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RESEARCHARTICLE A new weighting factor in combining belief function DeyunZhou1,QianPan1*,GyanChhipi-Shrestha2,XiaoyangLi1,KunZhang1, KasunHewage2,RehanSadiq2 1 SchoolofElectronicsandInformation,NorthwesternPolytechnicalUniversity,Xi’an,China,2 Schoolof Engineering,UniversityofBritishColumbiaOkanagan,Kelowna,BC,Canada *[email protected],[email protected] a1111111111 a1111111111 Abstract a1111111111 a1111111111 a1111111111 Dempster-Shaferevidencetheoryhasbeenwidelyusedinvariousapplications.However, tosolvetheproblemofcounter-intuitiveoutcomesbyusingclassicalDempster-Shafercom- binationruleisstillanopenissuewhilefusingtheconflictingevidences.Manyapproaches basedondiscountedevidenceandweightedaverageevidencehavebeeninvestigatedand havemadesignificantimprovements.Nevertheless,alloftheseapproacheshaveinherent OPENACCESS flaws.Inthispaper,anewweightingfactorisproposedtoaddressthisproblem.First,a Citation:ZhouD,PanQ,Chhipi-ShresthaG,LiX, modifieddissimilaritymeasurementisproposedwhichischaracterizedbybothdistance ZhangK,HewageK,etal.(2017)Anewweighting factorincombiningbelieffunction.PLoSONE12 andconflictbetweenevidences.Second,ameasurementofinformationvolumeofeachevi- (5):e0177695.https://doi.org/10.1371/journal. dencebasedonDengentropyisintroduced.Thentwokindsofweightderivedfromafore- pone.0177695 mentionedmeasurementarecombinedtoobtainanewweightingfactorandaweighted Editor:YongDeng,SouthwestUniversity,CHINA averagemethodbasedonthenewweightingfactorisproposed.Numericalexamplesare Received:December19,2016 usedtoillustratethevalidityandeffectivenessoftheproposedmethod.Intheend,thenew methodisappliedtoareal-lifeapplicationofriverwaterqualitymonitoring,whicheffectively Accepted:May2,2017 identifythemajorlanduseactivitiescontributingtoriverpollution. Published:May25,2017 Copyright:©2017Zhouetal.Thisisanopen accessarticledistributedunderthetermsofthe CreativeCommonsAttributionLicense,which permitsunrestricteduse,distribution,and Introduction reproductioninanymedium,providedtheoriginal authorandsourcearecredited. Dempster-Shafer(D-S)evidencetheoryprovidesareasonableandefficientwaytodealwith theinformationwhichisuncertainanddiscordant.Ithasbeenextensivelyusedinvarious DataAvailabilityStatement:ThedatainExample applicationsrelatedtodecision-makingsuchasinformationfusion[1–3],uncertainreasoning 5couldbeobtainedinthepublishedpaperof "Combinationofsourcesofevidencewithdifferent [4],faultdiagnosis[5],riskanalysis[6–9],cognitivemap[10],targetrecognitionandassocia- discountingfactorsbasedonanewdissimilarity tion[11–13].UnliketheprobabilitytheoryandBayesiantheory,theD-Sevidencetheory measure."Thedataintheapplicationof requiresfewpriorconditionsandknowledgewheninformationisprocessed.Forexample,the identificationofmajorlanduseactivities EvidentialReasoning(ER)algorithmisageneralizedBayesianinferenceprocessandtheER contributingtoriverpollutionarefrom"Riverwater rulerevealsthatthecombineddegreeofjointsupportforapropositionfromtwopiecesof qualityassessmentoftheManaharariverbyusing independentevidenceconstitutestwopartsingeneral[14–16].Whenthereisnoprioriinfor- macroinvertebratesasbiologicalindicators, KathmanduValley"studyareavailableinFigshare