DS证据理论合成算法Word格式.docx
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DS证据理论合成算法Word格式.docx
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Abstract
Dempster-Shafertheoryoffersanalternativetotraditionalprobabilistictheoryforthemathematicalrepresentationofuncertainty.Thesignificantinnovationofthisframeworkisthatitallowsfortheallocationofaprobabilitymasstosetsorintervals.Dempster-Shafertheorydoesnotrequireanassumptionregardingtheprobabilityoftheindividualconstituentsofthesetorinterval.Thisisapotentiallyvaluabletoolfortheevaluationofriskandreliabilityinengineeringapplicationswhenitisnotpossibletoobtainaprecisemeasurementfromexperiments,orwhenknowledgeisobtainedfromexpertelicitation.Animportantaspectofthistheoryisthecombinationofevidenceobtainedfrommultiplesourcesandthemodelingofconflictbetweenthem.ThisreportsurveysanumberofpossiblecombinationrulesforDempster-Shaferstructuresandprovidesexamplesoftheimplementationoftheserulesfordiscreteandinterval-valueddata.
ACKNOWLEDGEMENTS
TheauthorswishtothankBillOberkampf,JonHelton,andMartyPilchofSandiaNationalLaboratoriesfortheirmanycriticaleffortsinsupportofthisprojectandthedevelopmentofthisreportinparticular.Inaddition,theinitiativeofCliffJoslyntoorganizetheworkshoponnewmethodsinuncertaintyquantificationatLosAlamosNationalLaboratories(February,2002wasextremelyhelpfultothefinaldraftofthispaper.Finally,wewouldliketothankGeorgeKlirofBinghamtonUniversityforhisencouragementovertheyears.
TABLEOFCONTENTSABSTRACT(3
ACKNOWLEDGEMENTS(4
TABLEOFCONTENTS(5
LISTOFFIGURES(6
LISTOFTABLES(7
1.1:
INTRODUCTION(8
1.2:
TYPESOFEVIDENCE(10
2.1:
DEMPSTER-SHAFERTHEORY(13
2.2:
RULESFORTHECOMBINATIONOFEVIDENCE(15
3:
DEMONSTRATIONOFCOMBINATIONRULES(27
3.1:
Datagivenbydiscretevalues(27
3.2:
Datagivenbyintervals(31
4:
CONCLUSIONS(46
REFERENCES(50
APPENDIXA.................................................................................................................A-1
Figure1:
Consonantevidenceobtainedfrommultiplesources(11
Figure2:
Consistentevidenceobtainedfrommultiplesensors(11
Figure3:
Arbitraryevidenceobtainedfrommultiplesensors(12
Figure4:
Disjointevidenceobtainedfrommultiplesensors(12
Figure5:
ThePossibleValuesofkinInagaki’sUnifiedCombinationRule(22
Figure6:
Thevalueofm(BasafunctionofkinInagaki’srule(30
Figure7:
ThegcdfofA(32
Figure8:
ThegcdfofB(32
Figure9:
ThegcdfofC(33
Figure10:
Thegcdf’sofA,B,andCwithoutanycombinationoperation(33
Figure11:
ThegcdfofthecombinationofAandBusingDempster’srule(34
Figure12:
ThecombinationofAandBusingYager’srule(35
Figure13:
ThegcdfofthecombinationofAandCusingYager’srule(35
Figure14:
TheInagakicombinationofAandBfork=0(36
Figure15:
TheInagakicombinationofAandBwherek=1(36
Figure16:
TheZhangcombinationofAandB(38
Figure17:
ThemixtureofAandB(39
Figure18:
ThemixtureofAandC(40
Figure19:
ThegcdfofthecombinationofAandBusingconvolutivex-averaging(41
Figure20:
TheComparisonofCombinationsofAandBwithDempster’sruleandConvolutiveX-Averaging(41
Figure21:
ThegcdfoftheCombinationofAandCusingConvolutivex-Averaging(42
Figure22:
ComparisonofYager’sruleandConvolutivex-averagingforAandC(43
Figure23:
TheDisjunctiveConsensusPoolingofAandB(44
Figure24:
ThegcdffortheDisjunctiveConsensusPoolingofAandC(44
Figure25:
ImportantIssuesintheCombinationofEvidence(48
Table1:
DempsterCombinationofExpert1andExpert2(28
Table2:
UnionsobtainedbyDisjunctiveConsensusPooling(31
Table3:
Theinterval-baseddataforAandthebasicprobabilityassignments(31
Table4:
Theinterval-baseddataforBandthebasicprobabilityassignments(32
Table5:
Theinterval-baseddataforCandthebasicprobabilityassignments(33
Table6:
CombinationofAandBwithDempster’sRule(34
Table7:
ThecombinationofthemarginalswithZhang’srule(37
Table8:
Thelengthoftheintervalsandtheirintersections(37
Table9:
CalculationoftheMeasureofIntersection(37
Table10:
Theproductofmandr(A,B(38
Table11:
TherenormalizedmasseswithZhang’srule(38
Table12:
Table13:
Table14:
TheCombinationofAandBusingConvolutivex-Averaging(40
Table15:
TheCombinationofAandCusingConvolutivex-Averaging(42
Table16:
TheDisjunctiveConsensusPoolingofAandB(43
Table17:
CalculationsfortheDisjunctiveConsensusPoolingofAandC(44
Table18:
TheCombinationofAandBComparisonTable(45
Table19:
TheCombinationofAandCComparisonTable(46
Table20:
CombinationRulesandTheirAlgebraicProperties(47
INTRODUCTION
Onlyveryrecently,thescientificandengineeringcommunityhasbeguntorecognizetheutilityofdefiningmultipletypesofuncertainty.Inpartthegreaterdepthofstudyintothescopeofuncertaintyismadepossiblebythesignificantadvancementsincomputationalpowerwenowenjoy.Assystemsbecomecomputationallybetterequippedtohandlecomplexanalyses,weencounterthelimitationsofapplyingonlyonemathematicalframework(traditionalprobabilitytheoryusedtorepresentthefullscopeofuncertainty.Thedualnatureofuncertaintyisdescribedwiththefollowingdefinitionsfrom[Helton,1997]:
AleatoryUncertainty–thetypeofuncertaintywhichresultsfromthefactthatasystemcanbehaveinrandomways
alsoknownas:
Stochasticuncertainty,TypeAuncertainty,Irreducibleuncertainty,Variability,Objectiveuncertainty
EpistemicUncertainty-thetypeofuncertaintywhichresultsfromthelackofknowledgeaboutasystemandisapropertyoftheanalystsperformingtheanalysis.
