3Classification:BasicConceptsandTechniques
Humanshaveaninnateabilitytoclassifythingsintocategories,e.g.,mundanetaskssuchasfilteringspamemailmessagesormorespecializedtaskssuchasrecognizingcelestialobjectsintelescopeimages(seeFigure3.1 ).Whilemanualclassificationoftensufficesforsmallandsimpledatasetswithonlyafewattributes,largerandmorecomplexdatasetsrequireanautomatedsolution.
Figure3.1.ClassificationofgalaxiesfromtelescopeimagestakenfromtheNASAwebsite.
Thischapterintroducesthebasicconceptsofclassificationanddescribessomeofitskeyissuessuchasmodeloverfitting,modelselection,andmodelevaluation.Whilethesetopicsareillustratedusingaclassificationtechniqueknownasdecisiontreeinduction,mostofthediscussioninthischapterisalsoapplicabletootherclassificationtechniques,manyofwhicharecoveredinChapter4 .
3.1BasicConceptsFigure3.2 illustratesthegeneralideabehindclassification.Thedataforaclassificationtaskconsistsofacollectionofinstances(records).Eachsuchinstanceischaracterizedbythetuple( ,y),where isthesetofattributevaluesthatdescribetheinstanceandyistheclasslabeloftheinstance.Theattributeset cancontainattributesofanytype,whiletheclasslabelymustbecategorical.
Figure3.2.Aschematicillustrationofaclassificationtask.
Aclassificationmodelisanabstractrepresentationoftherelationshipbetweentheattributesetandtheclasslabel.Aswillbeseeninthenexttwochapters,themodelcanberepresentedinmanyways,e.g.,asatree,aprobabilitytable,orsimply,avectorofreal-valuedparameters.Moreformally,wecanexpressitmathematicallyasatargetfunctionfthattakesasinputtheattributeset andproducesanoutputcorrespondingtothepredictedclasslabel.Themodelissaidtoclassifyaninstance( ,y)correctlyif .
Table3.1 showsexamplesofattributesetsandclasslabelsforvariousclassificationtasks.Spamfilteringandtumoridentificationareexamplesofbinaryclassificationproblems,inwhicheachdatainstancecanbecategorizedintooneoftwoclasses.Ifthenumberofclassesislargerthan2,asinthe
f(x)=y
galaxyclassificationexample,thenitiscalledamulticlassclassificationproblem.
Table3.1.Examplesofclassificationtasks.
Task Attributeset Classlabel
Spamfiltering Featuresextractedfromemailmessageheaderandcontent
spamornon-spam
Tumoridentification
Featuresextractedfrommagneticresonanceimaging(MRI)scans
malignantorbenign
Galaxyclassification
Featuresextractedfromtelescopeimages elliptical,spiral,orirregular-shaped
Weillustratethebasicconceptsofclassificationinthischapterwiththefollowingtwoexamples.
3.1.ExampleVertebrateClassificationTable3.2 showsasampledatasetforclassifyingvertebratesintomammals,reptiles,birds,fishes,andamphibians.Theattributesetincludescharacteristicsofthevertebratesuchasitsbodytemperature,skincover,andabilitytofly.Thedatasetcanalsobeusedforabinaryclassificationtasksuchasmammalclassification,bygroupingthereptiles,birds,fishes,andamphibiansintoasinglecategorycallednon-mammals.
Table3.2.Asampledataforthevertebrateclassificationproblem.VertebrateName
BodyTemperature
SkinCover
GivesBirth
AquaticCreature
AerialCreature
HasLegs
Hibernates ClassLabel
human warm-
blooded
hair yes no no yes no mammal
3.2.ExampleLoanBorrowerClassificationConsidertheproblemofpredictingwhetheraloanborrowerwillrepaytheloanordefaultontheloanpayments.Thedatasetusedtobuildthe
blooded
python cold-blooded scales no no no no yes reptile
salmon cold-blooded scales no yes no no no fish
whale warm-blooded
hair yes yes no no no mammal
frog cold-blooded none no semi no yes yes amphibian
komodo cold-blooded scales no no no yes no reptile
dragon
bat warm-blooded
hair yes no yes yes yes mammal
pigeon warm-blooded
feathers no no yes yes no bird
cat warm-blooded
fur yes no no yes no mammal
leopard cold-blooded scales yes yes no no no fish
shark
turtle cold-blooded scales no semi no yes no reptile
penguin warm-blooded
feathers no semi no yes no bird
porcupine warm-blooded
quills yes no no yes yes mammal
eel cold-blooded scales no yes no no no fish
salamander cold-blooded none no semi no yes yes amphibian
classificationmodelisshowninTable3.3 .Theattributesetincludespersonalinformationoftheborrowersuchasmaritalstatusandannualincome,whiletheclasslabelindicateswhethertheborrowerhaddefaultedontheloanpayments.
