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Computing the Least Median of Squares Estimator in Time O(n d)

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Computational Science and Its Applications – ICCSA 2005 (ICCSA 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3480))

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Abstract

In modern statistics, the robust estimation of parameters of a regression hyperplane is a central problem, i. e., an estimation that is not or only slightly affected by outliers in the data. In this paper we will consider the least median of squares (LMS) estimator. For n points in d dimensions we describe a randomized algorithm for LMS running in O(n d) time and O(n) space, for d fixed, and in time O(d 3 (2n)d) and O(dn) space, for arbitrary d.

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References

  1. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  2. Chan, T.M.: Geometric applications of a randomized optimization technique. Discrete & Computational Geometry 22, 547–567 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chan, T.M.: Low-dimensional linear programming with violations. In: IEEE Symposium on Foundations of Computer Science, pp. 570–579 (2002)

    Google Scholar 

  4. Chien, H., Steiger, W.: Some geometric lower bounds. In: ISAAC: 6th International Symposium on Algorithms and Computation, pp. 72–81 (1995)

    Google Scholar 

  5. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press & McGraw-Hill (2001)

    Google Scholar 

  6. Donoho, D.L., Huber, J.: The notion of breakdown point. In: Bickel, P., Doksum, K., Hodges, J.L. (eds.) A Festschrift for Erich L. Lehmann, pp. 157–184 (1985)

    Google Scholar 

  7. Eades, P., McKay, B.: An algorithm for generating subsets of fixed size with a strong minimal change property. Information Processing Letters 19, 131–133 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  8. Edelsbrunner, H., Guibas, L.J.: Topologically sweeping an arrangement. Journal Computer and System Sciences 38, 165–194 (1989); Corrigendum in 42, 249–251 (1991)

    Google Scholar 

  9. Edelsbrunner, H., Souvaine, D.L.: Computing least median of squares regression line and guided topological sweep. Journal of the American Statistical Association 85, 115–119 (1990)

    Article  MATH  Google Scholar 

  10. Erickson, J., Har-Peled, S., Mount, D.M.: On the least median square problem. In: ACM Symposium on Computational Geometry (2004)

    Google Scholar 

  11. Gajentaan, A., Overmars, M.H.: On a class of O(n 2) problems in computational geometry. Computational Geometry 5, 165–185 (1995)

    Google Scholar 

  12. Karp, R.M.: Probabilistic recurrence relations. Journal of the Association of Computer Machinery 41(6), 1136–1150 (1994)

    MATH  MathSciNet  Google Scholar 

  13. Li, Y., Xu, L.-Q., Morrison, G., Nightingale, C., Morphett, J.: Robust panorama from mpeg video. In: IEEE International Conference on Multimedia and Expo 2003 (ICME 2003), pp. 81–84 (2003)

    Google Scholar 

  14. Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: Quantile approximation for robust statistical estimation and k-enclosing problems. International Journal of Computational Geometry and Applications 10, 593–608 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Mount, D.M., Netanyahu, N.S., Romanik, K., Silverman, R., Wu, A.Y.: A practical approximation algorithm for the LMS line estimator. In: Symposium on Discrete Algorithms, pp. 473–482 (1997)

    Google Scholar 

  16. Olson, C.F.: An approximation algorithm for least median of squares regression. Information Processing Letters 63(5), 237–241 (1997)

    Article  MathSciNet  Google Scholar 

  17. Pizarro, O., Singh, H.: Toward large-area mosaicing for underwater scientific applications. IEEE Journal of Oceanic Engineering 28(4), 651–672 (2003)

    Article  Google Scholar 

  18. Plets, H., Vynckier, C.: An analysis of the incidence of the vega phenomenon among main-sequence and post main-sequence stars. Astronomy and Astrophysics 343, 496–506 (1999)

    Google Scholar 

  19. Rafalin, E., Souvaine, D., Streinu, I.: Topological sweep in degenerate cases. In: Mount, D.M., Stein, C. (eds.) ALENEX 2002. LNCS, vol. 2409, pp. 155–165. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  20. Rousseeuw, P.J., Leroy, A.M.: Robust Regression and Outlier Detection. Wiley, Chichester (1987)

    Book  MATH  Google Scholar 

  21. Stromberg, A.J.: Computing the exact least median of squares estimate and stability diagnostics in multiple linear regression. SIAM Journal on Scientific Computing 14(6), 1289–1299 (1993)

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Lehrstuhl Informatik 2, Universität Dortmund, Germany

    Thorsten Bernholt

Authors
  1. Thorsten Bernholt
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Editor information

Editors and Affiliations

  1. Department of Mathematics and Computer Science, University of Perugia, via Vanvitelli, 1, I-06123, Perugia, Italy

    Osvaldo Gervasi

  2. Department of Computer Science, University of Calgary, 2500 University Drive N.W, T2N 1N4, Calgary, AB, Canada

    Marina L. Gavrilova

  3. William Norris Professor, Head of the Computer Science and Engineering Department, University of Minnesota, USA

    Vipin Kumar

  4. Department of Chemistry, University of Perugia, Via Elce di Sotto, 8, P.O. Box, I-06123, Perugia, Italy

    Antonio Laganà

  5. Institute of High Performance Computing, IHCP, 1 Science Park Road, 01-01 The Capricorn, Singapore Science Park II, 117528, Singapore

    Heow Pueh Lee

  6. School of Computing, Soongsil University, Seoul, Korea

    Youngsong Mun

  7. Clayton School of IT, Monash University, 3800, Clayton, Australia

    David Taniar

  8. OptimaNumerics Ltd, P.O. Box, Belfast, United Kingdom

    Chih Jeng Kenneth Tan

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© 2005 Springer-Verlag Berlin Heidelberg

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Bernholt, T. (2005). Computing the Least Median of Squares Estimator in Time O(n d). In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2005. ICCSA 2005. Lecture Notes in Computer Science, vol 3480. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11424758_72

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  • DOI: https://doi.org/10.1007/11424758_72

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-25860-5

  • Online ISBN: 978-3-540-32043-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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