mation,theERrulewillreducetotheD-Scombinationrule.Moreover,thecombinationrule (https://doi.org/10.6084/m9.figshare.4980803). oftheD-Sevidencetheorysatisfiessomeofmathematicalproperties,suchascommutativity andassociativity.However,counter-intuitiveresultsmayoccurredbythenormalizationstep Funding:ThisworkwassupportedbytheNational NaturalScienceFoundationofChina(GrantNo. oftheclassicalD-Scombinationrulewhencollectedsourcesofevidencehighlyconflictwith PLOSONE|https://doi.org/10.1371/journal.pone.0177695 May25,2017 1/20 Anewweightingfactorincombiningbelieffunction 61401363)(K.Z.);NaturalSciencesand eachother,aspointedoutbyZadeh[17].TheeffectivenessoftheD-Sevidencetheorywillbe EngineeringResearchCouncilofCanada(NSERC) considerablyreducedbythisdeficiency. CollaborativeResearchandDevelopmentGrants Evidently,itiscrucialtohandletheevidenceswithhighconflict.Inthelastfewyears,many (GrantNo.CRDPJ446638-12)(K.H.);Scienceand researchershavecarriedoutcomprehensiveresearchandhaveappliedaseriesofmodifica- TechnologyonAvionicsIntegrationLaboratoryand tionstotheconventionalevidencecombinationrule[18–24].Ingeneral,theexistingmethods AeronauticalScienceFoundation(GrantNo. 20155153034)(K.Z.);FundamentalResearch canbedividedintotwokindsofsolutions:revisalofthecombinationruleandrevisalofthe FundsfortheCentralUniversities(GrantNo. evidences,fortheproblemofhighlevelconflictevidencefusion.Proponentsofthefirstargu- 3102016AXXX005,3102015BJIIJGZ009)(K.Z.). mentpresentthatillogicalresultsarecausedbyinappropriatedistributionoftheconflictinfor- Thefundershadnoroleinstudydesign,data mation.Therefore,themodifiedmethodsbasedontherevisalofthecombinationrulemainly collectionandanalysis,decisiontopublish,or focusonalteringtheassignmentofconflictinformation[2,18,21,25–27].Amongthem,solu- preparationofthemanuscript. tionsofthetransferablebeliefmodel(TBM)andDezert-Smarandachetheory(DSmT)are Competinginterests:Theauthorshavedeclared morepopular.TheTBMdevelopsamethodtotransferthebasicbeliefassignments(BBAs)to thatnocompetinginterestsexist. probabilities,butthemethodonlycanbeusedinclosedworld[25,28].DSmTextendsthe assignmentuniverseofBBAsfromapowersettoasuperpowersetwhichismorethorough andcomplete,andcorrespondingcombinationmodelsandrulesaredevelopedaswell[27]. Nonetheless,themodifiedcombinationruleshavelimitationsinsomesituation.Forexample, mostofthemodifiedcombinationrulesarenotcommutativeandassociativeandaretime consumingwhendealingwithalargeamountofevidences. Theothermodification,revisaloftheevidences,preprocessesconflictevidencesbefore combinationprocess.ThefavorablemathematicalpropertiesoftheD-Scombinationruleare reservedintheimprovementastheydonotchangetheD-Scombinationrule.Manyrelated workhavebeenproposedtosupportthismodificationmethod[22–24,29–32].Murphy[29] generatesanewevidencebyaveragingNevidenceswithequalweightsandthencombineit withN-1times.Basedonthisidea,Deng[22]proposesaweightedaveragingmethodtoobtain thenewevidence.Besidestheweightedaveragingmethod,thediscountingmethodalsoplays animportantroleinpreprocessingconflictevidences.Theweightingfactorofboththe weightedaveragemethodanddiscountingmethodcanbeidentifiedbyevidencedistance, whichisusuallyusedtodescribetheconflictordissimilarity[33–35].Liu[35]arguesthatthe