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Studia Geophysica et Geodaetica

, Volume 63, Issue 4, pp 485–508 | Cite as

Variance-covariance component estimation for structured errors-in-variables models with cross-covariances

  • Zhipeng LvEmail author
  • Lifen Sui
Article
  • 24 Downloads

Abstract

In this contribution, an iterative algorithm for variance-covariance component estimation based on the structured errors-in-variables (EIV) model is proposed. We introduce the variable projection principle and derive alternative formulae for the structured EIV model by applying Lagrange multipliers, which take the form of a least-squares solution and are easy to implement. Then, least-squares variance component estimation (LS-VCE) is applied to estimate different (co)variance components in a structured EIV model. The proposed algorithm includes the estimation of covariance components, which is not considered in other recently proposed approaches. Finally, the estimability of the (co)variance components of the EIV stochastic model is discussed in detail. The efficacy of the proposed algorithm is demonstrated through two applications: multiple linear regression and auto-regression, on simulated datasets or on a real dataset with some assumptions.

Keywords

variable projection principle structured total least-squares STLS covariance component estimability analysis 

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Notes

Acknowledgments

We would like to acknowledge useful comments of the Associate Editor, P.J.G. Teunissen, and the anonymous reviewers that improved the presentation of this paper. This research was supported by the National Natural Science Foundation of PR China (Grant Nos 41674016, 41274016 and 41904039).

