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Bayesian Chain Classiers for Multidimensional Classication Julio H Bayesian Chain Classiers for Multidimensional Classication Julio H

Bayesian Chain Classiers for Multidimensional Classication Julio H - PDF document

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Bayesian Chain Classiers for Multidimensional Classication Julio H - PPT Presentation

Zaragoza L Enrique Sucar Eduardo F Morales Concha Bielza and Pedro Larra naga Computer Science Department National Institute for Astrophysics Optics and Electronics Puebla Mexico jzaragoza esucar emorales inaoepmx Computational Intelligence Group Te ID: 22892

Zaragoza Enrique Sucar

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BayesianChainClassi“ersforMultidimensionalClassi“cationJulioH.Zaragoza,L.EnriqueSucar,EduardoF.MoralesConchaBielzaPedroLarraComputerScienceDepartment,NationalInstituteforAstrophysics,OpticsandElectronics,Puebla,Mexicojzaragoza,esucar,emoralesComputationalIntelligenceGroup,TechnicalUniversityofMadrid,Madrid,Spainmcbielza,pedro.larranaga Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence Figure1:AnexampleofaBayesianChainClassi“erwhereeachintermediatenodeonthechainisanašveBayesianclas-si“erwhichhasasattributesonlyitsparentclasses()anditscorrespondingfeatures(featuresalongthechain,butonlytheparentsvariablesintheclassBN,asinaBNeveryvariableisindependentofitsnon-descendantsgivenitsparents.Thus,thenumberofadditionalfeaturesisrestrictedevenfordomainswithalargenumberofclasses.Finally,asforchainclassi“ers,thepredictedclasssetisobtainedbycombiningtheoutputsofalltheclassi“ersinthechain.InthispaperwepresentthesimplestversionofaBayesianChainClassi“er,asthedependencystructurebetweenclassvariablesisrestrictedtoadirectedtree.Thus,eachclassvari-ableinthetreehasatmostoneparent,soonlyoneadditionalfeatureisincorporatedtoeachbaseclassi“erinthechain.Additionally,thebaseclassi“ersarenašveBayes.AsininReadetal.,2009,wecombineseveralchainclassi“ersinaclas-si“erensemble,bychangingtherootnodeinthetree.ThisbasicBayesianChainClassi“erishighlyef“cientintermsoflearningandclassi“cationtimes,anditoutperformsothermorecomplexMDCsasdemonstratedintheexperiments.2MultidimensionalClassi“ersAspreviouslyintroduced,inthiscontributionwewillpresentanapproachtoclassi“cationproblemswithclassvariables,.Inthisframework,themulti-dimensionalclas-problemcorrespondstosearchingforafunctionthatassignstoeachinstancerepresentedbyavectorofavectorofclassvalues×···××···×Weassumethatforallandallarediscrete,andthatrespectivelyrepresenttheirsamplespaces.Underalossfunction,thefunctionshouldassigntoeachinstancethemostlikelycombinationofclasses,thatargmaxThisassignmentamountstosolvingatotalabductionin-ferenceproblemandcorrespondstothesearchforthemostprobableexplanation(MPE),aproblemthathasbeenprovedtobeanNP-hardproblemforBayesiannetworksorksShimony,3RelatedWorkInthissectionwebrie”yreviewthemainapproachesformultidimensionalclassi“cation.Thereviewisorganizedintothreesubsections,discussingresearchinmulti-labelclassi“-cation,multidimensionalBayesiannetworksclassi“ers,andchainclassi“ers,respectively.3.1Multi-labelClassi“cationInmulti-labelclassi“cationdomainseachinstanceisassoci-atedwithasubsetoflabels(presentintheinstance)fromasetoflabels.Takingthenotationintroducedinprevioussectionsintoaccount,thismulti-labelclassi“cationproblemcanbeseenasaparticularcaseofamultidimensionalclassi-“cationproblemwhereallclassvariablesarebinary,thatisAnoverviewofmulti-labelclassi“cationisgivenininTsoumakasandKatakis,2007,wheretwomaincategoriesaredistinguished:(a)problemtransformationmethods,and(b)algorithmadaptationmethods.Methodsin(a)transformthemulti-labelclassi“cationproblemintoeitheroneormoresingle-labelclassi“cationproblems.Methodsin(b)extendspeci“clearningalgorithmstohandlemulti-labeldatadi-rectly.Forexample,decisiontreestreesVensetal.,2008,sup-portvectormachinesmachinesBoutelletal.,2004-nearestneigh-neigh-ZhangandZhou,2007,neuralnetworksorksZhangandZhou,2006,andahybridoflogisticregressionandandChengandHullermeier,2009havebeenproposed.3.2MultidimensionalBayesianNetworkAmultidimensionalBayesiannetworkclassi“er(MBC)overaset,ofdiscreterandomvariablesisaBayesiannetwork,whereisanacyclicdirectedgraphwithvertexesisasetofparameters,whereisavaluefortheset,parentsvariablesofde“nesajointproba-bilitydistributionovergivenby:Thesetofvertexesispartitionedintotwosets,ofclassvariablesand,offeaturevariablesThesetofarcsisalsopartitionedintothreesets,,suchthatiscomposedofthearcsbe-tweentheclassvariables,iscomposedofthearcsbetweenthefeaturevariablesand“nally,iscomposedofthearcsfromtheclassvariablestothefeaturevariables.Thecorrespondinginducedsubgraphscalledrespectivelyclass,featureandbridgesubgraphs.Differentgraphicalstructuresfortheclassandfeaturesub-graphsmayleadtodifferentfamiliesofMBCs.MBCs.vanderGaaganddeWaal,2006treesforbothsubgraphsbysearch-ingforthemaximumweightedundirectedspanningtreeandtransformingitintoadirectedtreeusingChowandLiusal-al-1968].Thebridgesubgraphisgreedilylearntina wrapperway,tryingtoimprovethepercentageofcorrectlyclassi“edinstances.instances.deWaalandvanderGaag,2007isatheoreticalworkfor“ndingtheconditionsfortheoptimalre-coveryofpolytreestructuresinbothsubgraphs.subgraphs.Rodr´šguezandLozano,2008extendpolytreestostructuresforclassandfeaturessubgraphs.Learningthesestructuresiscarriedoutusingamulti-objectivegenetical-gorithmwheretheindividualsarepermittedstructurescodedwiththreesubstrings,onepersubgraph.SimplermodelsareusedinanapplicationforheartwallmotionpredictionpredictionQazietal.,2007:adirectedacyclicgraphfortheclasssubgraph,anemptygraphforthefeatures,andabridgesubgraphwherefeaturesreceivearcsfromsomeclassvariables,withoutshar-inganyofthem.Finally,,Bielzaetal.,2011presentthemostgeneralmod-elssinceanyBayesiannetworkstructureisallowedinthethreesubgraphs.Learningfromdataalgorithmscoverallthepossibilities:wrapper,“lterandhybridscore+searchstrate-gies.Moreover,sincethecomputationoftheMPEinvolvesahighcomputationalcost,severalcontributionsaredesignedtoalleviateit.3.3ChainClassi“ersReadandothersothers2009]introducechainclassi“ersasanal-ternativemethodformulti-labelclassi“cationthatincorpo-ratesclassdependencies,whileittriestokeepthecompu-tationalef“ciencyofthebinaryrelevanceapproach.Chainclassi“ersconsistofbinaryclassi“erswhicharelinkedinachain,suchthateachclassi“erincorporatestheclasspredictedbythepreviousclassi“ersasadditionalattributes.Thus,thefeaturevectorforeachbinaryclassi“er,,isex-tendedwiththelabels