Table3.3.Asampledatafortheloanborrowerclassificationproblem.
ID HomeOwner MaritalStatus AnnualIncome Defaulted?
1 Yes Single 125000 No
2 No Married 100000 No
3 No Single 70000 No
4 Yes Married 120000 No
5 No Divorced 95000 Yes
6 No Single 60000 No
7 Yes Divorced 220000 No
8 No Single 85000 Yes
9 No Married 75000 No
10 No Single 90000 Yes
Aclassificationmodelservestwoimportantrolesindatamining.First,itisusedasapredictivemodeltoclassifypreviouslyunlabeledinstances.Agoodclassificationmodelmustprovideaccuratepredictionswithafastresponsetime.Second,itservesasadescriptivemodeltoidentifythecharacteristicsthatdistinguishinstancesfromdifferentclasses.Thisisparticularlyusefulforcriticalapplications,suchasmedicaldiagnosis,whereit
isinsufficienttohaveamodelthatmakesapredictionwithoutjustifyinghowitreachessuchadecision.
Forexample,aclassificationmodelinducedfromthevertebratedatasetshowninTable3.2 canbeusedtopredicttheclasslabelofthefollowingvertebrate:
Inaddition,itcanbeusedasadescriptivemodeltohelpdeterminecharacteristicsthatdefineavertebrateasamammal,areptile,abird,afish,oranamphibian.Forexample,themodelmayidentifymammalsaswarm-bloodedvertebratesthatgivebirthtotheiryoung.
Thereareseveralpointsworthnotingregardingthepreviousexample.First,althoughalltheattributesshowninTable3.2 arequalitative,therearenorestrictionsonthetypeofattributesthatcanbeusedaspredictorvariables.Theclasslabel,ontheotherhand,mustbeofnominaltype.Thisdistinguishesclassificationfromotherpredictivemodelingtaskssuchasregression,wherethepredictedvalueisoftenquantitative.MoreinformationaboutregressioncanbefoundinAppendixD.
Anotherpointworthnotingisthatnotallattributesmayberelevanttotheclassificationtask.Forexample,theaveragelengthorweightofavertebratemaynotbeusefulforclassifyingmammals,astheseattributescanshowsamevalueforbothmammalsandnon-mammals.Suchanattributeistypicallydiscardedduringpreprocessing.Theremainingattributesmightnotbeabletodistinguishtheclassesbythemselves,andthus,mustbeusedin
VertebrateName
BodyTemperature
SkinCover
GivesBirth
AquaticCreature
AerialCreature
HasLegs
Hibernates ClassLabel
gilamonster
cold-blooded scales no no no yes yes ?
concertwithotherattributes.Forinstance,theBodyTemperatureattributeisinsufficienttodistinguishmammalsfromothervertebrates.WhenitisusedtogetherwithGivesBirth,theclassificationofmammalsimprovessignificantly.However,whenadditionalattributes,suchasSkinCoverareincluded,themodelbecomesoverlyspecificandnolongercoversallmammals.Findingtheoptimalcombinationofattributesthatbestdiscriminatesinstancesfromdifferentclassesisthekeychallengeinbuildingclassificationmodels.
3.2GeneralFrameworkforClassificationClassificationisthetaskofassigninglabelstounlabeleddatainstancesandaclassifierisusedtoperformsuchatask.Aclassifieristypicallydescribedintermsofamodelasillustratedintheprevioussection.Themodeliscreatedusingagivenasetofinstances,knownasthetrainingset,whichcontainsattributevaluesaswellasclasslabelsforeachinstance.Thesystematicapproachforlearningaclassificationmodelgivenatrainingsetisknownasalearningalgorithm.Theprocessofusingalearningalgorithmtobuildaclassificationmodelfromthetrainingdataisknownasinduction.Thisprocessisalsooftendescribedas“learningamodel”or“buildingamodel.”Thisprocessofapplyingaclassificationmodelonunseentestinstancestopredicttheirclasslabelsisknownasdeduction.Thus,theprocessofclassificationinvolvestwosteps:applyingalearningalgorithmtotrainingdatatolearnamodel,andthenapplyingthemodeltoassignlabelstounlabeledinstances.Figure3.3 illustratesthegeneralframeworkforclassification.
Figure3.3.Generalframeworkforbuildingaclassificationmodel.
Aclassificationtechniquereferstoageneralapproachtoclassification,e.g.,thedecisiontreetechniquethatwewillstudyinthischapter.Thisclassificationtechniquelikemostothers,consistsofafamilyofrelatedmodelsandanumberofalgorithmsforlearningthesemodels.InChapter4 ,wewillstudyadditionalclassificationtechniques,includingneuralnetworksandsupportvectormachines.
