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Vague集间新距离的分析和定义
Introduction
Distancemeasurementisacrucialaspectofmanyfieldssuchasstatisticsandpatternrecognition.Distancemeasuresareusedtoevaluatethesimilaritybetweentwoobjectsortoclustersimilarobjectstogether.Inmachinelearning,thechoiceofdistancemeasureisanimportantdecisionasitcangreatlyimpacttheperformanceofthemodel.Vague-settheoryisarelativelynewareaofstudythatgeneralizestheclassicalsettheorybyintroducingtheconceptofvaguemembership.Inthispaper,weexploretheconceptofvague-settheoryandhowitcanbeusedtodefineanewdistancemeasurebetweensets.
Vague-SetTheory
Vague-settheorywasintroducedbyLotfiZadehin1997asageneralizationofclassicalsettheory.Inclassicalsettheory,asetisdefinedbyitsmembershipfunction,whichmapseachelementtoabinaryvalueofeither0or1.Ifanelementbelongstotheset,itsmembershipvalueis1,andifitdoesnotbelongtotheset,itsmembershipvalueis0.Invague-settheory,themembershipfunctionisafuzzyfunctionthatmapselementstoavaluebetween0and1thatrepresentsthedegreeofmembership.Thus,invague-settheory,anelementcanbelongtoasettoacertaindegree,ratherthanbeingeitherinthesetornotintheset.
Vague-SetDistanceMeasures
Classicalsettheoryprovidesseveralwell-knowndistancemeasuressuchastheJaccarddistance,Euclideandistance,andHammingdistance.However,thesedistancemeasuresarenotsuitableforvaguesetsastheydonottakeintoaccountthedegreeofmembership.Therefore,severaldistancemeasureshavebeenproposedforvaguesets.Inthispaper,wefocusontheconceptofthenewvague-setdistancemeasurebasedonvagueinter-distance.
VagueInter-distance
Thevagueinter-distancebetweentwosetsAandBisdefinedasthesumofthevaguedistancesbetweentheelementsofAandBthatdonotbelongtotheintersectionofAandB.
Mathematically,thevagueinter-distancebetweenAandBcanbegivenas:
VID(A,B)=∑(x∈A∩B)'μA(x)'μB(x)
whereμA(x)andμB(x)arethemembershipvaluesoftheelementxinAandB,respectively,andthesymbol'denotesnegation.ThevaguedistancebetweentheelementsofAandBthatbelongtotheintersectionofAandBisnotconsideredinthecalculationofthevagueinter-distanceastheseelementsarecomm