conflictcoefficientkintheevidencetheoryisinadequatetoreflectthedegreeofconflictand dissimilaritybetweenevidences,andheutilizesatwo-dimensioncell<evidenceconflict,evi- dencedistance>tomeasurethedissimilaritybetweenevidences.Thecellisindeedmorecom- prehensiveandadequatethanthesinglecoefficientkwhendescribingthedissimilarity,butit alsohasintrinsicshortcomingsinpracticalsituation.Forinstance,theconflicttolerance thresholdεislargelysubjectiveanddependsontheperceptionofadecisionmaker.Inaddi- tion,thecellisnotsyncretizedinthecombinationrule.Adissimilaritymeasureisproposed onthebasisofHamacherT-conormfusionrulesgivenbyLiu,whoconsidersnotonlytheevi- denceconflictanddistancebutalsocombinesitinthecombinationruleasadiscount[36]. Nevertheless,therearetwofoldlimitationsassociatedwiththemathematicalmodeling.First, sincetheconflictfactorsonlyusethemaximalsubjectiveprobabilityoftheBBAs,itcannot solvethesituationrelatedtothepropositionswithequalbeliefvalues,whichareinvestigated inSection3.Second,combiningtheevidenceonebyonehasalowconvergencerate. Besidesevidenceconflictanddistance,theevidencevolumeisanothercriteriontomeasure theimportanceofanevidence[37].Ifanevidencehasmoreinformation,itshouldhavea greaterimpactonthefinalaggregatedresult.Dengentropy[38],asageneralizationofShan- nonentropy,canmeasuretheevidenceinformationvolumeundertheframeworkofD-Sevi- dencetheory.Inthispaper,Dengentropyandmodifieddissimilaritymeasureareusedto formanewweightingfactor.Thenthenewcombinationruleofevidenceiscarriedoutbased onthenewweightingfactor,whichhasimprovedtheversatilityandhasafastconvergence rate. PLOSONE|https://doi.org/10.1371/journal.pone.0177695 May25,2017 2/20 Anewweightingfactorincombiningbelieffunction Thispaperisorganizedasfollows.Section2describessomebasicconceptsrelatedtothe D-Sevidencetheoryanddissimilaritymeasure.Section3presentsproblemsofexistingconflict coefficients,especiallythelimitationsofLiu’smethod.Section4investigatesthenewweighting factorofmodifieddissimilarityandDengentropy,andsomeexamplesandanalysisarepre- sentedtoshowthesuperiorityandeffectivenessofproposedmethod.InSection5,thepro- posedmethodisusedinareal-lifeapplicationoftheidentificationofwaterpollutionsources. Finally,conclusionsaredrawninSection6. Preliminaries 2.1BasicsofD-Sevidencetheory Definition1.SupposeΘbeanonemptyfinitesetofmutuallyexclusivealternativesanddefined asframeofdiscernment.SetofallthepossiblesubsetsofΘ,denotedby2Θ,iscalledpowerset. Themappingm:2Θ![0,1]isdefinedasthebasicbeliefassignment(BBA)(alsoknownas basicprobabilityassignment,BPA)[39,40].TheBBAsatisfies P mðAÞ¼1 ð1Þ A(cid:18)Y mð;Þ¼0 ð2Þ wherem(A)reflectsthestrengthofeachofevidencesupportforthepropositionAintheframe ofdiscernment,and;denotestheemptysetofΘ.Aiscalledthefocalelement,ifm(A)>0. Definition2.ThebelieffunctionBel(A)andplausibilityfunctionPl(A)fromaBBAare definedas P BelðAÞ¼ mðBÞ ð3Þ B(cid:18)A P PlðAÞ¼ mðBÞ ð4Þ B\A6¼; whereBel(A)representstheamountofbeliefthatdefinitelysupportA,andthePl(A)couldbe viewedastheamountofbeliefthatpotentiallyplacedinA. Definition3.Letm andm betwoBBAsdefinedonthesameframeΘ.D-Sevidencetheory 1 2 combinationruleisexpressedas 8P < m ðBÞm ðCÞ B\C¼A 1 2 A6¼; mðAÞ¼ 