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Bayesian inference for the Errors-In-Variables model

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Abstract

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.

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References

  • Akyilmaz O., 2007. Total least squares solution of coordinate transformation. Surv. Rev., 39, 68–80.

    Article  Google Scholar 

  • Amiri-Simkooei A., 2013. Application of least squares variance component estimation to errors-invariables models. J. Geodesy, 87, 935–944.

    Article  Google Scholar 

  • 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 

  • Amiri-Simkooei A. and Jazaeri S., 2013. Data-snooping procedure applied to errors-in-variables models. Stud. Geophys. Geod., 57, 426–441.

    Article  Google Scholar 

  • Bauwens L., Lubrano M. and Richard J.-F., 1999. Bayesian Inference in Dynamic Econometric Models. Oxford University Press, Oxford, U.K.

    Google Scholar 

  • Bolfarine H. and Rodrigues J., 2007. Bayesian inference for an extended simple regression measurement error model using skewed priors. Bayesian Anal., 2, 349–364.

    Article  Google Scholar 

  • Dellaportas P. and Stephens D.A., 1995. Bayesian analysis of errors-in-variables regression models. Biometrics, 51, 1085–1095.

    Article  Google Scholar 

  • Fang X., 2011. Weighted Total Least Squares Solutions for Applications in Geodesy. PhD Thesis. Leibniz University, Hannover, Germany.

    Google Scholar 

  • Fang X., 2013. Weighted Total Least Squares: necessary and sufficient conditions, fixed and random parameters. J. Geodesy, 87, 733–749.

    Article  Google Scholar 

  • Fang X., 2014a. A structured and constrained Total Least-Squares solution with cross-covariances. Stud. Geophys. Geod., 58, 1–16.

    Article  Google Scholar 

  • Fang X., 2014b. On non-combinatorial weighted Total Least Squares with inequality constraints. J. Geodesy, 88, 805–816.

    Article  Google Scholar 

  • Fang X., 2014c. A total least squares solution for geodetic datum transformations. Acta Geod. Geophys., 49, 189–207.

    Article  Google Scholar 

  • Fang X., 2015. Weighted total least-squares with constraints: a universal formula for geodetic symmetrical transformations. J. Geodesy, 89, 459–469.

    Article  Google Scholar 

  • Felus F., 2004. Application of Total Least Squares for Spatial Point Process Analysis. J. Surv. Eng., 130, 126–133.

    Article  Google Scholar 

  • Felus F. and Burtch R., 2009. On symmetrical three-dimensional datum conversion. GPS Solut., 13, 65–74.

    Article  Google Scholar 

  • Florens J.-P., Mouchart M. and Richard J.-F., 1974. Bayesian inference in error-in-variables models. J. Multivar. Anal., 4, 419–452.

    Article  Google Scholar 

  • Golub G. and Van Loan C., 1980. An analysis of the Total least-squares problem. SIAM J. Numer. Anal., 17, 883–893.

    Article  Google Scholar 

  • 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 

  • Grafarend E.W. and Schaffrin B., 1993. Ausgleichungsrechnung in linearen Modellen. BIWissenschaftsverlag, Mannheim, Germany (in German).

    Google Scholar 

  • Huang H.-J., 2010. Bayesian Analysis of Errors-in-Variables Growth Curve Models. PhD Thesis. University of California, Riverside, CA.

    Google Scholar 

  • Koch K., 1986. Maximum likelihood estimate of variance components. Bull. Geod., 60, 329–338.

    Article  Google Scholar 

  • Koch K., 1999. Parameter Estimation and Hypothesis Testing in Linear Models. 2nd Edition. Springer-Verlag, Berlin, Germany.

    Book  Google Scholar 

  • Koch K., 2007. Introduction to Bayesian Statistics. 2nd Edition. Springer-Verlag, Berlin, Germany.

    Google Scholar 

  • Markovsky I. and Van Huffel S., 2007. Overview of total least-squares methods. Signal Process., 87, 2283–2302.

    Article  Google Scholar 

  • 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, 279–291

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • Neitzel F., 2010. Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation. J. Geodesy, 84, 751–762.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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 

  • Reilly P.M. and Patino-Lea H., 1981. A Bayesian study of the error-in-variables model. Technometrics, 23, 221–231.

    Article  Google Scholar 

  • Schaffrin B., 1997. Reliability measures for correlated observations. J. Surv. Eng.-ASCE, 123, 126–137.

    Article  Google Scholar 

  • Schaffrin B., 2006. A note on Constrained Total Least-Squares estimation. Linear Alg. Appl., 417, 245–258.

    Article  Google Scholar 

  • Schaffrin B. and Felus Y., 2008. On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms. J. Geodesy, 82, 373–383.

    Article  Google Scholar 

  • Schaffrin B. and Wieser A., 2008. On weighted total least-squares adjustment for linear regression. J. Geodesy, 82, 415–421.

    Article  Google Scholar 

  • Shan J., 1989. A fast recursive method for repeated computation of the reliability matrix QvvP. Photogrammetria, 43, 337–346.

    Article  Google Scholar 

  • Shen Y., Li B.F. and Chen Y., 2010. An iterative solution of weighted total least-squares adjustment. J. Geodesy, 85, 229–238.

    Article  Google Scholar 

  • 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 

  • 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 

  • Teunissen P.J.G., 1988. The nonlinear 2D symmetric Helmert transformation: an exact nonlinear least squares solution, Bull. Geod., 62, 1–15.

    Article  Google Scholar 

  • Xu P., Liu Y., Shen Y. and Fukuda Y., 2007. Estimability analysis of variance and covariance components. J. Geodesy, 81, 593–602.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Xu P., 2016. The effect of errors-in-variables on variance component estimation. J. Geodesy, DOI: 10.1007/s00190-016-0902-0

    Google Scholar 

  • Zellner A., 1971. An Introduction to Bayesian Inference in Econometrics. John Wiley & Sons.

    Google Scholar 

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Authors and Affiliations

  1. School of Geodesy and Geomatics, Wuhan University, Wuhan, China

    Xing Fang, Wenxian Zeng & Yibin Yao

  2. School of Earth Science, Ohio State University, Columbus, Ohio, USA

    Xing Fang

  3. College of Surveying and Geo-Informatics, Tongji University, Shanghai, China

    Bofeng Li

  4. Geodetic Institute, Leibniz University, Hannover, Germany

    Hamza Alkhatib

Authors
  1. Xing Fang
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  2. Bofeng Li
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  3. Hamza Alkhatib
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  4. Wenxian Zeng
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  5. Yibin Yao
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Correspondence to Wenxian Zeng.

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Fang, X., Li, B., Alkhatib, H. et al. Bayesian inference for the Errors-In-Variables model. Stud Geophys Geod 61, 35–52 (2017). https://doi.org/10.1007/s11200-015-6107-9

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  • DOI: https://doi.org/10.1007/s11200-015-6107-9

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