Skip to main content
SpringerLink
Log in

Application of Pareto optimality to linear models with errors-in-all-variables

  • Original Article
  • Published:
Journal of Geodesy Aims and scope Submit manuscript

Abstract

In some geodetic and geoinformatic parametric modeling, the objectives to be minimized are often expressed in different forms, resulting in different parametric values for the estimated parameters at non-zero residuals. Sometimes, these objectives may compete in a Pareto sense, namely a small change in the parameters results in the increase of one objective and a decrease of the other, as frequently occurs in multiobjective problems. Such is the case with errors-in-all-variables (EIV) models, e.g., in the geodetic and photogrammetric coordinate transformation problems often solved using total least squares solution (TLS) as opposed to ordinary least squares solution (OLS). In this contribution, the application of Pareto optimality to solving parameter estimation for linear models with EIV is presented. The method is tested to solve two well-known geodetic problems of linear regression and linear conformal coordinate transformation. The results are compared with those from OLS, Reduced Major Axis Regression (TLS solution), and the least geometric mean deviation (GMD) approach. It is shown that the TLS and GMD solutions applied to the EIV models are just special cases of the Pareto optimal solution, since both of them belong to the Pareto-set of the problems. The Pareto balanced optimum (PBO) solution as a member of this Pareto optimal solution set has special features and is numerically equal to the GMD solution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Akyilmaz O (2007) Total least squares solution of coordinate transformation. Survey Rev 39(303):68–80. doi:10.1179/003962607X165005

    Google Scholar 

  • Angus D, Deller A et al.: Computational intelligence in radio astronomy: using computational intelligence techniques to tune geodesy models. In: Li, X (eds) SEAL 2008. LNCS, vol 5361, pp. 615–624. Springer, Berlin (2008)

    Google Scholar 

  • Awange JL et al.: Algebraic geodesy and geoinformatics. Springer, Berlin (2010)

    Book  Google Scholar 

  • Cai J, Grafarend E (2009) Systematical analysis of the transformation between Gauss–Krueger-Coordinate/DHDN and UTM-Coordinate/ETRS89 in Baden-Württemberg with different estimation methods. In: Drewes H (eds) Geodetic reference frames. International Association of Geodesy Symposia 134. doi:10.1007/978-3-642-00860-3_32

  • Censor Y: Pareto optimality in multiobjective problems. Appl Math Optim 4, 41–59 (1977)

    Article  Google Scholar 

  • Coello CA: A comprehensive survey of evolutionary-based multiobjective optimization techniques. Knowl Inf Syst 1(3), 269–308 (2003)

    Google Scholar 

  • Doicu A, Trautmann T, Schreier F (2010) Numerical regularization for atmospheric inverse problems. Springer, pp 251–270. doi:10.1007/978-3-642-05439-6_8

  • Ehrgott M: Multicriteria optimization. Springer, Berlin (2005)

    Google Scholar 

  • Felus YA, Schaffrin B (2005) Performing similarity transformations using the errors-in-variable model. ASPRS Ann. Conference, Baltimore, Maryland

  • Fišerová E, Hron K: Total least squares solution for compositional data using linear models. J Appl Stat 37(7), 1137–1152 (2010). doi:10.1080/02664760902914532

    Article  Google Scholar 

  • Geisler J, Trächtler A (2009) Control of the Pareto optimality of systems with unknown disturbances. In: IEEE international conference on control and automation, Christchurch, New Zealand, December 9–11, 2009, pp 695–700

  • Golub GH, Van Loan CF: An analysis of the total least-squares problem. SIAM J Numer Anal 17(6), 883–893 (1980)

    Article  Google Scholar 

  • Gruna R (2010) Evolutionary multiobjective optimization. Wolfram Demonstration Project. www.wolfram.com

  • Haneberg W: Computational geosciences with Mathematica. Springer, Berlin (2004)

    Book  Google Scholar 

  • Hochman HM, Rodgers JD: Pareto optimal redistribution. Am Econ Rev 59(4 Part 1), 542–557 (1969)

    Google Scholar 

  • Hu L, Lin Y, Guo Y (2010) Space registration algorithm based on constrictive total least squares. In: International conference on intelligent computation technology and automation, ICICTA, vol 3, pp 359–362

  • Huband S, Hingston P, Barone L, While L: A review of multiobjective test problems and scalable test problem toolkit. IEEE Trans Evol Comput 10(N0), 477–506 (2006)

    Article  Google Scholar 

  • Huffel SV, Lemmerling P: Total least square and errors-in-variables modeling: analysis, algorithms and applications. Kluwer, Dordrecht (2002)

    Google Scholar 

  • Knowles, J, Corne, D, Deb, K (eds): Multiobjective problem solving from nature. Springer, Berlin (2008)

    Google Scholar 

  • Koch, KR (eds): Parameter estimation and hypothesis testing in linear models. Springer, Berlin (1999)

    Google Scholar 

  • Konak A, Coit DW, Smith AE: Multi-objective optimization using genetic algorithms: a tutorial. Reliab Eng Syst Safety 91, 992–1007 (2006)