1(cid:0) k ð5Þ : 0 A¼; with P k¼ m ðBÞm ðCÞ ð6Þ B\C¼; 1 2 wherekisnamedasconflictcoefficienttomeasurethedegreeofconflictbetweentwoBBAs. Thecombinationruleisoutofworkwhenk=1. Zadeh[17]presentsafamousexamplethattheD-Scombinationrulewillproduceanunex- pectedresult.SupposeaframeisΘ={A,B,C}andtwoBBAsaregivenas m :m ðAÞ¼0:99;m ðBÞ¼0:01 1 1 1 m :m ðBÞ¼0:01;m ðCÞ¼0:99 2 2 2 bytheD-Scombinationrule,theaggregatedresultisk=0.9999,m(A)=m(C)=0andm(B)= 1,whichisobviouslycounter-intuitiveandunreasonable. PLOSONE|https://doi.org/10.1371/journal.pone.0177695 May25,2017 3/20 Anewweightingfactorincombiningbelieffunction 2.2Jousselmedistance Jousselmedistance[32],consideringboththemassandcardinalityoffocalelementsofeach BBA,iscommonlyusedasthemeasureofdissimilarity. Definition4.Letm andm betwoBBAsonthesameframeΘ,containingNmutually 1 2 exclusiveandexhaustivepropositions.TheJousselmedistancebetweenm andm aredefined 1 2 as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dm1 ¼ 0:5(cid:3)ðkm k2þkm k2(cid:0) 2hm ;m iÞ ð7Þ Jm2 1 2 1 2 wherekm k2=hm ,m i,km k2=hm ,m iandhm ,m iisgivenby 1 1 1 2 2 2 1 2 P P jA \Bj hm ;m i¼ 2N 2Nm ðAÞm ðBÞ i j ð8Þ 1 2 i¼1 j¼1 1 i 2 j jA [Bj i j withA andB aretheelementsofthepowerset2Θ.|A \B|and|A [B|denotethecardinality i j i j i j intersectionsetandunionsetofA andB. i j 2.3Probabilistic-baseddistance SincetheprobalilistictransformationhasanabilitytoconvertaBBAfromthefocalelements intoaprobabilitymeasureofdistinctatomic,itprovidesaprobabilistic-baseddistancetomea- surethedissimilarityoftwoevidences[41]. Definition5.LetmbeaBBAonaframeΘ,andtheprobabilisticexpressionofasingleton elementBinΘcouldbeobtainedbypignisticprobabilityfunction P mðAÞ BetP ðBÞ¼ ð9Þ m A22Y;B(cid:18)A jAj where|A|isthecardinalityofpropositionA.If|A|=1,thenB=AandBetP(B)=BetP(A)=m (A). Definition6.Letm andm betwoBBAsonthesameframeΘandletBetP andBetP be 1 2 m1 m2 theresultsofpignisticprobabilitytransformationofm andm ,theprobabilistic-baseddis- 1 2 tancedifBetPm2 isdefinedas m1 difBetPm2 ¼max ðjBetP ðAÞ(cid:0) BetP ðAÞjÞ ð10Þ m1 A2Y m1 m2 andtheMurkowskidistance[36]proposedbyLiuisdefinedas P distPm2 ¼ 0:5(cid:3)ðjBetP ðAÞ(cid:0) BetP ðAÞjÞ ð11Þ m1 Ai2Y m1 i m2 i 2.4Combinatorialdissimilaritymeasure Somecompounddissimilaritymeasuresarepresentedbasedontheconflictcoefficient,evi- dencedistanceandprobabilistic-baseddistance. Definition7.Letm andm betwoBBAsonthesameframeΘ,andacombinatorialdissimi- 1 2 laritymeasurebasedontheconflictcoefficientkm2andprobabilistic-baseddistancedifBetPm2 m1 m1 isdefinedas cfm1 ¼hkm2;difBetPm2i ð12Þ m2 m1 m1 m1andm2areinconflict,iffbothkmm21 >εanddifBetPmm12 >ε.ε2[0,1]denotesthethreshold ofconflicttolerance,andidentifiedaccordingtodifferentapplications[35]. PLOSONE|https://doi.org/10.1371/journal.pone.0177695 May25,2017 4/20 Anewweightingfactorincombiningbelieffunction Definition8.Letm andm betwoBBAsonthesameframeΘ,andacombinatorialdissimi- 1 2 laritymeasure[42]basedontheconflictcoefficientkm2andJousselmedistancedm2 isdefined m1 Jm1 as 1 k ¼ (cid:2)ðkm2þdm2Þ ð13Þ d 2 m1 Jm1 Definition9.Letm andm betwoBBAsonthesameframeΘandletBetP andBetP be 1 2 m1 m2 theresultsofpignisticprobabilitytransformationofm andm .Thenacombinatorialdissimi- 1 2 