Acouplenotesonterminology.First,theterms“classifier”and“model”areoftentakentobesynonymous.Ifaclassificationtechniquebuildsasingle,
globalmodel,thenthisisfine.However,whileeverymodeldefinesaclassifier,noteveryclassifierisdefinedbyasinglemodel.Someclassifiers,suchask-nearestneighborclassifiers,donotbuildanexplicitmodel(Section4.3 ),whileotherclassifiers,suchasensembleclassifiers,combinetheoutputofacollectionofmodels(Section4.10 ).Second,theterm“classifier”isoftenusedinamoregeneralsensetorefertoaclassificationtechnique.Thus,forexample,“decisiontreeclassifier”canrefertothedecisiontreeclassificationtechniqueoraspecificclassifierbuiltusingthattechnique.Fortunately,themeaningof“classifier”isusuallyclearfromthecontext.
InthegeneralframeworkshowninFigure3.3 ,theinductionanddeductionstepsshouldbeperformedseparately.Infact,aswillbediscussedlaterinSection3.6 ,thetrainingandtestsetsshouldbeindependentofeachothertoensurethattheinducedmodelcanaccuratelypredicttheclasslabelsofinstancesithasneverencounteredbefore.Modelsthatdeliversuchpredictiveinsightsaresaidtohavegoodgeneralizationperformance.Theperformanceofamodel(classifier)canbeevaluatedbycomparingthepredictedlabelsagainstthetruelabelsofinstances.Thisinformationcanbesummarizedinatablecalledaconfusionmatrix.Table3.4 depictstheconfusionmatrixforabinaryclassificationproblem.Eachentry denotesthenumberofinstancesfromclassipredictedtobeofclassj.Forexample, isthenumberofinstancesfromclass0incorrectlypredictedasclass1.Thenumberofcorrectpredictionsmadebythemodelis andthenumberofincorrectpredictionsis .
Table3.4.Confusionmatrixforabinaryclassificationproblem.
PredictedClass
ActualClass
fijf01
(f11+f00)(f10+f01)
Class=1 Class=0
Class=1 f11 f10
Althoughaconfusionmatrixprovidestheinformationneededtodeterminehowwellaclassificationmodelperforms,summarizingthisinformationintoasinglenumbermakesitmoreconvenienttocomparetherelativeperformanceofdifferentmodels.Thiscanbedoneusinganevaluationmetricsuchasaccuracy,whichiscomputedinthefollowingway:
Accuracy=
Forbinaryclassificationproblems,theaccuracyofamodelisgivenby
Errorrateisanotherrelatedmetric,whichisdefinedasfollowsforbinaryclassificationproblems:
Thelearningalgorithmsofmostclassificationtechniquesaredesignedtolearnmodelsthatattainthehighestaccuracy,orequivalently,thelowesterrorratewhenappliedtothetestset.WewillrevisitthetopicofmodelevaluationinSection3.6 .
Class=0 f01 f00
Accuracy=NumberofcorrectpredictionsTotalnumberofpredictions. (3.1)
Accuracy=f11+f00f11+f10+f01+f00. (3.2)
Errorrate=NumberofwrongpredictionsTotalnumberofpredictions=f10+f01f11(3.3)
3.3DecisionTreeClassifierThissectionintroducesasimpleclassificationtechniqueknownasthedecisiontreeclassifier.Toillustratehowadecisiontreeworks,considertheclassificationproblemofdistinguishingmammalsfromnon-mammalsusingthevertebratedatasetshowninTable3.2 .Supposeanewspeciesisdiscoveredbyscientists.Howcanwetellwhetheritisamammaloranon-mammal?Oneapproachistoposeaseriesofquestionsaboutthecharacteristicsofthespecies.Thefirstquestionwemayaskiswhetherthespeciesiscold-orwarm-blooded.Ifitiscold-blooded,thenitisdefinitelynotamammal.Otherwise,itiseitherabirdoramammal.Inthelattercase,weneedtoaskafollow-upquestion:Dothefemalesofthespeciesgivebirthtotheiryoung?Thosethatdogivebirtharedefinitelymammals,whilethosethatdonotarelikelytobenon-mammals(withtheexceptionofegg-layingmammalssuchastheplatypusandspinyanteater).
Thepreviousexampleillustrateshowwecansolveaclassificationproblembyaskingaseriesofcarefullycraftedquestionsabouttheattributesofthetestinstance.Eachtimewereceiveananswer,wecouldaskafollow-upquestionuntilwecanconclusivelydecideonitsclasslabel.Theseriesofquestionsandtheirpossibleanswerscanbeorganizedintoahierarchicalstructurecalledadecisiontree.Figure3.4 showsanexampleofthedecisiontreeforthemammalclassificationproblem.Thetreehasthreetypesofnodes:
Arootnode,withnoincominglinksandzeroormoreoutgoinglinks.Internalnodes,eachofwhichhasexactlyoneincominglinkandtwoormoreoutgoinglinks.Leaforterminalnodes,eachofwhichhasexactlyoneincominglinkandnooutgoinglinks.