Subjectiveuncertainty,TypeBuncertainty,Reducibleuncertainty,StateofKnowledgeuncertainty,Ignorance
Traditionally,probabilitytheoryhasbeenusedtocharacterizebothtypesofuncertainty.Itiswellrecognizedthataleatoryuncertaintyisbestdealtwithusingthefrequentistapproachassociatedwithtraditionalprobabilitytheory.However,therecentcriticismsoftheprobabilisticcharacterizationofuncertaintyclaimthattraditionalprobabilitytheoryisnotcapableofcapturingepistemicuncertainty.TheapplicationoftraditionalprobabilisticmethodstoepistemicorsubjectiveuncertaintyisoftenknownasBayesianprobability.Aprobabilisticanalysisrequiresthatananalysthaveinformationontheprobabilityofallevents.Whenthisisnotavailable,theuniformdistributionfunctionisoftenused,justifiedbyLaplace’sPrincipleofInsufficientReason.[Savage,1972]Thiscanbeinterpretedthatallsimpleeventsforwhichaprobabilitydistributionisnotknowninagivensamplespaceareequallylikely.Takeforanexampleasystemfailurewheretherearethreepossiblecomponentsthatcouldhavecausedthistypeoffailure.Anexpertinthereliabilityofonecomponentassignsaprobabilityoffailureofthatcomponentwith0.3(ComponentA.Theexpertknowsnothingabouttheothertwopotentialsourcesoffailure(ComponentsBandC.AtraditionalprobabilisticanalysisfollowingthePrincipleofInsufficientReason,couldassignaprobabilityoffailureof0.35toeachofthetworemainingcomponents(BandC.Thiswouldbeaveryprecisestatementabouttheprobabilityoffailureofthesetwocomponentsinthefaceofcompleteignoranceregardingthesecomponentsonthepartoftheexpert.
Anadditionalassumptioninclassicalprobabilityisentailedbytheaxiomofadditivitywhereallprobabilitiesthatsatisfyspecificpropertiesmustaddto1.Thisforcestheconclusionthatknowledgeofaneventnecessarilyentailsknowledgeofthecomplementofanevent,i.e.,knowledgeoftheprobabilityofthelikelihoodoftheoccurrenceofaneventcanbetranslatedintotheknowledgeofthelikelihoodofthateventnotoccurring.Ifanexpertbelievesthatasystemmayfailduetoaparticular
componentwithalikelihoodof0.3,doesthatnecessarilymeanthattheexpertbelievesthatthesystemwillnotfailduetothatcomponentof0.7?
Thisarticulatesthechallengeofmodelinganyuncertaintyassociatedwithanexpert’ssubjectivebelief.ThoughtheassumptionsofadditivityandthePrincipleofInsufficientReasonmaybeappropriatewhenmodelingtherandomeventsassociatedwithaleatoricuncertainty,theseconstraintsarequestionablewhenappliedtoanissueofknowledgeorbelief.
Asaconsequenceoftheseconcerns,appliedmathematicianshaveinvestigatedmanymoregeneralrepresentationofuncertaintytocopewithparticularsituationsinvolvingepistemicuncertainty.Examplesofthesetypesofsituationsinclude:
1.Whenthereislittleinformationonwhichtoevaluateaprobabilityor
2.Whenthatinformationisnonspecific,ambiguous,orconflicting.
Analysisofthesesituationscanberequired,foranexampleinriskassessment,thoughprobabilitytheorylackstheabilitytohandlesuchinformation.Whereitisnotpossibletocharacterizeuncertaintywithaprecisemeasuresuchasapreciseprobability,itisreasonabletoconsiderameasureofprobabilityasanintervaloraset.
Thischaracterizationofameasureofprobabilityasanintervalorsethasthreeimportantimplications:
1.Itisnotnecessarytoelicit
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