()ofallpreviousclassi“ersinthechain.Eachclassi“erinthechainistrainedtolearntheas-sociationoflabelgiventhefeaturesaugmentedwithallpreviousbinarypredictionsinthechain,.Forclassi“cation,itstartsat,andpropagatesalongthechainsuchthatfor)itpredicts.Asinthebinaryrelevanceapproach,theclassvectorisdeterminedbycombiningtheoutputsofallthebinaryclassi“ersinthechain.chain.Readetal.,2009combineseveralchainclassi“ersbychangingtheorderforthelabels,buildinganensembleofchainclassi“ers.Thus,chainclassi“ersaretrained,byvaryingthetrainingdataandtheorderoftheclassesinthechain(botharesetrandomly).The“nallabelvectorisob-tainedusingavotingscheme;eachlabelreceivesanumberofvotesfromthechainclassi“ers,andathresholdisusedtodeterminethe“nalpredictedmulti-labelset.Recently,Dembczynskietal.al.2010]presentprobabilisticchainclassi“ers(PCCs),bybasicallyputtingchainclassi“ersunderaprobabilisticframework.Usingthechainrule,theprobabilityofthevectorofclassvaluesgiventhefeaturevectorcanbewrittenas:Givenafunctionthatprovidesanapproximationoftheprobabilityof,theyde“neaprobabilisticchainclas-si“eras:PCCestimatesthejointprobabilityoftheclasses,pro-vidingbetterestimatesthanthechainclassi“ers,butwithamuchhighercomputationalcomplexity.Infact,theexperi-mentsreportedbybyDembczynskietal.,2010arelimitedto10classes.AsshownbybyDembczynskietal.,2010,amethodthatconsidersclassdependenciesunderaprobabilisticframeworkcanhaveasigni“cantimpactontheperformanceofmultidi-mensionalclassi“ers.However,bothMBCsandPCCshaveahighcomputationalcomplexity,whichlimitstheirapplica-bilitytohighdimensionalproblems.Inthefollowingsectionwedescribeanalternativeprobabilisticmethodwhichalsoincorporatesclassdependenciesbutatthesametimeisveryef“cient.4BayesianChainClassi“ersGivenamultidimensionalclassi“cationproblemwithclasses,aBayesianChainClassi“er(BCC)usesoneperclass,linkedinachain.Theobjectiveofthisproblemcanbeposedas“ndingajointdistributionoftheclassesgiventheattributesrepresentstheparentsofclass.Inthisset-ting,achainclassi“ercanbeconstructedbyinducing“rsttheclassi“ersthatdonotdependonanyotherclassandthenproceedwiththeirsons.ABayesianframeworkallowsusto:€Createa(partial)orderofclassesinthechainclassi-“erbasedonthedependenciesbetweenclassesgiventhefeatures.Assumingthatthesedependenciescanberep-resentedasaBayesiannetwork(directedacyclicgraph),thechainstructureisde“nedbythestructureoftheBN,suchthatwecanthenstartbuildingclassi“ersfortheclasseswithoutparents,andcontinuewiththeirchildrenclasses,andsoon.€Considerconditionalindependenciesbetweenclassestocreatesimplerclassi“ers.Inthiscase,constructsi“ersconsideringonlytheparentclassesofeachclass.Foralargenumberofclassesthiscanbeahugereduc-tionasnormallywecanexpecttohavealimitednumberofparentsperclass.Ingeneral,whenweinduceaBayesiannetworktorep-resenttheabovejointdistribution,itisnotalwayspossibleto“nddirectionsforallthelinks.Inthatcasewecanhavedifferentordersdependingonthechosendirections.Intheworstcasethenumberofpossibledirectionsgrowsexponen-tiallywiththenumberoflinks(undirectedlinks).Inpracticeweexpecttohaveonlyalimitednumberofundi-rectedlinks.Inthatcasewecanobtain(asubsetof)the Figure2:ExampleofaMaximumWeightSpanningTreeof