Everyleafnodeinthedecisiontreeisassociatedwithaclasslabel.Thenon-terminalnodes,whichincludetherootandinternalnodes,containattributetestconditionsthataretypicallydefinedusingasingleattribute.Eachpossibleoutcomeoftheattributetestconditionisassociatedwithexactlyonechildofthisnode.Forexample,therootnodeofthetreeshowninFigure3.4 usestheattribute todefineanattributetestconditionthathastwooutcomes,warmandcold,resultingintwochildnodes.
Figure3.4.Adecisiontreeforthemammalclassificationproblem.
Givenadecisiontree,classifyingatestinstanceisstraightforward.Startingfromtherootnode,weapplyitsattributetestconditionandfollowtheappropriatebranchbasedontheoutcomeofthetest.Thiswillleaduseithertoanotherinternalnode,forwhichanewattributetestconditionisapplied,ortoaleafnode.Oncealeafnodeisreached,weassigntheclasslabelassociatedwiththenodetothetestinstance.Asanillustration,Figure3.5
tracesthepathusedtopredicttheclasslabelofaflamingo.Thepathterminatesataleafnodelabeledas .
Figure3.5.Classifyinganunlabeledvertebrate.Thedashedlinesrepresenttheoutcomesofapplyingvariousattributetestconditionsontheunlabeledvertebrate.Thevertebrateiseventuallyassignedtothe class.
3.3.1ABasicAlgorithmtoBuildaDecisionTree
Manypossibledecisiontreesthatcanbeconstructedfromaparticulardataset.Whilesometreesarebetterthanothers,findinganoptimaloneiscomputationallyexpensiveduetotheexponentialsizeofthesearchspace.Efficientalgorithmshavebeendevelopedtoinduceareasonablyaccurate,
albeitsuboptimal,decisiontreeinareasonableamountoftime.Thesealgorithmsusuallyemployagreedystrategytogrowthedecisiontreeinatop-downfashionbymakingaseriesoflocallyoptimaldecisionsaboutwhichattributetousewhenpartitioningthetrainingdata.OneoftheearliestmethodisHunt'salgorithm,whichisthebasisformanycurrentimplementationsofdecisiontreeclassifiers,includingID3,C4.5,andCART.ThissubsectionpresentsHunt'salgorithmanddescribessomeofthedesignissuesthatmustbeconsideredwhenbuildingadecisiontree.
Hunt'sAlgorithmInHunt'salgorithm,adecisiontreeisgrowninarecursivefashion.Thetreeinitiallycontainsasinglerootnodethatisassociatedwithallthetraininginstances.Ifanodeisassociatedwithinstancesfrommorethanoneclass,itisexpandedusinganattributetestconditionthatisdeterminedusingasplittingcriterion.Achildleafnodeiscreatedforeachoutcomeoftheattributetestconditionandtheinstancesassociatedwiththeparentnodearedistributedtothechildrenbasedonthetestoutcomes.Thisnodeexpansionstepcanthenberecursivelyappliedtoeachchildnode,aslongasithaslabelsofmorethanoneclass.Ifalltheinstancesassociatedwithaleafnodehaveidenticalclasslabels,thenthenodeisnotexpandedanyfurther.Eachleafnodeisassignedaclasslabelthatoccursmostfrequentlyinthetraininginstancesassociatedwiththenode.
Toillustratehowthealgorithmworks,considerthetrainingsetshowninTable3.3 fortheloanborrowerclassificationproblem.SupposeweapplyHunt'salgorithmtofitthetrainingdata.ThetreeinitiallycontainsonlyasingleleafnodeasshowninFigure3.6(a) .ThisnodeislabeledasDefaulted=No,sincethemajorityoftheborrowersdidnotdefaultontheirloanpayments.Thetrainingerrorofthistreeis30%asthreeoutofthetentraininginstanceshave
theclasslabel .Theleafnodecanthereforebefurtherexpandedbecauseitcontainstraininginstancesfrommorethanoneclass.
Figure3.6.Hunt'salgorithmforbuildingdecisiontrees.
LetHomeOwnerbetheattributechosentosplitthetraininginstances.Thejustificationforchoosingthisattributeastheattributetestconditionwillbediscussedlater.TheresultingbinarysplitontheHomeOwnerattributeisshowninFigure3.6(b) .AllthetraininginstancesforwhichHomeOwner=Yesarepropagatedtotheleftchildoftherootnodeandtherestarepropagatedtotherightchild.Hunt'salgorithmisthenrecursivelyappliedtoeachchild.Theleftchildbecomesaleafnodelabeled ,since
Defaulted=Yes
Defaulted=No
allinstancesassociatedwiththisnodehaveidenticalclasslabel.Therightchildhasinstancesfromeachclasslabel.Hence,
wesplititfurther.TheresultingsubtreesafterrecursivelyexpandingtherightchildareshowninFigures3.6(c) and(d) .
