Abstract
Recent studies have extensively discussed total least squares (TLS) algorithms for solving the errors-in-variables (EIV) model with equality constraints but rarely investigated the inequality-constrained EIV model. The most existing inequality-constrained TLS algorithms assume that all the elements in the coefficient matrix are random and independent and that their numerical efficiency is significantly limited due to combinatorial difficulty. To solve the above issues, we formulate a partial EIV model with inequality constraints of both unknown parameters and the random elements of the coefficient matrix. Based on the formulated EIV model, the inequality-constrained TLS problem is transformed into a linear complementarity problem through linearization. In this way, the inequality-constrained TLS method remains applicable even when the elements of the coefficient matrix are subject to inequality constraints. Furthermore, the precision of the constrained estimates is put forward from a frequentist point of view. Three numerical examples are presented to demonstrate the efficiency and superiority of the proposed algorithm. The application is accomplished by preserving the structure of random coefficient matrix and satisfying the constraints simultaneously, without any combinatorial difficulty.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adcock RJ (1877) Note on the method of least squares. Analyst 4:183–184
Amiri-Simkooei AR (2013) Application of least squares variance component estimation to errors-in-variables models. J Geod 87:935–944
Choi YJ, Kim JY, Sumg KM (1997) A robust algorithm of total least squares method. IEICE Trans Fundam E80-A(7):1336–1339
De Moor B (1990) Total linear least squares with inequality constraints. In: Proceedings of ESAT-SISTA report 1990-2, Department of Electrical Engineering, Katholieke Universiteit Leuven, Belgium
Deming WE (1931) The application of least squares. Phil Mag 11:146–158
Deming WE (1934) On the application of least squares–II. Phil Mag 17:804–829
Fang X (2011) Weighted total least squares solutions for applications in geodesy. PhD dissertation, Department of Geodesy and Geoinformatics, Leibniz University Hannover, Germany
Fang X (2013) Weighted total least squares: necessary and sufficient conditions, fixed and random parameters. J Geod 87:733–749
Fang X (2014a) On non-combinatorial weighted total least squares with inequality constraints. J Geod 88:805–816
Fang X (2014b) A structured and constrained total least-squares solution with cross-covariances. Stud Geophys Geod 58:1–16
Felus Y, Schaffrin B (2005) Performing similarity transformations using the error-in-variables model. In: Proceedings of the ASPRS meeting, Washington
Gerhold GA (1969) Least-squares adjustment of weighted data to a general linear equation. Am J Phys 37:156–1619
Golub GH, van Loan CF (1980) An analysis of the total least squares problem. SIAM J Numer Anal 17:883–893
Koch KR (1999) Parameter estimation and hypothesis testing in linear models. Springer, Berlin
Kojima M, Megiddo N, Noma T, Yoshise A (1991) A unified approach to interior point algorithms for linear complementarity problems. Springer, Berlin
Liew CK (1976) Inequality constrained least-squares estimation. J Am Stat Assoc 71:746–751
Mahboub V (2012) On weighted total least-squares for geodetic transformation. J Geod 86:359–367
Mahboub V, Sharifi MA (2013) On weighted total least squares with linear and quadratic constraints. J Geod 87:607–608
Mahboub V (2014) Variance component estimation in errors-in-variables models and a rigorous total least-squares approach. Studia Geophysica et Geodaetica 58:17–40
Markovsky I, van Huffel S (2006) On weighted structured total least squares. Large Scale Sci Comput Lect Notes Comput Sci 3743:695–702
Neitzel F (2010) Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation. J Geod 84:751–762
Pearson K (1901) On lines and planes of closest fit to systems of points in space. Philos Mag 2:559–572
Peng JH, Guo CX, Zhang HP, Song SL (2006) An aggregate constraint method for inequality-constrained least squares problem. J Geod 79:705–713
Roese-Koerner L, Devaraju B, Sneeuw N, Schuh WD (2012) A stochastic framework for inequality constrained estimation. J Geod 86:1005–1018
Roese-Koerner L, Schuh WD (2014) Convex optimization under inequality constraints in rank-deficient systems. J Geod 88:415–426
Schaffrin B, Felus YA (2005) On total least-squares adjustment with constraints. IAG Symp 128:417–421
Schaffrin B (2006) A note on constrained total least-squares estimation. Linear Algebra Appl 417:245–258
Schaffrin B, Felus YA (2008) On the multivariate total least-squares approach to empirical coordinate transformations, three algorithms. J Geod 82:373–383
Schaffrin B, Wieser A (2008) On weighted total least-squares adjustment for linear regression. J Geod 82:415–421
Schaffrin B (2009) TLS-collocation: the total-least squares approach to EIV-models with stochastic prior information. In: Presented at the 18th international, workshop on matrices and statistics, Smolenice Castle, Solvakia
Shen YZ, Li BF, Chen Y (2011) An iterative solution of weighted total least-squares adjustment. J Geod 85:229–238
Snow K (2012) Topics in total least-squares adjustment within the errors-in-variables model: singular cofactor matrices and prior information. PhD dissertation, School of Earth Science, The Ohio State University, USA
Van Huffel S, Vandewalle J (1991) The total least squares problem: computational aspects and analysis. SIAM, Philadelphia
Waston GA (2007) Robust counterparts of errors-in-variables problems. Comput Stat Data Anal 52:1080–1089
Xu PL (1998) Mixed integer geodetic observation and interger programming with applications to GPS ambiguity resolution. J Geod Soc Japan 44:169–187
Xu PL, Cannon E, Lachapelle G (1999) Stabilizing ill-conditioned linear complementarity problems. J Geod 73:204–213
Xu PL, Liu JN, Shi C (2012) Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis. J Geod 86:661–675
Xu PL, Liu JN, Zeng WX, Shen YZ (2014) Effects of errors-in-variables on weighted least squares estimation. J Geod 88:705–716
Xu PL, Liu JN (2014) Variance components in errors-in-variables models: estimability, stability and bias analysis. J Geod 88:719–734
Zhang SL, Tong XH, Zhang K (2013) A solution to EIV model with inequality constraints and its geodetic applications. J Geod 87:89–99
Zhu JJ, Santerre R, Chang XW (2005) A Bayesian method for linear, inequality-constrained adjustment and its application to GPS positioning. J Geod 78:528–534
Acknowledgments
The first author thanks her supervisor, Dr. Peiliang Xu, for suggesting this topic to investigate and for his suggestion to use LCP methods to solve problems of this kind. This research was supported by the National Natural Science Foundation of China (41474006; 41404005; 41231174; 41274022).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zeng, W., Liu, J. & Yao, Y. On partial errors-in-variables models with inequality constraints of parameters and variables. J Geod 89, 111–119 (2015). https://doi.org/10.1007/s00190-014-0775-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00190-014-0775-z