laritymeasurebasedontheHamacherT-conormfusionrules[43]isdefinedas distPm2 þConfPm2 DismPm2≜TðdistPm2;ConfPm2Þ¼ m1 m1 ð14Þ m1 m1 m1 1þdistPm2 (cid:1)ConfPm2 m1 m1 whereConfPm2 denotestheconflictcoefficientbasedonthepignisticprobability, m1 ( 0; if BetPðXmaxÞ\BetPðXmaxÞ6¼; ConfPm2 ¼ m1 m2 ð15Þ m1 BetPðXmaxÞ(cid:1)BetPðXmaxÞ; otherwise m1 m2 whereBetPðXmaxÞ¼arg max BetP ðxÞ;i¼1;2 m1 x2Y mi 2.5Dengentropy Dengentropy,asageneralizationofShannonentropy,provideasolutiontomeasuretheinfor- mationvolumeofaBBA.ItisobservedthattheDengentropyandShannonentropycorre- spondtoanuncertaindegreeofmeasurement[38]. Definition10.LetmbeaBBAontheframeΘandtheDengentropyofmisdefinedas P mðAÞ E ¼(cid:0) mðAÞlog i ð16Þ d i i 2jAij(cid:0) 1 whereA isapropositioninBBAm,and|A|isthecardinalityofA.TheDengentropywill i i i becomeidenticaltoShannonentropyif|A|=1,thatis i P mðAÞ P E ¼(cid:0) mðAÞlog i ¼(cid:0) mðAÞlogmðAÞ ð17Þ d i i 2jAij(cid:0) 1 i i i LimitationsofexitingdissimilaritymeasurementsbetweenBBAs Example1.Letm ,m andm bethreeBBAsonthesameframeΘwithfourpropositionsΘ= 1 2 3 {A ,A ,A ,A }.ThethreeBBAsaregivenas 1 2 3 4 m : m ðA Þ¼m ðA Þ¼m ðA Þ¼m ðA Þ¼0:25 1 1 1 1 2 1 3 1 4 m : m ðA Þ¼m ðA Þ¼m ðA Þ¼m ðA Þ¼0:25 2 2 1 2 2 2 3 2 4 m : m ðA Þ¼m ðA Þ¼m ðA Þ¼1=3 3 3 1 3 2 3 3 wecangettheconflictcoefficientskm2 ¼0:75andkm3 ¼0:67byusingEq(6)betweenthe m1 m1 BBAs.Theresultshowsthatthedegreeofconflictbetweenm andm isbiggerthanthedegree 1 2 ofconflictbetweenm andm ,andtheybothareinrelativehighconflict.Infact,thereisno 1 3 conflictintuitivelybetweenm andm becausetheyarethesame.ByusingtheEq(10)andEq 1 2 PLOSONE|https://doi.org/10.1371/journal.pone.0177695 May25,2017 5/20 Anewweightingfactorincombiningbelieffunction (12),wecangetthedifBetPmm12 ¼0andcf(m1,m2)=h0.75,0i,whichillustratesthatm1andm2 areconsistentandmeasurementofcoefficientkcannotmeasurethedegreeofconflictbetween theevidencesinthissituation.AlthoughthecombinedmeasurementimpliesthattheD-S combinationruleshouldbeused,itcannotconcludethathowmuchtheerrorwillbeconduct byusingthecombinationrule.Therefore,thecombinationrulehasalimitationintermsof providinganexplicitexpressionandcannotbeuseddirectlyinthecombinationrule. Example2.Letm andm betwoBBAsonthesameframeΘ={A ,A ,...,A },suchthat 1 2 1 2 2n 1 m : m ðA Þ¼m ðA Þ¼(cid:1)(cid:1)(cid:1)¼m ðA Þ¼ 1 1 1 1 2 1 n n (cid:0) (cid:1) (cid:0) (cid:1) 1 m : m A ¼m A ¼(cid:1)(cid:1)(cid:1)¼m ðA Þ¼ 2 2 nþ1 2 nþ2 2 2n n Itisobviousthatthem andm aretotallycontrarytoeachotherastheysupportthedifferent 1 2 propositions.Thedifferentdissimilaritiesbetweenm andm aredisplayedinFig1. 1 2 FromFig1,itisevidentthatthevaluesofthed,k ,anddifBetPare1,whenn=1,which J d areintuitive.Butwhenn>1,thevaluesofd anddifBetPtendto0,andk tendsto0.6,in- J d dicatingthatm andm aregettingcloserandlessconflictwiththeincreaseofn,whichare 1 2 counter-intuitiveandabnormal.OnlytheDismPkeeps1withtheincreasewithn,meaning thatthem andm aretotallyindisagreementwitheachother.Therefore,d,k ,anddifBetP 1 2 J d cannotbeusedasmeasurementofthedissimilaritybetweenBBAsinthisexample. Fig1.Differentdissimilaritymeasurements. https://doi.org/10.1371/journal.pone.0177695.g001 PLOSONE|https://doi.org/10.1371/journal.pone.0177695 May25,2017 6/20 Anewweightingfactorincombiningbelieffunction