    Article  Google Scholar 

  • Lin JG: Multiple-objective problems—Pareto-optimal solutions by method of proper equality constraints. IEEE Trans Autom Control 21, 641–650 (1976)

    Article  Google Scholar 

  • Marler RT, Arora JS: Survey of multi-objective optimization methods for engineering. Struct Multidisc Optim 26, 369–395 (2004)

    Article  Google Scholar 

  • Mikhail EM, Bethel JS, McGlone CJ: Introduction to modern photogrammetry. Wiley, New York (2001)

    Google Scholar 

  • Neitzel F (2010) Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation. J Geod. doi:10.1007/s00190-010-0408-0

  • Paris Q (2004) Robust estimators of errors-in-variables models, part I. Working paper no. 04-007, Department of Agricultural and Resource Economics, University of California, Davis

  • Pressl B, Mader C, Wieser M (2010) User-specific web-based route planning. In: Miesenberger K et al (eds) ICCHP 2010, Part I. LNCS, vol 6179. Springer, Berlin, pp 280–287

  • Schaffrin B: A note on constrained total least-squares estimation. Linear Algebra Appl 417, 245–258 (2006). doi:10.1016/j.laa.2006.03.044

    Article  Google Scholar 

  • Schaffrin B: Correspondence, coordinate transformation. Surv Rev 40(307), 102 (2008)

    Google Scholar 

  • Schaffrin B, Felus YA (2008a) Multivariate total least-squares adjustment for empirical affine transformations. In: VI Hotine-Marussi Symposium on Theoretical and Computational Geodesy International Association of Geodesy Symposia, 2008, vol 132, Part III, pp 238–242. doi:10.1007/978-3-540-74584-6_38

  • Schaffrin B, Felus YA: On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms. J Geod 82(6), 373–383 (2008b). doi:10.1007/s00190-007-0186-5

    Article  Google Scholar 

  • Schaffrin B, Felus YA: An algorithmic approach to the total least-squares problem with linear and quadratic constraints. Stud Geophys 53(1), 1–16 (2009). doi:10.1007/s11200-009-0001-2

    Article  Google Scholar 

  • Schaffrin B, Snow K: Total least-squares regularization of Tykhonov type and an ancient racetrack in Corinth. Linear Algebra Appl 432(2010), 2061–2076 (2009). doi:10.1016/j.laa.2009.09.014

    Google Scholar 

  • Schaffrin B, Wieser A: On weighted total least-squares adjustment for linear regression. J Geod 82(7), 415–421 (2008). doi:10.1007/s00190-007-0190-9

    Article  Google Scholar 

  • Shanker AP, Zebker H: Edgelist phase unwrapping algorithm for time series InSAR analysis. J Opt Soc Am A 27, 605–612 (2010)

    Article  Google Scholar 

  • Sonnier DL: A Pareto-optimality based routing and wavelength assignment algorithm for WDM networks. J Comput Sci Colleges Archive 25(5), 118–123 (2010)

    Google Scholar 

  • Tofallis C: Model fitting for multiple variables by minimising the geometric mean deviation. In: Van Huffel, S, Lemmerling, P (eds) Total least squares and errors-in-variables modeling: algorithms, analysis and applications, Kluwer, Dordrecht (2002)

    Google Scholar 

  • Warr PG: Pareto optimal redistribution and private charity. J Public Econ 19(1), 131–138 (1982). doi:10.1016/0047-2727(82)90056-1

    Article  Google Scholar 

  • Werth S, Güntner A (2010) Calibration of a global hydrological model with GRACE data. System Earth via geodetic-geophysical space techniques, advanced technologies in Earth Sciences, 2010, Part 5, pp 417–426. doi:10.1007/978-3-642-10228-8_3

  • Wessel P (2000) Geologic data analysis. http://www.higp.hawaii.edu/~cecily/courses/gg313/DA_book/index.html

  • Wilson PB, Macleod MD (1993) Low implementation cost IIR digital filter design using genetic algorithms. In: IEE/IEEE workshop on natural algorithms in signal processing, pp 1–8

  • Zitler E, Thiele L: Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE Trans Evol Comput 3(4), 257–271 (1999)

    Article  Google Scholar 

  • Zwanzig S (2006) On an application of deconvolution techniques to local linear regression with errors in variables. Department of Mathematics, Uppsala University, U.U.D.M. Report 2006:12

Download references

Author information

Authors and Affiliations

  1. Department of Photogrammetry and Geoinformatics, Budapest University of Technology and Economy, 1521, Budapest, Hungary

    B. Paláncz

  2. Western Australian Centre for Geodesy and The Institute for Geoscience Research, Curtin University, Perth, Australia

    J. L. Awange

  3. Geodetic Institute, Karlsruhe Institute of Technology (KIT), Engler-Strasse 7, 76131, Karlsruhe, Germany

    J. L. Awange

Authors
  1. B. Paláncz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. L. Awange
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. L. Awange.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Paláncz, B., Awange, J.L. Application of Pareto optimality to linear models with errors-in-all-variables. J Geod 86, 531–545 (2012). https://doi.org/10.1007/s00190-011-0536-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00190-011-0536-1

Keywords

Search

Navigation