Abstract
A method for variance component estimation (VCE) in errors-in-variables (EIV) models is proposed, which leads to a novel rigorous total least-squares (TLS) approach. To achieve a realistic estimation of parameters, knowledge about the stochastic model, in addition to the functional model, is required. For an EIV model, the existing TLS techniques either do not consider the stochastic model at all or assume approximate models such as those with only one variance component. In contrast to such TLS techniques, the proposed method considers an unknown structure for the stochastic model in the adjustment of an EIV model. It simultaneously predicts the stochastic model and estimates the unknown parameters of the functional model. Moreover the method shows how an EIV model can support the Gauss-Helmert model in some cases. To make the VCE theory into EIV model more applicable, two simplified algorithms are also proposed. The proposed methods can be applied to linear regression and datum transformation. We apply these methods to these examples. In particular a 3-D non-linear close to identical similarity transformation is performed. Two simulation studies besides an experimental example give insight into the efficiency of the algorithms.
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Acar M., Özlüdemir M., Akyilmaz O., Celik R. and Ayan T., 2006. Deformation analysis with Total Least Squares. Nat. Hazards Earth Syst. Sci., 6, 663–669.
Akaike H., 1974. A new look at the statistical model identification. IEEE Trans. Autom. Control., 19, 716–723.
Baarda W., 1968. A Testing Procedure for Use in Geodetic Networks. Publications on Geodesy, 2, No. 5, Netherlands Geodetic Commission, Delft, The Netherlands.
Fang X., 2011. Weighted Total Least Squares Solutions for Applications in Geodesy. PhD Thesis, Leibniz University, Hannover, Germany.
Fang X., 2013. Weighted total least squares: necessary and sufficient conditions, fixed and random parameters. J. Geodesy, 87, 733–748, DOI: 10.1007/s00190-013-0643-2.
Felus A., 2004. Application of total least squares for spatial point process analysis. J. Surv. Eng., 130(3), 126–133.
Felus Y.A. and Felus M., 2009. On Choosing the Right Coordinate Transformation Method. https://www.fig.net/pub/fig2009/papers/ts04c/ts04c_felus_felus_3313.pdf.
Golub G. and Van Loan C., 1980. An analysis of the Total Least Squares problem. SIAM J. Num. Anal., 17, 883–893.
Grafarend E.W., 1984. Variance-covariance component estimation of Helmert type in the Gauss-Helmert model. Zeitschrift fur Vermessungswesen, 109, 34–44.
Grafarend E.W., 1985. Variance-covariance component estimation: theoretical results and geodetic applications. Statistics & Decisions, Suppl. 2, 407–441.
Helmert F.R., 1907. Die Ausgleichungsrechnung nach der Methode der kleinsten Quadrate. 2. Auflage. B.G. Teubner, Leipzig/Berlin, Germany (in German).
Koch K.R., 1978. Schätzung von Varianzkomponenten. Allgemeine Vermessungs Nachrichten, 85, 264–269 (in German).
Koch K.R., 1999. Parameter Estimation and Hypothesis Testing in Linear Models. Springer Verlag, Berlin, Germany.
Magnus J.R., 1988. Linear Structures. Griffin’s Statistical Monographs 42, Oxford University Press, New York.
Mahboub V., 2012. On weighted total least-squares for geodetic transformations. J. Geodesy, 86, 359–367, DOI: 10.1007/s00190-012-0524-5.
Mahboub V., Amiri-Simkooei A.R. and Sharifi M.A., 2012. Iteratively reweighted total leastsquares: A robust estimation in errors-in-variables models. Surv. Rev., 45, 92–99, DOI: 10.1179/1752270612Y.0000000017.
Mahboub V. and Sharifi M.A., 2013. On weighted total least-squares with linear and quadratic constraints. J. Geodesy, 87, 279–286, DOI: 10.1007/s00190-012-0598-8.
Mallows C.L. 1973., Some comments on Cp. Technometrics, 15, 661–675, DOI: 10.2307/1267380.