Hunt'salgorithm,asdescribedabove,makessomesimplifyingassumptionsthatareoftennottrueinpractice.Inthefollowing,wedescribetheseassumptionsandbrieflydiscusssomeofthepossiblewaysforhandlingthem.
1. SomeofthechildnodescreatedinHunt'salgorithmcanbeemptyifnoneofthetraininginstanceshavetheparticularattributevalues.Onewaytohandlethisisbydeclaringeachofthemasaleafnodewithaclasslabelthatoccursmostfrequentlyamongthetraininginstancesassociatedwiththeirparentnodes.
2. Ifalltraininginstancesassociatedwithanodehaveidenticalattributevaluesbutdifferentclasslabels,itisnotpossibletoexpandthisnodeanyfurther.Onewaytohandlethiscaseistodeclareitaleafnodeandassignittheclasslabelthatoccursmostfrequentlyinthetraininginstancesassociatedwiththisnode.
DesignIssuesofDecisionTreeInductionHunt'salgorithmisagenericprocedureforgrowingdecisiontreesinagreedyfashion.Toimplementthealgorithm,therearetwokeydesignissuesthatmustbeaddressed.
1. Whatisthesplittingcriterion?Ateachrecursivestep,anattributemustbeselectedtopartitionthetraininginstancesassociatedwithanodeintosmallersubsetsassociatedwithitschildnodes.Thesplittingcriteriondetermineswhichattributeischosenasthetestconditionand
Defaulted=No
howthetraininginstancesshouldbedistributedtothechildnodes.ThiswillbediscussedinSections3.3.2 and3.3.3 .
2. Whatisthestoppingcriterion?Thebasicalgorithmstopsexpandinganodeonlywhenallthetraininginstancesassociatedwiththenodehavethesameclasslabelsorhaveidenticalattributevalues.Althoughtheseconditionsaresufficient,therearereasonstostopexpandinganodemuchearliereveniftheleafnodecontainstraininginstancesfrommorethanoneclass.Thisprocessiscalledearlyterminationandtheconditionusedtodeterminewhenanodeshouldbestoppedfromexpandingiscalledastoppingcriterion.TheadvantagesofearlyterminationarediscussedinSection3.4 .
3.3.2MethodsforExpressingAttributeTestConditions
Decisiontreeinductionalgorithmsmustprovideamethodforexpressinganattributetestconditionanditscorrespondingoutcomesfordifferentattributetypes.
BinaryAttributes
Thetestconditionforabinaryattributegeneratestwopotentialoutcomes,asshowninFigure3.7 .
Figure3.7.Attributetestconditionforabinaryattribute.
NominalAttributes
Sinceanominalattributecanhavemanyvalues,itsattributetestconditioncanbeexpressedintwoways,asamultiwaysplitorabinarysplitasshowninFigure3.8 .Foramultiwaysplit(Figure3.8(a) ),thenumberofoutcomesdependsonthenumberofdistinctvaluesforthecorrespondingattribute.Forexample,ifanattributesuchasmaritalstatushasthreedistinctvalues—single,married,ordivorced—itstestconditionwillproduceathree-waysplit.Itisalsopossibletocreateabinarysplitbypartitioningallvaluestakenbythenominalattributeintotwogroups.Forexample,somedecisiontreealgorithms,suchasCART,produceonlybinarysplitsbyconsideringall
waysofcreatingabinarypartitionofkattributevalues.Figure3.8(b)illustratesthreedifferentwaysofgroupingtheattributevaluesformaritalstatusintotwosubsets.
2k−1−1
Figure3.8.Attributetestconditionsfornominalattributes.
OrdinalAttributes
Ordinalattributescanalsoproducebinaryormulti-waysplits.Ordinalattributevaluescanbegroupedaslongasthegroupingdoesnotviolatetheorderpropertyoftheattributevalues.Figure3.9 illustratesvariouswaysofsplittingtrainingrecordsbasedontheShirtSizeattribute.ThegroupingsshowninFigures3.9(a) and(b) preservetheorderamongtheattributevalues,whereasthegroupingshowninFigure3.9(c) violatesthispropertybecauseitcombinestheattributevaluesSmallandLargeintothesamepartitionwhileMediumandExtraLargearecombinedintoanotherpartition.
Figure3.9.Differentwaysofgroupingordinalattributevalues.
ContinuousAttributes
Forcontinuousattributes,theattributetestconditioncanbeexpressedasacomparisontest(e.g., )producingabinarysplit,orasarangequeryoftheform ,for producingamultiwaysplit.ThedifferencebetweentheseapproachesisshowninFigure3.10 .Forthebinarysplit,anypossiblevaluevbetweentheminimumandmaximumattributevaluesinthetrainingdatacanbeusedforconstructingthecomparisontest .However,itissufficienttoonlyconsiderdistinctattributevaluesinthetrainingsetascandidatesplitpositions.Forthemultiwaysplit,anypossiblecollectionofattributevaluerangescanbeused,aslongastheyaremutuallyexclusiveandcovertheentirerangeofattributevaluesbetweentheminimumandmaximumvaluesobservedinthetrainingset.OneapproachforconstructingmultiwaysplitsistoapplythediscretizationstrategiesdescribedinSection2.3.6 onpage63.Afterdiscretization,anewordinalvalueisassignedtoeachdiscretizedinterval,andtheattributetestconditionisthendefinedusingthisnewlyconstructedordinalattribute.
