Advertisement

EIV Models and Pareto Optimalitity

  • Joseph L. Awange
  • Béla Paláncz
Chapter

Abstract

In some geospatial 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 is often the case in multiobjective problems. Such is often 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 Chapter, 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 to 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. 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.

Keywords

Pareto Front Very Long Baseline Interferometry Multiobjective Problem Local Optimization Method Local Optimization Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 8.
    Akyilmaz O (2007) Total least squares solution of coordinate transformation. Surv Rev 39:68–80CrossRefGoogle Scholar
  2. 15.
    Akyilmaz O (2007) Total least squares solution of coordinate transformation. Surv Rev 39(303):68–80(13). doi:10.1179/003962607X165005CrossRefGoogle Scholar
  3. 16.
    Angus D, Deller A (2008) Computational intelligence in radio astronomy: using computational intelligence techniques to tune geodesy models. In: Li X et al (eds) SEAL 2008. LNCS, vol 5361. Springer, Berlin/Heidelberg, pp 615–624Google Scholar
  4. 108.
    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 (ed) Geodetic reference frames, international association of geodesy symposia, vol 134. doi:10.1007/978-3-642-00860-3_32Google Scholar
  5. 109.
    Censor Y (1977) Pareto optimality in multiobjective problems. Appl Math Optim 4:41–59CrossRefGoogle Scholar
  6. 110.
    Coello CA (2003) A comprehensive survey of evolutionary-based multiobjective optimization techniques. Knowl Inf Syst 1(3):269–308CrossRefGoogle Scholar
  7. 118.
    Gau C-Y, Stadtherr MA (2002) Deterministic global optimization for error-in-variables parameter estimation. AIChE J 48(6):1192–1197CrossRefGoogle Scholar
  8. 121.
    Chernov N, Lesort C (2008) Least squares fitting of circles. J Comput Vis 80:167–188CrossRefGoogle Scholar
  9. 152.
    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_8Google Scholar
  10. 156.
    Esposito WR, Floudas CA (1998) Global parameter estimation in nonlinear algebraic models via global optimization. Comput Chem Eng 22(1):24, 213–220(8)Google Scholar
  11. 157.
    Ehrgott M (2005) Multicriteria optimization. Springer, Berlin/HeidelbergGoogle Scholar
  12. 159.
    Felus YA, Schaffrin B (2005) Performing similarity transformations using the errors-in-variable model. In: ASPRS Annual Conference, BaltimoreGoogle Scholar
  13. 160.
    Fišerová E, Hron K (2010) Total least squares solution for compositional data using linear models. J Appl Stat 37(7):1137–1152. doi:10.1080/02664760902914532CrossRefGoogle Scholar
  14. 183.
    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, 9–11 Dec 2009, pp 695–700Google Scholar
  15. 184.
    Gruna R (2010) Evolutionary multiobjective optimization, Wolfram Demonstration Project. www.wolfram.com Google Scholar
  16. 185.
    Gander W, Golub GH, Strebel R (1994) Least-squares fitting of circles and ellipses. BIT No 43:558–578CrossRefGoogle Scholar
  17. 191.
    Golub GH, Van Loan CF (1980) An analysis of the total least-squares problem. SIAM J Numer Anal 17(6):883–893CrossRefGoogle Scholar
  18. 194.
    Golub GH, Van Loan (1980) An analysis of the total least-squares problem. SIAM J Numer Anal 17(6):883–893CrossRefGoogle Scholar
  19. 247.
    Hochman HM, Rodgers JD (1969) Pareto optimal redistribution. Am Econ Rev 59(4):542–557. Part 1Google Scholar
  20. 248.
    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, Changsha, vol 3, pp 359–362Google Scholar
  21. 249.
    Huband S, Hingston P, Barone L, While L (2006) A review of multiobjective test problems and scalable test problem toolkit. IEEE Trans Evol Comput 10(N0):477–506CrossRefGoogle Scholar
  22. 293.
    Knowles J, Corne D, Deb K (eds) (2008) Multiobjective problem solving from nature. Springer, Berlin/HeidelbergGoogle Scholar
