The probabilistic constraints in the support vector machine

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Available online at www.sciencedirect.com
Applied Mathematics and Computation 194 (2007) 467–479
www.elsevier.com/locate/amc
The probabilistic constraints in the support vector machine
a, b b
*
Hadi Sadoghi Yazdi , Sohrab Effati , Zahra Saberi
a
Engineering Department, Tarbiat Moallem University of Sabzevar, Sabzevar, Iran
b
Department of Mathematics, Tarbiat Moallem University of Sabzevar, Sabzevar, Iran
Abstract
In this paper, a new support vector machine classifier with probabilistic constrains is proposed which presence proba-
bility of samples in each class is determined based on adistribution function. Noise is caused incorrect calculation of sup-
port vectors thereupon margin can not be maximized. In the proposed method, constraints boundaries and constraints
occurrence have probability density functions which it help for achieving maximum margin. Experimental results show
superiority of the probabilistic constraints support vector machine (PC-SVM) relative to standard SVM.
2007 Elsevier Inc. All rights reserved.
Keywords: Probabilistic constraints; Support vector machine; Margin maximization
1. Introduction
Suchlearningonlyaimsatminimizingtheclassificationerrorinthetrainingphase,anditcannotguarantee
thelowesterrorrateinthetesting phase.Instatisticallearningtheory,thesupportvectormachine(SVM)has
beendevelopedforsolvingthisbottleneck.Supportvectormachines(SVMs)asoriginallyintroducedbyVap-
nikwithintheareaofstatisticallearningtheoryandstructuralriskminimization[1]andcreateaclassifierwith
minimized VC dimension. It have proven to work successfully on wide range applications of nonlinear clas-
sificationandfunctionestimationsuchasopticalcharacterrecognition[2,3],textcategorization[4],facedetec-
tioninimages[5],vehicletrackinginvideosequence[6],nonlinearequalizationincommunicationsystems[7],
and generating of fuzzy rule based system using SVM framework [8,9].
Basically, the support vector machine is a linear machine with some very nice properties. It is not possible
for such a set of training data to construct a separating hyperplane without encountering classification error.
In this case a set of slack variable are used for samples that reduce confidence interval. In this case, it may be
formulatedtoadualproblemformandsoslackvariableisnotappearedinthedualproblemandisconverted
toseparableform.Mainmotivationofthispaperrelyonprobabilisticconstraintsandobtainedresultsinclude
asymmetricmargindependontoprobabilitydensityfunctionofthedataclassesandimportanceofeachsam-
ples in determination of hyperplane parameters.
*
Corresponding author.
E-mail address: sadoghi@sttu.ac.ir (H.S. Yazdi).
0096-3003/$ - see front matter 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2007.04.109