Figure3:UsingtheMaximumWeightedSpanningTreeofClasseswithnode3asrootfordeterminingthechainingor-der.possibleordersandbuildanensembleofchainclassi“erswithdifferentorders.Givenanewinstance,wedeterminewitheachchainclassi“er,anduseavotingschemetooutputasetofclasses.Wecansimplifytheproblembyconsideringthemarginaldependenciesbetweenclasses(asa“rstapproximation)toobtainanorderinthechainclassi“erandtheninduceclas-si“ersconsideringsuchorder.Additionally,wecansimplifyevenfurthertheproblembyconsideringonlyoneparentperclass.Thiscanbesolvedbyobtainingtheskeletonofatree-structuredBNfortheclassesusingChowandLiusalgorithm(1968),thatis,amaximumweightundirectedspanningtreeMWST)(seeFig.2).ChowandLiusalgorithmdoesnotgiveusthedirectionsofthelinks,however,wecanbuilddirectedtreesbytakingeachclass(node)asrootofatreeandassigningdirectionstothearcsstartingfromthisrootnodetobuildadirectedtree(Fig.3).Thechainingorderoftheclassi“ersisgivenbytraversingthetreefollowinganancestralordering.Forclasseswebuildclassi“ersintheordergivenbythedifferenttreesandthencombinetheminanensemble(ifisverylargewecanlimitthenumberofchainsbyselectingarandomsubsetoftrees).Therearemanydifferentchoicesforrepresentingeachclassi“er,oneofthesimplestandfastesttobuildisthenašveBayesclassi“er(NBCs);althoughotherclassi“erscouldbeusedaswell.WithnašveBayesclassi“ersinthechain,weneedtoconsideronlytheclassparent,,andthefea-turevector,,asattributesforeachNBC,seeFig.1.Wecansummarizeouralgorithmasfollows.Givenamul-tidimensionalclassi“cationproblemwith1.Buildanundirectedtreetoapproximatethedependencystructureamongclassvariables.2.Createordersforthechainclassi“ersbytakingeachclassastherootofthetreeandassigningtherestofthelinksinorder.3.Foreachclassi“erineachchain,buildanNBCwiththeasrootandonlytheparentsandalltheattributesaschildren,takingadvantageofconditionalindependenceproperties.Toclassifyanewinstancecombinetheoutputofthechains,usingasimplevotingscheme.Thisisaveryfastandeasytobuildensembleofchainclas-si“ers,whichrepresentsthesimplestalternativeforaBCC.Other,morecomplexalternativescanbeexploredby:(i)con-sideringconditionaldependenciesbetweenclasses,(ii)build-ingmorecomplexclassdependencystructures,and(iii)usingotherbaseclassi“ers.InthenextsectionwepresenttheexperimentalresultswherewecompareBCCswithotherstateoftheartmulti-dimensionalclassi“ers.5ExperimentsandResultsTheproposedmethodwastestedondifferentbenchmarkmultidimensionaldatasets;eachofthemwithdifferentdi-mensionsrangingfromlabels,andfromaboutexamplestomorethan.Allclassvariablesofthedatasetsarebinary,however,insomeofthedatasetsthefeaturevariablesarenumeric.Inthesecasesweusedastatic,global,supervisedandtop-downdiscretizationalgo-algo-Cheng-Jungetal.,2008.ThedetailsofthedatasetsaresummarizedinTable1.Table1:Multidimensionaldatasetsusedintheexperimentsandassociatedstatistics.isthesizeofthedataset,thenumberofbinaryclassesorlabels,isthenumberofindicatesnumericattributes. No. Dataset N d m Type 1 Emotions 593 6 72 Music 2 Scene 2407 6 294 Vision 3 Yeast 2417 14 103 Biology 4 TMC2007 28596 22 500 Text 5 Medical 978 45 1449 Text 6 Enron 1702 53 1001 Text 7 MediaMill 43907 101 120 Media 8 Bibtex 7395 159 1836 Text 9 Delicious 16105 983 500 Text