Multiclass Distance-Weighted Discrimination

Abstract

In this article, we extend the binary distance-weighted discrimination (DWD) to the multiclass case. In addition to the usual extensions that combine several binary DWD classifiers, we propose a global multiclass DWD (MDWD) that finds a single classifier that considers all classes at once. Our theoretical results show that MDWD is Fisher consistent, even in the particularly challenging case when there is no dominating class, that is, a class with probability bigger than 0.5. The performance of different multiclass DWD methods is assessed through simulation studies and application to real microarray datasets. Comparison with the support vector machines is also provided. MATLAB implementation of the proposed methods is given in the online supplementary materials.

Key Words:

• Classification
• Fisher consistency
• Optimization

ACKNOWLEDGMENT

The authors thank the editor, the associate editor, and three referees for many helpful comments and suggestions that led to a much improved presentation. Hanwen Huang is supported in part by NIH (National Institutes of Health) grant UL1 RR024148. Yufeng Liu is supported in part by NSF (National Science Foundation) grant DMS-0747575 and NIH grant 5R01CA149569-03. Charles M. Perou is supported in part by NCI (National Cancer Institute) Breast SPORE program P50-CA58223-09A1, NIH grant RO1-CA138255, and the Breast Cancer Research Foundation. D. Neil Hayes is supported in part by Cancer Center Core Support grant CA016086-36 and NCI grant 5-U24-CA126544-03. Michael J. Todd is supported in part by NSF grant DMS-0513337 and ONR (Office of Naval Research) grant N00014-08-1-0036. J. S. Marron is supported in part by NCI grants 5-P50-CA58223-17 and 5-RO1-CA138255-04 and NSF grant DMS-0854908.