  23. 294.
    Koch KR (1999) Parameter estimation and hypothesis testing in linear models. Springer, Berlin/HeidelbergCrossRefGoogle Scholar
  24. 295.
    Konak A, Coit DW, Smith AE (2006) Multi-objective optimization using genetic algorithms: a tutorial. Reliab Eng Syst Saf 91:992–1007. ElsevierGoogle Scholar
  25. 317.
    Lin JG (1976) Multiple-objective problems – Pareto-optimal solutions by method of proper equality constraints. IEEE Trans Autom Control AC-21:641–650Google Scholar
  26. 352.
    Mikhail EM, Bethel JS, McGlone CJ (2001) Introduction to modern photogrammetry. Wiley, New York/ChichesterGoogle Scholar
  27. 384.
    Nievergelt Y (2000) A tutorial history of least squares with applicatons to astronomy and geodesy. J Comput Appl Math 121:37–72CrossRefGoogle Scholar
  28. 389.
    Neitzel F (2010) Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation. J Geod. doi:10.1007/s00190-010-0408-0Google Scholar
  29. 390.
    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/Heidelberg, pp 280–287Google Scholar
  30. 409.
    Petriciolet AB, Bravo-Sánchez UI, Castillo-Borja F, Zapiain-Salinas JG, Soto-Bernal (2007) The performance of simulated annealing in parameter estimation for vapor-liquid equilibrium modeling. Braz J Chem Eng 24(01):151–162Google Scholar
  31. 442.
    Schaffrin B, Snow K (2009) Total least-squares regularization of Tykhonov type and an ancient racetrack in Corinth. Linear Algebra Appl 432(2010):2061–2076. doi:10.1016/j.laa.2009.09.014CrossRefGoogle Scholar
  32. 443.
    Schaffrin B, Felus YA (2009) An algorithmic approach to the total least-squares problem with linear and quadratic constraints. Studia Geophysica 53(1):1–16. doi:10.1007/s11200-009-0001-2CrossRefGoogle Scholar
  33. 444.
    Schaffrin B, Wieser A (2008) On weighted total least-squares adjustment for linear regression. J Geod 82(7):415–421. doi:10.1007/s00190-007-0190-9CrossRefGoogle Scholar
  34. 445.
    Schaffrin B, Felus YA (2008) 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_38Google Scholar
  35. 446.
    Schaffrin B, Felus YA (2008) On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms. J Geod 82(6):373–383. doi:10.1007/s00190-007-0186-5CrossRefGoogle Scholar
  36. 447.
    Schaffrin B (2006) A note on constrained total least-squares estimation. Linear Algebra Appl 417(1):245–258. doi:10.1016/j.laa.2006.03.044CrossRefGoogle Scholar
  37. 452.
    Shanker AP, Zebker H (2010) Edgelist phase unwrapping algorithm for time series InSAR analysis. J Opt Soc Am A 27:605–612CrossRefGoogle Scholar
  38. 453.
    Sonnier DL (2010) A Pareto-optimality based routing and wavelength assignment algorithm for WDM networks. J Comput Sci Coll Arch 25(5):118–123Google Scholar
  39. 488.
    See Ref. [487]Google Scholar
  40. 494.
    van Huffel S, Lemmerling P (eds) (2002) Total least squares techniques and errors-in-variables modeling analysis, algorithms and applications. Kluwer Academic, DordrechtGoogle Scholar
  41. 506.
    Warr PG (1982) Pareto optimal redistribution and private charity. J Public Econ 19(1):131–138. doi:10.1016/0047-2727(82)90056-1CrossRefGoogle Scholar
  42. 507.
    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, 417–426. doi:10.1007/978-3-642-10228-8_3Google Scholar
  43. 508.
    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, Chelmsford, pp 4/1–4/8Google Scholar
  44. 552.
    Zitler E, Thiele L (1999) Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE Trans Evol Comput 3(4):257–271CrossRefGoogle Scholar
  45. 554.
    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 2006:12Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Joseph L. Awange
    • 1
  • Béla Paláncz
    • 2
  1. 1.Curtin UniversityPerthAustralia
  2. 2.Budapest University of Technology and EconomicsBudapestHungary

Personalised recommendations