SinceDismPconsidersnotonlythedistancebutalsotheconflictbetweenBBAs,themea- surementbasedontheHamacherT-conormfusionrulesprovidesageneralmethodofthedis- similarity.However,ithasdificiencyasshowninExample3. Example3.Letm andm betwoBBAsonthesameframeofdiscernmentΘ={A ,A ,..., 1 2 1 2 A }.Fornotationconciseness1,2,andsoforthhavebeenusedtodenoteA ,A ,andsoforth 20 1 2 intheframe.ThetwopairsofBBAsareshownas 1stPair: m : m ð2;3;4Þ¼0:05;m ð7Þ¼0:05;m ðYÞ¼0:1;m ðDÞ¼0:8 1 1 1 1 1 m m ð1;2;3;4;5Þ¼1 2: 2 2ndPair: m : m ð2;3;4Þ¼0:05;m ð7Þ¼0:05;m ðYÞ¼0:1;m ðDÞ¼0:8 1 1 1 1 1 m : m ð1;2;3;4;5Þ¼0:5; m ð6;7;8;9;10Þ¼0:5 2 2 2 wheretheΔisasubsetofΘ.Thisexampleconsiders20casesofthesubsetΔ,whichincreases byaddinganewelementateachcasefromΔ={1}toΔ={1,2,...,20}.Thecomparisonofthe dissimilaritymeasurementsbetweenm andm ofthetwopairsareshowninFigs2and3 1 2 respectively. FromFigs2and3,itcanbeseenthatthedistPandDismPareverycloseandfollowthe sametrend,sincetheDismParemainlydecidedbythedistPandConfP.AscalculatedinEq (15),themaximalpignisticprobabilitiesinbothm andm alwayshaveintersection.SoConfP 1 2 issmallwhentheΔincreasesfrom{1}to{1,2,...,20}.However,itisquitedistinctsituationof thetwopairsofBBAs.Inthe1stpair,bothBBAsdistributetheirmajorbelieftothesameele- mentswhenthecasesfrom1to6,whichcausethattheConfPkeeps0.Thisisreasonableasthe m onlyhasonefocalelementm (1,2,3,4,5)=1whichcorrespondstoclassicalconflictcoeffi- 2 2 cientk.Inthe2ndpair,m hastwoequalfocalelementsm (1,2,3,4,5)=0.5andm (6,7,8,9,10)= 2 2 2 0.5.Asthecasefrom1to5,thereshouldbeanotabledissimilaritybetweenm andm ,which 1 2 isshownaskinFig3.Nevertheless,thepignisticprobabilitytransformationdividesthebelief equallytoeachsinglepropositionasBetP ð1Þ¼BetP ð2Þ¼(cid:1)(cid:1)(cid:1)¼BetP ð10Þ¼0:1,indi- m2 m2 m2 catingthattheConfPconsidersthedissimilarityas0.Therefore,thedissimilaritymeasuresof ConfPandDismPareillogicalinthissituation.Althoughtheclassicalconflictcoefficientkcould depictsthedissimilarityfromcases1to5,itcannotreflectthevarietyofdivergencedegreeas thecaseincreases.NeitherdoesthedifBetP. Combiningbelieffunctionwithanewweightingfactor 4.1Amodifieddissimilaritymeasure Inthissection,amodifieddissimilaritymeasureisproposedwhichisbasedontheHamacher T-conormfusionrulestodescribethedissimilaritybetweenBBAs.Thedissimilaritymeasure- mentbasedonHamacherT-conormrulessatisfytwoimportantpropertiesofcommutativity andmonotonicity.Thecommutativitycouldensurethatthedissimilaritymatrixissymmetri- calandnomatterthefusionorderoftwoevidencesis,theirdissimilarityiscoincident.The monotonicityprovidesthatdissimilaritymeasurementhassinglevariationtrendinaspecific interval,whichiseasytocomparethedissimilaritybetweenevidences. PLOSONE|https://doi.org/10.1371/journal.pone.0177695 May25,2017 7/20 Anewweightingfactorincombiningbelieffunction Fig2.Comparisonofdissimilaritymeasuresofthe1stpairevidence. https://doi.org/10.1371/journal.pone.0177695.g002 Definition11.Letm andm betwoBBAsonthesameframeΘ.Themodifieddissimilarity 1 2 measureisdefinedas (cid:16) (cid:17) distPm2 þkm2 MDismPm2≜T distPm2;km2 ¼ m1 m1 ð18Þ m1 m1 m1 1þdistPm2 (cid:1)km2 m1 m1 wherekm2 istheclassicalconflictcoefficient. m1 P km2 ¼ m ðAÞm ðAÞ ð19Þ m1 Ai\Aj¼; 1 i 2 j