First,wecomparedBCCsagainstdifferentstate-of-the-artmethods(showninTable2)usingtheYeastdatasets.AlgorithmisthebasicBinaryRelevanceanceTsoumakasandKatakis,2007.AlgorithmsaremethodsexplicitlydesignedforlearningMBCs.Algo-usegreedysearchapproachesthatlearnagen-eralBayesiannetwork,oneguidedbythetheCooperandHerskovits,1992(“lterapproach),andtheotherguidedbyaperformanceevaluationmetric,asde“nedininBielzaetal.,2011(wrapperapproach).Algorithmisamulti-labellazylearningapproachnamednamedZhangandZhou,,derivedfromthetraditionalK-nearestneighboralgo-rithm.Inthisexperimentforthewassetinthedatasets,andintheYeast Thedatasetscanbefoundatmu-lan.sourceforge.net/datasets.html,mlkd.csd.auth.gr/multilabel.htmlandwww.cs.waikato.ac.nz/jmr30/#„datasets. dataset.Sinceitisunfeasibletocomputethemutualinforma-tionoftwofeaturesgivenalltheclassvariables,asrequiredinindeWaalandvanderGaag,2007,theimplementationofthepolytree-polytreelearningalgorithmusesthemarginalmutualinformationofpairsoffeatures.AlgorithmfromTable2isourBayesianChainClassi“er.Table2:Algorithmsusedintheexperiments. No. Algorithm[Reference] 1 binaryrelevanceanceTsoumakasandKatakis,2007 2 vanderGaaganddeWaal,2006 3 deWaalandvanderGaag,2007 4 Bielzaetal.,2011 5 Bielzaetal.,2011 6 Bielzaetal.,2011 7 K2BNBNCooperandHerskovits,1992 8 wrapperBNBNBielzaetal.,2011 9 ZhangandZhou,2006 10 BayesianChainClassi“ers(BCC) Forthepurposeofcomparisonweusedtwodifferentmul-tidimensionalperformancemeasuresmeasuresBielzaetal.,2011Meanaccuracy(accuracyperlabelorperclass)overtheclassvariables: Accd=1 ddj=1Accj=1 ddj=11 Notethatdenotestheclassvalueoutputtedbythemodelforcaseisitstruevalue.Globalaccuracy(accuracyperexample)overthedimensionalclassvariable: isthed-dimensionalvectorofclassvaluesandotherwise.Therefore,wecallforatotalcoincidenceonallofthecomponentsofthevectorofpredictedclassesandthevectorofrealTheestimationmethodforperformanceevaluationisfoldcross-validation.ResultsforaccuracyandacomparisonofthedifferentmethodsareshowninTable3.Table4showstheaveragerankingsofthealgorithmsusedforcomparisonandthatofourmethod.Ascanbeseenfromthistable,ingeneral,theperformanceofourmethodisbetterthattheothermethodsusedintheseexperiments.Secondly,weperformedexperimentswiththeEnronBibtexGiventhecomplexityofthesedatasets,inparticularinthenumberofclasses,theycannotbetestedwiththeothermeth-ods;sointhiscasewecomparedBayesianChainClassi-“ersandEnsemblesofBCCs(EBCCs).InTable5weshowtheMeanandGlobalaccuraciesperdatasetforBCCsandEBCCs,respectively.WeobservethatformostofthedatasetsTable3:Performancemetrics(mean±std.deviation)andrank(inbrackets)ofthealgorithmsusing-foldcross-validation. Dataset MeanAccuracy GlobalAccuracy Emotions binaryrelevance 0.7762±0.1667(7) 0.2860±0.0452(8) tree-tree 0.8300±0.0151(3) 0.3844±0.0398(1) polytree-polytree 0.8209±0.0243(5) 0.3776±0.0622(3) pure“lter 0.7548±0.0280(9) 0.2866±0.0495(7) purewrapper 0.8333±0.0123(2) 0.3708±0.0435(4) hybrid 0.8210±0.0170(4) 0.3557±0.0435(5) K2BN 0.7751±0.0261(8) 0.2812±0.0799(9) wrapperBN 0.7985±0.0200(6) 0.3033±0.0752(6) ML-KNN 0.6133±0.0169(10) 0.0254±0.0120(10) BCC 0.8417±0.0231(1) 0.3822±0.0631(2) Scene binaryrelevance 0.8236±0.0250(2) 0.2898±0.0149(3) tree-tree 0.7324±0.0359(9) 0.1857±0.0977(8) polytree-polytree 0.7602±0.0663(8) 0.2643±0.1915(6) pure“lter 0.7726±0.0700(6) 0.3067±0.1991(1) purewrapper 0.7765±0.0580(4) 0.2688±0.1642(5) hybrid 0.7229±0.0442(10) 0.1570±0.1018(9) K2BN 0.7689±0.0692(7) 