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A practical approximation algorithm for the LTS estimator

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  • Mount, David M.
  • Netanyahu, Nathan S.
  • Piatko, Christine D.
  • Wu, Angela Y.
  • Silverman, Ruth

Abstract

The linear least trimmed squares (LTS) estimator is a statistical technique for fitting a linear model to a set of points. It was proposed by Rousseeuw as a robust alternative to the classical least squares estimator. Given a set of n points in Rd, the objective is to minimize the sum of the smallest 50% squared residuals (or more generally any given fraction). There exist practical heuristics for computing the linear LTS estimator, but they provide no guarantees on the accuracy of the final result. Two results are presented. First, a measure of the numerical condition of a set of points is introduced. Based on this measure, a probabilistic analysis of the accuracy of the best LTS fit resulting from a set of random elemental fits is presented. This analysis shows that as the condition of the point set improves, the accuracy of the resulting fit also increases. Second, a new approximation algorithm for LTS, called Adaptive-LTS, is described. Given bounds on the minimum and maximum slope coefficients, this algorithm returns an approximation to the optimal LTS fit whose slope coefficients lie within the given bounds. Empirical evidence of this algorithm’s efficiency and effectiveness is provided for a variety of data sets.

Suggested Citation

  • Mount, David M. & Netanyahu, Nathan S. & Piatko, Christine D. & Wu, Angela Y. & Silverman, Ruth, 2016. "A practical approximation algorithm for the LTS estimator," Computational Statistics & Data Analysis, Elsevier, vol. 99(C), pages 148-170.
    • Handle: RePEc:eee:csdana:v:99:y:2016:i:c:p:148-170
    DOI: 10.1016/j.csda.2016.01.016
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    References listed on IDEAS

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    1. Hofmann, Marc & Kontoghiorghes, Erricos John, 2010. "Matrix strategies for computing the least trimmed squares estimation of the general linear and SUR models," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 3392-3403, December.
    2. Hawkins, Douglas M., 1994. "The feasible solution algorithm for least trimmed squares regression," Computational Statistics & Data Analysis, Elsevier, vol. 17(2), pages 185-196, February.
    3. Rousseeuw, Peter J., 1991. "A diagnostic plot for regression outliers and leverage points," Computational Statistics & Data Analysis, Elsevier, vol. 11(1), pages 127-129, January.
    4. Torti, Francesca & Perrotta, Domenico & Atkinson, Anthony C. & Riani, Marco, 2012. "Benchmark testing of algorithms for very robust regression: FS, LMS and LTS," Computational Statistics & Data Analysis, Elsevier, vol. 56(8), pages 2501-2512.
    5. Mount, David M. & Netanyahu, Nathan S. & Romanik, Kathleen & Silverman, Ruth & Wu, Angela Y., 2007. "A practical approximation algorithm for the LMS line estimator," Computational Statistics & Data Analysis, Elsevier, vol. 51(5), pages 2461-2486, February.
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