References

  1. Amiri-Simkooei A.R., 2007. Least-Squares Variance Component Estimation: Theory and GPS Applications. PhD Thesis. Delft University of Technology, Delft, The Netherlands.Google Scholar
  2. Amiri-Simkooei A.R., Teunissen P.J.G. and Tiberius C., 2009. Application of least-squares variance component estimation to GPS observables. J. Surv. Eng., 135, 149–160.Google Scholar
  3. Amiri-Simkooei A.R. 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.R., 2013. Application of least-squares variance component estimation to errors-in-variables models. J. Geodesy, 87, 935–944.Google Scholar
  5. Amiri-Simkooei A.R., 2016. Non-negative least-squares variance component estimation with application to GPS time series. J. Geodesy, 90, 451–466.Google Scholar
  6. Amiri-Simkooei A.R., 2018. Parameter estimation in 3D affine and similarity transformation: implementation of variance component estimation. J. Geodesy, 92, 1285–1297.Google Scholar
  7. Bjorck A., 1996. Numerical Methods for Least-Squares Problems. SIAM, Philadelphia, PA.Google Scholar
  8. Fang X., 2011. Weighted Total Least-Squares Solution for Application In Geodesy. PhD Thesis. Leibniz University Hannover, 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.Google Scholar
  10. Fang X., 2014. A structured and constrained total least-squares solution with cross-covariances. Stud. Geophys. Geod., 58, 1–16.Google Scholar
  11. Fuller W.A., 1987. Measurement Error Models. Wiley, New York, NY.Google Scholar
  12. Golub G. and Van Loan C., 1980. An analysis of the total least-squares problem. SIAM J. Numer. Anal, 17, 883–893.Google Scholar
  13. Helmert F.R., 1907. Die Ausgleichungsrechnung nach der Methode der kleinsten Quadrate. 2. Auflage. B.G. Teubner, Leipzig/Berlin, Germany (in German).Google Scholar
  14. Jazaeri S., Amiri-Simkooei A.R. and Sharifi M.A., 2014. Iterative algorithm for weighted total least-squares adjustment. Surv. Rev., 46, 19–27.Google Scholar
  15. Koch K.R., 1986. Maximum likelihood estimate of variance components. Bull. Geod., 60, 329–338.Google Scholar
  16. Koch K.R., 1999. Parameter Estimation and Hypothesis Testing in Linear Models. Springer-Verlag, Berlin, Germany.Google Scholar
  17. Mahboub V., 2012. On weighted total least-squares for geodetic transformations. J. Geodesy, 86, 359–367.Google Scholar
  18. Mahboub V., 2014. Variance component estimation in errors-in-variables models and a rigorous total least-squares approach. Stud. Geophys. Geod., 58, 17–40.Google Scholar
  19. Mann M.E. and Emanuel K.A., 2006. Atlantic hurricane trends linked to climate change. Eos Trans. AGU, 87, DOI: 10.1029/2006EO240001.Google Scholar
  20. Markovsky I., Van Huffel S. and Pintelon R., 2005. Block-Toeplitz/Hankel structured total least-squares. SIAM J. Matrix Anal. A, 26, 1083–1099.Google Scholar
  21. Markovsky I. and Van Huffel S., 2006. On weighted structured total least-squares. In: Lirkov I., Margenov S. and Waśniewski J. (Eds), Large-Scale Scientific Computing. Lecture Notes in Computer Science 3743. Springer-Verlag, Berlin, Heidelberg, Germany.Google Scholar
  22. Moghtased-Azar K., Tehranchi R. and Amiri-Simkooei A.R., 2014. An alternative method for non-negative estimation of variance components. J. Geodesy, 88, 427–439.Google Scholar
  23. 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. Instrum., 22, 215–217.Google Scholar
  24. Pincus R., 1974. Estimability of parameters of the covarlance matrix and variance components. Math. Operationsforsch. Stat., 5, 245–248.Google Scholar
  25. Rao C.R., 1971. Estimation of variance and covariance components-MINQUE theory. J. Multivar. Anal., 1, 257–275.Google Scholar
  26. Rao C.R. and Kleffe J., 1988. Estimation of Variance Components and Applications. North-Holland, Amsterdam, The Netherlands.Google Scholar
  27. Schaffrin B., 1981. Best invariant covariance component estimators and its application to the generalize multivariate adjustment of heterogeneous deformation observations. Bull. Geod., 55, 73–85.Google Scholar
  28. Schaffrin B. and Felus Y.A., 2008. On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms. J. Geodesy, 82, 373–383.Google Scholar
  29. Schaffrin B. and Wieser A., 2008. On weighted total least-squares adjustment for linear regression. J. Geodesy, 82, 415–421.Google Scholar
  30. Shen Y.Z., Li, B.F. and Chen Y., 2011. An iterative solution of weighted total least-squares adjustment. J. Geodesy, 85, 229–238.Google Scholar
  31. Shi Y., Xu P.L. and Liu J.N., 2015. Alternative formulae for parameter estimation in partial errors-in-variables models. J. Geodesy, 89, 13–16.Google Scholar
  32. Snow K., 2012. Topics in Total Least-Squares Adjustment within the Errors-in-Variables Model: Singular Cofactor Matrices and Prior Information. PhD Thesis. The Ohio State University, Columbus, OH.Google Scholar
  33. Tong X.H., Jin Y.M. and Li L.Y., 2011. An improved weighted total least-squares method with applications in linear fitting and coordinate transformation. J. Surv. Eng., 137, 120–128.Google Scholar
  34. Teunissen P.J.G., 1984. A note on the use of Gauss' formula in nonlinear geodetic adjustments. Stat. Descis., 2, 455–466.Google Scholar
  35. Teunissen P.J.G., 1985. The Geometry of Geodetic Inverse Linear Mapping and Nonlinear Adjustment. Publications on Geodesy, 8(1), Netherlands Geodetic Commission, Delft, The Netherlands.Google Scholar
  36. Teunissen P.J.G., 1988a. The nonlinear 2D symmetric Helmert transformation: an exact nonlinear least-squares solution. Bull. Geod., 62, 1–15.Google Scholar
  37. Teunissen P.J.G. and Knickmeyer E. H., 1988b. Nonlinearity and least-squares. CISM J., 42, 321–330, DOI: 10.1139/geomat-1988-0027.Google Scholar
  38. Teunissen P.J.G., 1989. A note on the bias in the symmetric Helmert transformation. In: Kejlso E., Poder K. and Tscherning C.C. (Eds), Festchrift to Torben Krarup. Geodetical Institute, Odense, Denmark. Medelelse 58, 335–342.Google Scholar
  39. Teunissen P.J.G., 1990. Nonlinear least-squares. Manus. Geod., 15, 137–150.Google Scholar
  40. Teunissen P.J.G., 2000. Adjustment Theory: an Introduction. Mathematical Geodesy and Positioning, Delft University Press, Delft University of Technology, Delft, The Netherlands.Google Scholar
  41. Teunissen P.J.G. and Amiri-Simkooei A.R., 2008. Least-squares variance component estimation. J. Geodesy, 82, 65–82.Google Scholar
  42. Van Huffel S. and Vandewalle J., 1991. The Total Least-Squares Problem: Computational Aspects and Analysis. SIAM, Philadelphia, PA.Google Scholar
  43. Wang L.Y. and Xu G.Y., 2016. Variance component estimation for partial errors-in-variables models. Stud. Geophys. Geod., 60, 35–55.Google Scholar
  44. Wang X.Z., Yao Y.B., Qiu W.N., and Yao Y.B., 2006. Advanced Surveying Adjustment. Surveying and Mapping Press, Beijing, China (in Chinese).Google Scholar
  45. Xu P.L., Shen Y.Z., Fukuda Y. and Liu Y.M., 2006. Variance component estimation in linear inverse ill-posed models. J. Geodesy, 80, 69–81.Google Scholar
  46. Xu P.L., Liu Y.M., Shen Y.Z. and Fukuda Y.C., 2007. Estimability analysis of variance and covariance components. J. Geodesy, 81, 593–602.Google Scholar
  47. Xu P.L., Liu J.N. and Shi C., 2012. Total least-squares adjustment in partial errors-in-variables models: algorithm and statistical analysis. J. Geodesy, 86, 661–675.Google Scholar
  48. Xu P.L. and Liu J.N., 2014. Variance components in errors-in-variables models: estimability, stability and bias analysis. J. Geodesy, 88, 719–734.Google Scholar
  49. Xu P.L., 2016. The effect of errors-in-variables on variance component estimation. J. Geodesy, 90, 681–701.Google Scholar
  50. Yao Y.B., Xiong Z.H., Zhang B., Zhang L. and Kong J., 2017. A new method to solving AR parameters considering random errors of design matrix. Acta Geod. Cartogr. Sin., 46, 1795–1801 (in Chinese with English abstract).Google Scholar

Copyright information

© Inst. Geophys. CAS, Prague 2019

Authors and Affiliations

  1. 1.Institute of Surveying and MappingInformation Engineering UniversityZhengzhouChina

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