A<vvi≤A<vi+1 i=1,…,k,
A<v
Figure3.10.Testconditionforcontinuousattributes.
3.3.3MeasuresforSelectinganAttributeTestCondition
Therearemanymeasuresthatcanbeusedtodeterminethegoodnessofanattributetestcondition.Thesemeasurestrytogivepreferencetoattributetestconditionsthatpartitionthetraininginstancesintopurersubsetsinthechildnodes,whichmostlyhavethesameclasslabels.Havingpurernodesisusefulsinceanodethathasallofitstraininginstancesfromthesameclassdoesnotneedtobeexpandedfurther.Incontrast,animpurenodecontainingtraininginstancesfrommultipleclassesislikelytorequireseverallevelsofnodeexpansions,therebyincreasingthedepthofthetreeconsiderably.Largertreesarelessdesirableastheyaremoresusceptibletomodeloverfitting,aconditionthatmaydegradetheclassificationperformanceonunseeninstances,aswillbediscussedinSection3.4 .Theyarealsodifficulttointerpretandincurmoretrainingandtesttimeascomparedtosmallertrees.
Inthefollowing,wepresentdifferentwaysofmeasuringtheimpurityofanodeandthecollectiveimpurityofitschildnodes,bothofwhichwillbeusedtoidentifythebestattributetestconditionforanode.
ImpurityMeasureforaSingleNodeTheimpurityofanodemeasureshowdissimilartheclasslabelsareforthedatainstancesbelongingtoacommonnode.Followingareexamplesofmeasuresthatcanbeusedtoevaluatetheimpurityofanodet:
wherepi(t)istherelativefrequencyoftraininginstancesthatbelongtoclassiatnodet,cisthetotalnumberofclasses,and inentropycalculations.Allthreemeasuresgiveazeroimpurityvalueifanodecontainsinstancesfromasingleclassandmaximumimpurityifthenodehasequalproportionofinstancesfrommultipleclasses.
Figure3.11 comparestherelativemagnitudeoftheimpuritymeasureswhenappliedtobinaryclassificationproblems.Sincethereareonlytwoclasses, .Thehorizontalaxispreferstothefractionofinstancesthatbelongtooneofthetwoclasses.Observethatallthreemeasuresattaintheirmaximumvaluewhentheclassdistributionisuniform(i.e.,
)andminimumvaluewhenalltheinstancesbelongtoasingleclass(i.e.,either or equalsto1).Thefollowingexamplesillustratehowthevaluesoftheimpuritymeasuresvaryaswealtertheclassdistribution.
Entropy=−∑i=0c−1pi(t)log2pi(t), (3.4)
Giniindex=1−∑i=0c−1pi(t)2, (3.5)
Classificationerror=1−maxi[pi(t)], (3.6)
0log20=0
p0(t)+p1(t)=1
p0(t)+p1(t)=0.5p0(t) p1(t)
Figure3.11.Comparisonamongtheimpuritymeasuresforbinaryclassificationproblems.
Node Count
0
6
Node Count
1
5
Node Count
3
N1 Gini=1−(0/6)2−(6/6)2=0
Class=0 Entropy=−(0/6)log2(0/6)−(6/6)log2(6/6)=0
Class=1 Error=1−max[0/6,6/6]=0
N2 Gini=1−(1/6)2−(5/6)2=0.278
Class=0 Entropy=−(1/6)log2(1/6)−(5/6)log2(5/6)=0.650
Class=1 Error=1−max[1/6,5/6]=0.167
N3 Gini=1−(3/6)2−(3/6)2=0.5
Class=0 Entropy=−(3/6)log2(3/6)−(3/6)log2(3/6)=1
3
Basedonthesecalculations,node hasthelowestimpurityvalue,followedby and .Thisexample,alongwithFigure3.11 ,showstheconsistencyamongtheimpuritymeasures,i.e.,ifanode haslowerentropythannode ,thentheGiniindexanderrorrateof willalsobelowerthanthatof .Despitetheiragreement,theattributechosenassplittingcriterionbytheimpuritymeasurescanstillbedifferent(seeExercise6onpage187).