0.2883±0.1995(4) wrapperBN 0.7739±0.0492(5) 0.2277±0.1372(7) ML-KNN 0.8196±0.0092(3) 0.0311±0.0147(10) BCC 0.8260±0.0373(1) 0.2920±0.1218(2) Yeast binaryrelevance 0.7297±0.2380(9) 0.0890±0.0242(8) tree-tree 0.7728±0.0071(4) 0.1953±0.0208(1) polytree-polytree 0.7336±0.0182(8) 0.1431±0.0258(3) pure“lter 0.7480±0.0119(6) 0.0989±0.0342(7) purewrapper 0.7845±0.0131(1) 0.1410±0.0989(4) hybrid 0.7397±0.0114(7) 0.1200±0.0268(6) K2BN 0.7686±0.0112(5) 0.1299±0.0204(5) wrapperBN 0.7745±0.0049(3) 0.0550±0.0212(9) ML-KNN 0.6364±0.0196(10) 0.0062±0.0029(10) BCC 0.7771±0.0147(2) 0.1616±0.0875(2) Table4:AveragerankvaluesforeachalgorithmfromtheresultsinTable3. Algorithm MeanAcc. GlobalAcc. GlobalRank binaryrelevance 6.0000 6.3334 6.1667(7) tree-tree 5.3334 3.3334 4.3334(3) polytree-polytree 7.0000 4.0000 5.5000(4) pure“lter 7.0000 5.0000 6.0000(5) purewrapper 2.3334 4.3334 3.3334(2) hybrid 7.0000 6.6667 6.8333(9) K2BN 6.6667 6.0000 6.3333(8) wrapperBN 4.6667 7.3334 6.0000(5) ML-KNN 7.6667 10.000 8.8333(10) BCC 1.3333 2.0000 1.6667(1) thereisasigni“cantimprovementinmeanandglobalaccu-racywiththeensemble.Thenumberofiterationsontheen-semblesweresetto10.Aclassisdeterminedwhetherposi-tiveornegativebytakingthevaluewiththehighernumberofvotesintheensemble. Table5:GlobalandMeanaccuracyresultsfortheBayesianChainClassi“ers(BCC)andfortheEnsemblesofBayesianChainClassi“ers(EBCC). MeanAccuracy GlobalAccuracy Dataset BCC EBCC BCC EBCC TMC2007 0. 0. 0. 0. Medical 0. 0. 0. 0. Enron 0. 0. 0. 0. MediaMill 0. 0. 0. 0. Bibtex 0. 0. 0. 0. Delicious 0. 0. 0. 0. Intermsofcomputationalresources,thetrainingandclas-si“cationtimesforthedatasets(Yeast)arelessthanoneminute;andforthemorecomplexdatasetsareintheorderofhours6ConclusionsandFutureWorkInthispaperwehaveintroducedBayesianChainClassi“ersformultidimensionalclassi“cation.Theproposedapproachissimpleandeasytoimplement,andyetishighlycompeti-tiveagainstotherBayesianmultidimensionalclassi“ers.WeexperimentedwiththesimplestmodelforaBCC,consider-ingatreestructurefortheclassdependenciesandasimplešveBayesclassi“erasbaseclassi“er.Inthefuturewewillexplorealternativemodelsconsideringmorecomplexdepen-dencystructuresandothermorepowerfulbaseclassi“ers.Wealsoplantocompareourapproachwithotherclassi“erchainsusingdifferentmetricsanddatasets.7AcknowledgmentsTheauthorswishtoacknowledgeforthesup-portprovidedthroughProjectNo.95185(DyNaMo).Also,thisresearchhasbeenpartiallysupportedbytheSpanishMinistryofScienceandInnovation,projectsTIN2010-20900-C04-04,ConsoliderIngenio2010-CSD2007-00018andCajalBlueBrain.ReferencesencesBielzaetal.,2011C.Bielza,G.Li,andP.LarraMulti-dimensionalclassi“cationwithbayesiannetworks.InternationalJournalofApproximateReasoning,2011.2011.Boutelletal.,2004MatthewR.Boutell,JieboLuo,XipengShen,andChristopherM.Brown.Learningmulti-labelsceneclassi“cation.PatternRecognition37(9):1757…1771,2004.2004.ChengandHullermeier,2009WeiweiChengandEykeHullermeier.Combininginstance-basedlearningandlo-gisticregressionformulti-labelclassi“cation.Machine,76(23):211…225,2009.2009.Cheng-Jungetal.,2008TsaiCheng-Jung,LeeChien-I,andYangWei-Pang.Adiscretizationalgorithmbasedonclass-attributecontingencycoef“cient.InformationSci-,(178):714…731,2008. 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