Title
Robustness of reweighted least squares kernel based regression Robustness of reweighted least squares kernel based regression
Author
Faculty/Department
Faculty of Sciences. Mathematics and Computer Science
Publication type
article
Publication
New York, N.Y. ,
Subject
Mathematics
Source (journal)
Journal of multivariate analysis. - New York, N.Y.
Volume/pages
101(2009) :2 , p. 447-463
ISSN
0047-259X
ISI
000272526300014
Carrier
E
Target language
English (eng)
Full text (Publishers DOI)
Affiliation
University of Antwerp
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.
E-info
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