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Robustness of reweighted Least Squares Kernel Based Regression

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  • Debruyne, Michiel
  • Christmann, Andreas
  • Hubert, Mia
  • Suykens, Johan A.K.

Abstract

Kernel Based Regression (KBR) minimizes a convex risk over a possibly infinite dimensional reproducing kernel Hilbert space. Recently, it was shown that KBR with a least squares loss function may have some undesirable properties from a robustness point of view: even very small amounts of outliers can dramatically affect the estimates. KBR with other loss functions is more robust, but often gives rise to more complicated computations (e.g. for Huber or logistic losses). In classical statistics robustness is often improved by reweighting the original estimate. In this paper we provide a theoretical framework for reweighted Least Squares KBR (LS-KBR) and analyze its robustness. Some important differences are found with respect to linear regression, indicating that LS-KBR with a bounded kernel is much more suited for reweighting. In two special cases our results can be translated into practical guidelines for a good choice of weights, providing robustness as well as fast convergence. In particular a logistic weight function seems an appropriate choice, not only to downweight outliers, but also to improve performance at heavy tailed distributions. For the latter some heuristic arguments are given comparing concepts from robustness and stability.

Suggested Citation

  • Debruyne, Michiel & Christmann, Andreas & Hubert, Mia & Suykens, Johan A.K., 2010. "Robustness of reweighted Least Squares Kernel Based Regression," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 447-463, February.
  • Handle: RePEc:eee:jmvana:v:101:y:2010:i:2:p:447-463
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    References listed on IDEAS

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    1. Hee‐Seok Oh & Doug Nychka & Tim Brown & Paul Charbonneau, 2004. "Period analysis of variable stars by robust smoothing," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 53(1), pages 15-30, January.
    2. Tomaso Poggio & Ryan Rifkin & Sayan Mukherjee & Partha Niyogi, 2004. "General conditions for predictivity in learning theory," Nature, Nature, vol. 428(6981), pages 419-422, March.
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    Cited by:

    1. Huang, Xiaolin & Shi, Lei & Suykens, Johan A.K., 2014. "Asymmetric least squares support vector machine classifiers," Computational Statistics & Data Analysis, Elsevier, vol. 70(C), pages 395-405.
    2. De Brabanter, Kris & Suykens, Johan & De Moor, Bart, 2013. "Nonparametric Regression via StatLSSVM," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 55(i02).

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