Markovsky I., Rastello M., Premoli A., Kukush A. and van Huffel S., 2006. The element-wise weighted total least-squares problem. Comput. Statist. Data Anal., 50, 181–209.
Neitzel F., 2010. Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation. J. Geodesy, 84, 751–762.
Pukelsheim F., 1981. On the existence of unbiased nonnegative estimates of variance covariance components. Annals of Statistics, 9, 293–299 (http://www.math.uni-augsburg.de/stochastik/pukelsheim/1981b.pdf).
Rao C.R., 1971. Estimation of variance and covariance components — MINQUE theory. J. Multivar. Anal., 1, 257–275.
Schaffrin B., 1981. Best invariant covariance component estimators and its application to the generalized multivariate adjustment of heterogeneous deformation observations. Bull. Geod., 55, 73–85.
Schaffrin B., 1983. Varianz-Kovarianz-Komponenten-Schätzung bei der Ausgleichung heterogener Wiederholungsmessungen. Reihe C, 282. Deutsche Geodätische Kommission, Munchen, Germany (in German).
Schaffrin B., 1985. Das geodatische Datum mit stochastischer Vorinfformation. Reihe C, 313. Deustche Geodätische Kommission, Munchen, Germany (in German).
Schaffrin B., 2006. A note on constraint total least-squares estimation. Linear Alg. Appl., 417, 245–258.
Schaffrin B. and Wieser A., 2008. On weighted total least-squares adjustment for linear regression. J. Geodesy, 82, 415–421.
Schaffrin B. and Wieser A., 2009. Empirical affine reference frame transformations by weighted multivariate TLS adjustment. In: Drewes H (Ed.), Geodetic Reference Frames. International Association of Geodesy Symposia 134, Springer Verlag, Berlin, Germany, 213–218.
Snow K., 2012. Topics in Total Least-Squares Adjustment within the Errors-In-Variables Model: Singular Cofactor Matrices and Priori Information. PhD Thesis. Report No 502, Geodetic Science Program, School of Earth Sciences, The Ohio State University, Columbus, OH.
Teunissen P.J.G., 1984. Generalized Inverse, Adjustment the Datum Problem and S-Transformations. Department of Geodesy, Delft University of Technology, Delft, The Netherlands (http://saegnss1.curtin.edu.au/Publications/1984/Teunissen1984Generalized.pdf).
Teunissen P.J.G., 1988. The nonlinear 2D nymmetric Helmert transformation: An exact nonlinear least-squares solution. J. Geodesy, 62, 1–15.
Teunissen P.J.G., 1988. Towards a Least-Squares Framework for Adjusting and Testing of Both Functional and Stochastic Models. Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands (http://www.lr.tudelft.nl/mgp).
Teunissen P.J.G., 1989. First and second order moments of nonlinear least-squares estimators. J. Geodesy, 63, 253–262.
Teunissen P.J.G., 1990. Nonlinear least-squares. Manuscripta Geodaetica, 15, 137–150.
Teunissen P.J.G. and Amiri-Simkooei A.R., 2008. Least-squares variance component estimation. J. Geodesy. 82, 65–82, DOI: 10.1007/s00190-007-0157-x.
Teunissen P.J.G., Simons D.G. and Tiberius C.C.J.M., 2005. Probability and Observation Theory. Lecture Notes AE2-E01, Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands.
van Huffel S. and Vandewalle J., 1991. The Total Least-Squares Problem: Computational Aspects and Analysis. SIAM, Philadelphia.
Zhang J., Bock Y., Johnson H., Fang P., Williams S., Genrich J., Wdowinski S. and Behr J., 1997. Southern California permanent GPS geodetic array: Error analysis of daily position estimates and site velocitties. J. Geophys. Res., 102, 18035–18055.
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Mahboub, V. Variance component estimation in errors-in-variables models and a rigorous total least-squares approach. Stud Geophys Geod 58, 17–40 (2014). https://doi.org/10.1007/s11200-013-1150-x
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DOI: https://doi.org/10.1007/s11200-013-1150-x