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Comparison between various regression depth methods and the support vector machine to approximate the minimum number of misclassifications

Author

Listed:
  • Andreas Christmann

    (University of Dortmund)

  • Paul Fischer

    (University of Dortmund)

  • Thorsten Joachims

    (Institute for Autonomous Intelligent Systems)

Abstract

Summary The minimum number of misclassifications achievable with affine hyperplanes on a given set of labeled points is a key quantity in both statistics and computational learning theory. However, determining this quantity exactly is NP-hard, c.f. Höffgen, Simon and van Horn (1995). Hence, there is a need to find reasonable approximation procedures. This paper introduces two new approaches to approximating the minimum number of misclassifications achievable with affine hyperplanes. Both approaches are modifications of the regression depth method proposed by Rousseeuw and Hubert (1999) for linear regression models. Our algorithms are compared to the existing regression depth algorithm (c.f. Christmann and Rousseeuw, 1999) for various data sets. We also used a support vector machine approach, as proposed by Vapnik (1998), as a reference method.

Suggested Citation

  • Andreas Christmann & Paul Fischer & Thorsten Joachims, 2002. "Comparison between various regression depth methods and the support vector machine to approximate the minimum number of misclassifications," Computational Statistics, Springer, vol. 17(2), pages 273-287, July.
    • Handle: RePEc:spr:compst:v:17:y:2002:i:2:d:10.1007_s001800200106
    DOI: 10.1007/s001800200106
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    Citations

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    Cited by:

    1. Pavlo Mozharovskyi & Karl Mosler & Tatjana Lange, 2015. "Classifying real-world data with the $${ DD}\alpha $$ D D α -procedure," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 9(3), pages 287-314, September.
    2. Tatjana Lange & Karl Mosler & Pavlo Mozharovskyi, 2014. "Fast nonparametric classification based on data depth," Statistical Papers, Springer, vol. 55(1), pages 49-69, February.
    3. Nedret Billor & Asheber Abebe & Asuman Turkmen & Sai Nudurupati, 2008. "Classification Based on Depth Transvariations," Journal of Classification, Springer;The Classification Society, vol. 25(2), pages 249-260, November.
    4. Christmann, Andreas & Steinwart, Ingo, 2003. "On robustness properties of convex risk minimization methods for pattern recognition," Technical Reports 2003,15, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    5. Christmann, Andreas, 2004. "Regression depth and support vector machine," Technical Reports 2004,54, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    6. Dyckerhoff, Rainer & Mozharovskyi, Pavlo & Nagy, Stanislav, 2021. "Approximate computation of projection depths," Computational Statistics & Data Analysis, Elsevier, vol. 157(C).
    7. Christmann, Andreas, 2004. "On a strategy to develop robust and simple tariffs from motor vehicle insurance data," Technical Reports 2004,16, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    8. Rousseeuw, Peter J. & Christmann, Andreas, 2003. "Robustness against separation and outliers in logistic regression," Computational Statistics & Data Analysis, Elsevier, vol. 43(3), pages 315-332, July.
    9. Christmann, Andreas & Steinwart, Ingo & Hubert, Mia, 2007. "Robust learning from bites for data mining," Computational Statistics & Data Analysis, Elsevier, vol. 52(1), pages 347-361, September.
    10. Mia Hubert & Peter Rousseeuw & Pieter Segaert, 2017. "Multivariate and functional classification using depth and distance," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 11(3), pages 445-466, September.

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