A tutorial on [nu]-support vector machines

APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRYAppl. Stochastic Models Bus. Ind., 2005; 21:111–136Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/asmb.537A tutorial on n-support vector machines1 1 2, ,y,z.Pai-Hsuen Chen , Chih-Jen Lin and Bernhard Scholkopfo *1Department of Computer Science and Information Engineering, National Taiwan University, Taipei 106, Taiwan2.Max Planck Institute for Biological Cybernetics, Tuubingen, GermanySUMMARYWe briefly describe the main ideas of statistical learning theory, support vector machines (SVMs), andkernel feature spaces. We place particular emphasis on a description of the so-called n-SVM, includingdetails of the algorithm and its implementation, theoretical results, and practical applications. Copyright# 2005 John Wiley & Sons, Ltd.KEY WORDS: n-support vector machines; support vector regression; support vector implementation;statistical learning theory; positive definite kernels1. AN INTRODUCTORY EXAMPLESuppose we are given empirical dataðx ;yÞ;...;ðx ;yÞ2Xf 1gð 1Þ1 1 m mHere, the domainX is some non-empty set that the patterns x are taken from; the y are calledi ilabels or targets.Unlessstatedotherwise,indicesi andj willalwaysbeunderstoodtorunoverthetrainingset,i.e. i;j¼1;...;m:Note that we have not made any assumptions on the domainX other than it being a set. Inordertostudytheproblemoflearning,weneedadditionalstructure.Inlearning, ...