLS approach, which has been applied to the affine transformation by Schaffrin and Felus, resp. Schaffrin and Wieser with restrictions similar to \({{P}_{\rm A} =\left( {P_0 \otimes P_x}\right)}\). In addition, this algorithm for WTLS can be interpreted well in the geodetic literature since it is based on the perfect description of the inverse dispersion matrix (or variance–covariance). By using the algorithm of WTLS, one obtains more realistic results in some applications of transformation where a high precision is needed. Some empirical examples, resp. simulation studies give insight into the efficiency of the procedure.

This is a preview of subscription content, access via your institution.

References

  1. Cadzow JA (1988) Signal enhancement—a composite property mapping algorithm. IEEE Trans Acoust Speech Signal Process 36(1): 49–62

    Article  Google Scholar 

  2. Felus Y, Schaffrin B (2005) Performing similarity transformations using the errors-in-variables-model. In: Proceedings of the ASPRS meeting, Washington, DC, May 2005

  3. Golub G, van Loan C (1980) An analysis of the total least squares problem. SIAM J Num Anal 17: 883–893

    Article  Google Scholar 

  4. Magnus JR (1988) Linear structures. Charles Griffin and Company Ltd. Oxford University Press, London

    Google Scholar 

  5. Markovsky I, van Huffel S (2006) On weighted structured total least squares. Large-scale scientific computing. Lecture notes in computer science, vol 3743. Springer, Berlin, pp 695–702

  6. Markovsky I, Rastello M, Premoli A, Kukush A, van Huffel S (2006) The element-wise weighted total least-squares problem. Comput Stat Data Anal 50: 181–209

    Article  Google Scholar 

  7. Markovsky I, Sima D, van Huffel S (2010) Generalizations of the total least squares problem. Wiley Interdiscip Rev Comput Stat 2(2): 212–217

    Article  Google Scholar 

  8. Neri F, Saitta G, Chiofalo S (1989) An accurate and straightforward approach to line regression analysis of error-affected experimental data. J Phys Ser E: Sci Instr 22: 215–217

    Article  Google Scholar 

  9. Schaffrin B, Felus Y (2008) On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms. J Geod 82(2008): 353–383

    Google Scholar 

  10. Schaffrin B, Wieser A (2008) On weighted total least-squares adjustment for linear regression. J Geod 82(7): 415–421

    Article  Google Scholar 

  11. Schaffrin B, Wieser A (2009) Empirical affine reference frame transformations by weighted multivariate TLS adjustment. In: Drewes H (ed) International Association of geodesy symposia, vol 134. Geodetic reference frames. Springer, Berlin, pp 213–218

    Google Scholar 

  12. van Huffel S, Vandewalle J (1991) The total least-squares problem. Computational aspects and analysis. SIAM, Philadelphia

    Book  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Vahid Mahboub.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mahboub, V. On weighted total least-squares for geodetic transformations. J Geod 86, 359–367 (2012). https://doi.org/10.1007/s00190-011-0524-5

Download citation

Keywords

  • EIV model
  • Weighted total least-squares principle
  • Similarity transformation
  • Affine transformation