CollectiveImpurityofChildNodesConsideranattributetestconditionthatsplitsanodecontainingNtraininginstancesintokchildren, ,whereeverychildnoderepresentsapartitionofthedataresultingfromoneofthekoutcomesoftheattributetestcondition.Let bethenumberoftraininginstancesassociatedwithachildnode ,whoseimpurityvalueis .Sinceatraininginstanceintheparentnodereachesnode forafractionof times,thecollectiveimpurityofthechildnodescanbecomputedbytakingaweightedsumoftheimpuritiesofthechildnodes,asfollows:
3.3.ExampleWeightedEntropyConsiderthecandidateattributetestconditionshowninFigures3.12(a)and(b) fortheloanborrowerclassificationproblem.SplittingontheHomeOwnerattributewillgeneratetwochildnodes
Class=1 Error=1−max[6/6,3/6]=0.5
N1N2 N3
N1N2 N1N2
{v1,v2,⋯,vk}
N(vj)vj I(vj)
vj N(vj)/N
I(children)=∑j=1kN(vj)NI(vj), (3.7)
Figure3.12.Examplesofcandidateattributetestconditions.
whoseweightedentropycanbecalculatedasfollows:
SplittingonMaritalStatus,ontheotherhand,leadstothreechildnodeswithaweightedentropygivenby
Thus,MaritalStatushasalowerweightedentropythanHomeOwner.
IdentifyingthebestattributetestconditionTodeterminethegoodnessofanattributetestcondition,weneedtocomparethedegreeofimpurityoftheparentnode(beforesplitting)withtheweighteddegreeofimpurityofthechildnodes(aftersplitting).Thelargertheir
I(HomeOwner=yes)=03log203−33log233=0I(HomeOwner=no)=−37log237−47log247=0.985I(HomeOwner=310×0+710×0.985=0.690
I(MaritalStatus=Single)=−25log225−35log235=0.971I(MaritalStatus=Married)=−03log203−33log233=0I(MaritalStatus=Divorced)=−12log212−12log212=1.000I(MaritalStatus)=510×0.971+310×0+210×1=0.686
difference,thebetterthetestcondition.Thisdifference, ,alsotermedasthegaininpurityofanattributetestcondition,canbedefinedasfollows:
Figure3.13.SplittingcriteriafortheloanborrowerclassificationproblemusingGiniindex.
whereI(parent)istheimpurityofanodebeforesplittingandI(children)istheweightedimpuritymeasureaftersplitting.Itcanbeshownthatthegainisnon-negativesince foranyreasonablemeasuresuchasthosepresentedabove.Thehigherthegain,thepureraretheclassesinthechildnodesrelativetotheparentnode.Thesplittingcriterioninthedecisiontreelearningalgorithmselectstheattributetestconditionthatshowsthemaximumgain.NotethatmaximizingthegainatagivennodeisequivalenttominimizingtheweightedimpuritymeasureofitschildrensinceI(parent)isthesameforallcandidateattributetestconditions.Finally,whenentropyisused
Δ
Δ=I(parent)−I(children), (3.8)
I(parent)≥I(children)
astheimpuritymeasure,thedifferenceinentropyiscommonlyknownasinformationgain, .
Inthefollowing,wepresentillustrativeapproachesforidentifyingthebestattributetestconditiongivenqualitativeorquantitativeattributes.
SplittingofQualitativeAttributesConsiderthefirsttwocandidatesplitsshowninFigure3.12 involvingqualitativeattributes and .Theinitialclassdistributionattheparentnodeis(0.3,0.7),sincethereare3instancesofclass and7instancesofclass inthetrainingdata.Thus,
TheinformationgainsforHomeOwnerandMaritalStatusareeachgivenby
TheinformationgainforMaritalStatusisthushigherduetoitslowerweightedentropy,whichwillthusbeconsideredforsplitting.
BinarySplittingofQualitativeAttributesConsiderbuildingadecisiontreeusingonlybinarysplitsandtheGiniindexastheimpuritymeasure.Figure3.13 showsexamplesoffourcandidatesplittingcriteriaforthe and attributes.Sincethereare3borrowersinthetrainingsetwhodefaultedand7otherswhorepaidtheirloan(seeTableinFigure3.13 ),theGiniindexoftheparentnodebeforesplittingis
Δinfo
I(parent)=−310log2310−710log2710=0.881
Δinfo(HomeOwner)=0.881−0.690=0.191Δinfo(MaritalStatus)=0.881−0.686=0.195
If ischosenasthesplittingattribute,theGiniindexforthechildnodes and are0and0.490,respectively.TheweightedaverageGiniindexforthechildrenis
wheretheweightsrepresenttheproportionoftraininginstancesassignedtoeachchild.Thegainusing assplittingattributeis
.Similarly,wecanapplyabinarysplitontheattribute.However,since isanominalattributewith
threeoutcomes,therearethreepossiblewaystogrouptheattributevaluesintoabinarysplit.TheweightedaverageGiniindexofthechildrenforeachcandidatebinarysplitisshowninFigure3.13 .Basedontheseresults,
andthelastbinarysplitusing areclearlythebestcandidates,sincetheybothproducethelowestweightedaverageGiniindex.Binarysplitscanalsobeusedforordinalattributes,ifthebinarypartitioningoftheattributevaluesdoesnotviolatetheorderingpropertyofthevalues.