Themodifieddissimilaritystillsatisfiesthebasicpropertiesofcommutativityand monotonicity: (1)Commutativity: MDismPðx;yÞ¼MDismPðy;xÞ ð20Þ (2)Monotonicity: 0(cid:20)MDismPðx;yÞ(cid:20)MDismPðx0;yÞ(cid:20)MDismPðx0;y0Þ(cid:20)1 ð21Þ where0(cid:20)x(cid:20)x0(cid:20)1and0(cid:20)y(cid:20)y0(cid:20)1. PLOSONE|https://doi.org/10.1371/journal.pone.0177695 May25,2017 8/20 Anewweightingfactorincombiningbelieffunction Fig3.Comparisonofdissimilaritymeasuresofthe2ndpairevidence. https://doi.org/10.1371/journal.pone.0177695.g003 Themodifieddissimilaritymeasurementconsistsofadistancecoefficientandaconflict coefficientbetweentwoevidences.Asboththedistancecoefficientandtheconflictcoefficient liein[0,1],themodifiedmeasurementislargerthanitseithercomponents.Theevidences havelargedistanceandhighconflictwiththemajorityofotherevidenceswouldhavealarger dissimilaritymeasurementandviceversa. Itisobviousthatthemodifieddissimilaritymeasurereplacestheconflictcoefficient ConfPm2 withkm2.TheConfPm2 impliesthatthemainconflictresultsfromdiscordantproposi- m1 m1 m1 tionswhicharestronglysupportedbytwoBBAsrespectively.However,theConfPm2 cannot m1 handlethesituationthatoneBBAhasseveralpropositionswithequalbelief.Asweseeit,the conflictcoefficientshouldinvolveallconflictsexistedbetweenBBAsnomatterhowsmallthe extentofconflictsis.Furthermore,theMDismPnotonlymaintainsgoodfeaturesbutalso makesupforshortcomingsofDismP. Example4.ConsideringtwopairsofBBAsfromExample3withtheproposeddissimilarity measureofMDismP,theresultsareplottedinFigs4and5. Fortheresultsofthe1stpairillustratedinFig4,thedistP,DismPandMDismPareidentical. ThelinesshowavariationtendencyfromahighdissimilaritywhenΔ={1}totheminimum dissimilaritywhenΔ={1,2,3,4,5}andincreaseagainasΔincludesmoreelements.Thisis becausethem onlyhasonepropositionandwhenΔ={1,2,3,4,5},thepropositionswiththe 2 PLOSONE|https://doi.org/10.1371/journal.pone.0177695 May25,2017 9/20 Anewweightingfactorincombiningbelieffunction Fig4.Comparisonofdissimilarityofthe1stpairevidence. https://doi.org/10.1371/journal.pone.0177695.g004 maximumbeliefoftwoBBAsareaccordant.InFig5,whenm hastwopropositionswith 2 equalbeliefvalue,theresultsaredifferent.TheMDismPhasabiggervaluethanthedistPand DismPwhenΔfrom{1}to{1,2,3,4,5}.ThedissimilarityvalueofMDismPisnear0.8before cases6,whichmeansthetwoBBAsareincompatiblewitheachother.Butthedissimilarityval- uesofdistPandDismParelessthan0.6,whichseemsunreasonable.Basedontheanalysisof theaboveexamples,aconclusioncanbedrawnthatthemodifieddissimilaritymeasure MDismPcanefficientlyreflectsthedegreeofdissimilaritybetweenBBAs. 4.2Weightingfactors Inthissection,weproposeanovelmethodtodeterminetheweightingfactorsamongBBAs basedonthemodifieddissimilaritymeasureandDengentropy. Theweightdeterminationsarebasedontheprinciplethatifanevidenceissupportedby greaternumberofevidences,thispieceofevidenceshouldbemoreimportantandhavelarge effectonthefinalcombinationresults.Moreover,ifanevidencehasconsiderableinformation, italsoshouldbeweightedmore[37]. SupposeNevidences{m ,m ,...,m }areinthesameframeofdiscernmentΘandthe 1 2 N weightofeachevidenceismadeupofthedegreeofsimilarityandinformationvolume.The PLOSONE|https://doi.org/10.1371/journal.pone.0177695 May25,2017 10/20