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Leveraged least trimmed absolute deviations

Author

Listed:
  • Nathan Sudermann-Merx

    (BASF SE)

  • Steffen Rebennack

    (Karlsruhe Institute of Technology (KIT))

Abstract

The design of regression models that are not affected by outliers is an important task which has been subject of numerous papers within the statistics community for the last decades. Prominent examples of robust regression models are least trimmed squares (LTS), where the k largest squared deviations are ignored, and least trimmed absolute deviations (LTA) which ignores the k largest absolute deviations. The numerical complexity of both models is driven by the number of binary variables and by the value k of ignored deviations. We introduce leveraged least trimmed absolute deviations (LLTA) which exploits that LTA is already immune against y-outliers. Therefore, LLTA has only to be guarded against outlying values in x, so-called leverage points, which can be computed beforehand, in contrast to y-outliers. Thus, while the mixed-integer formulations of LTS and LTA have as many binary variables as data points, LLTA only needs one binary variable per leverage point, resulting in a significant reduction of binary variables. Based on 11 data sets from the literature, we demonstrate that (1) LLTA’s prediction quality improves much faster than LTS and as fast as LTA for increasing values of k and (2) that LLTA solves the benchmark problems about 80 times faster than LTS and about five times faster than LTA, in median.

Suggested Citation

  • Nathan Sudermann-Merx & Steffen Rebennack, 2021. "Leveraged least trimmed absolute deviations," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 43(3), pages 809-834, September.
    • Handle: RePEc:spr:orspec:v:43:y:2021:i:3:d:10.1007_s00291-021-00627-y
    DOI: 10.1007/s00291-021-00627-y
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    References listed on IDEAS

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    1. Steffen Rebennack & Vitaliy Krasko, 2020. "Piecewise Linear Function Fitting via Mixed-Integer Linear Programming," INFORMS Journal on Computing, INFORMS, vol. 32(2), pages 507-530, April.
    2. Steffen Rebennack & Josef Kallrath, 2015. "Continuous Piecewise Linear Delta-Approximations for Bivariate and Multivariate Functions," Journal of Optimization Theory and Applications, Springer, vol. 167(1), pages 102-117, October.
    3. Lembke B., 1918. "√ a. p," Journal of Economics and Statistics (Jahrbuecher fuer Nationaloekonomie und Statistik), De Gruyter, vol. 111(1), pages 709-712, February.
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    5. Steffen Rebennack & Josef Kallrath, 2015. "Continuous Piecewise Linear Delta-Approximations for Univariate Functions: Computing Minimal Breakpoint Systems," Journal of Optimization Theory and Applications, Springer, vol. 167(2), pages 617-643, November.
    6. Tableman, Mara, 1994. "The asymptotics of the least trimmed absolute deviations (LTAD) estimator," Statistics & Probability Letters, Elsevier, vol. 19(5), pages 387-398, April.
    7. C. Chatzinakos & L. Pitsoulis & G. Zioutas, 2016. "Optimization techniques for robust multivariate location and scatter estimation," Journal of Combinatorial Optimization, Springer, vol. 31(4), pages 1443-1460, May.
    8. Hawkins, Douglas M. & Olive, David, 1999. "Applications and algorithms for least trimmed sum of absolute deviations regression," Computational Statistics & Data Analysis, Elsevier, vol. 32(2), pages 119-134, December.
    9. Dodge, Yadolah, 1997. "LAD Regression for Detecting Outliers in Response and Explanatory Variables," Journal of Multivariate Analysis, Elsevier, vol. 61(1), pages 144-158, April.
    10. Bernholt, Thorsten, 2006. "Robust Estimators are Hard to Compute," Technical Reports 2005,52, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    11. Noam Goldberg & Steffen Rebennack & Youngdae Kim & Vitaliy Krasko & Sven Leyffer, 2021. "MINLP formulations for continuous piecewise linear function fitting," Computational Optimization and Applications, Springer, vol. 79(1), pages 223-233, May.
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