BinarySplittingofQuantitativeAttributesConsidertheproblemofidentifyingthebestbinarysplit fortheprecedingloanapprovalclassificationproblem.Asdiscussedpreviously,eventhough cantakeanyvaluebetweentheminimumandmaximumvaluesofannualincomeinthetrainingset,itissufficienttoonlyconsidertheannualincomevaluesobservedinthetrainingsetascandidatesplitpositions.Foreachcandidate ,thetrainingsetisscannedoncetocountthenumberofborrowerswithannualincomelessthanorgreaterthan alongwiththeirclassproportions.WecanthencomputetheGiniindexateachcandidatesplit
1−(310)2−(710)2=0.420.
N1 N2
(3/10)×0+(7/10)×0.490=0.343,
0.420−0.343=0.077
AnnualIncome≤τ
τ
ττ
positionandchoosethe thatproducesthelowestvalue.ComputingtheGiniindexateachcandidatesplitpositionrequiresO(N)operations,whereNisthenumberoftraininginstances.SincethereareatmostNpossiblecandidates,theoverallcomplexityofthisbrute-forcemethodis .ItispossibletoreducethecomplexityofthisproblemtoO(NlogN)byusingamethoddescribedasfollows(seeillustrationinFigure3.14 ).Inthismethod,wefirstsortthetraininginstancesbasedontheirannualincome,aone-timecostthatrequiresO(NlogN)operations.Thecandidatesplitpositionsaregivenbythemidpointsbetweeneverytwoadjacentsortedvalues:$55,000,$65,000,$72,500,andsoon.Forthefirstcandidate,sincenoneoftheinstanceshasanannualincomelessthanorequalto$55,000,theGiniindexforthechildnodewith isequaltozero.Incontrast,thereare3traininginstancesofclass and instancesofclassNowithannualincomegreaterthan$55,000.TheGiniindexforthisnodeis0.420.TheweightedaverageGiniindexforthefirstcandidatesplitposition, ,isequalto .
Figure3.14.Splittingcontinuousattributes.
Forthenextcandidate, ,theclassdistributionofitschildnodescanbeobtainedwithasimpleupdateofthedistributionforthepreviouscandidate.Thisisbecause,as increasesfrom$55,000to$65,000,thereisonlyone
τ
O(N2)
AnnualIncome<$55,000
τ=$55,0000×0+1×0.420=0.420
τ=$65,000
τ
traininginstanceaffectedbythechange.Byexaminingtheclasslabeloftheaffectedtraininginstance,thenewclassdistributionisobtained.Forexample,as increasesto$65,000,thereisonlyoneborrowerinthetrainingset,withanannualincomeof$60,000,affectedbythischange.Sincetheclasslabelfortheborroweris ,thecountforclass increasesfrom0to1(for
)anddecreasesfrom7to6(for),asshowninFigure3.14 .Thedistributionforthe
classremainsunaffected.TheupdatedGiniindexforthiscandidatesplitpositionis0.400.
ThisprocedureisrepeateduntiltheGiniindexforallcandidatesarefound.ThebestsplitpositioncorrespondstotheonethatproducesthelowestGiniindex,whichoccursat .SincetheGiniindexateachcandidatesplitpositioncanbecomputedinO(1)time,thecomplexityoffindingthebestsplitpositionisO(N)onceallthevaluesarekeptsorted,aone-timeoperationthattakesO(NlogN)time.TheoverallcomplexityofthismethodisthusO(NlogN),whichismuchsmallerthanthe timetakenbythebrute-forcemethod.Theamountofcomputationcanbefurtherreducedbyconsideringonlycandidatesplitpositionslocatedbetweentwoadjacentsortedinstanceswithdifferentclasslabels.Forexample,wedonotneedtoconsidercandidatesplitpositionslocatedbetween$60,000and$75,000becauseallthreeinstanceswithannualincomeinthisrange($60,000,$70,000,and$75,000)havethesameclasslabels.Choosingasplitpositionwithinthisrangeonlyincreasesthedegreeofimpurity,comparedtoasplitpositionlocatedoutsidethisrange.Therefore,thecandidatesplitpositionsat and
canbeignored.Similarly,wedonotneedtoconsiderthecandidatesplitpositionsat$87,500,$92,500,$110,000,$122,500,and$172,500becausetheyarelocatedbetweentwoadjacentinstanceswiththesamelabels.Thisstrategyreducesthenumberofcandidatesplitpositionstoconsiderfrom9to2(excludingthetwoboundarycases and
).
τ
AnnualIncome≤$65,000AnnualIncome>$65,000
τ=$97,500
O(N2)
τ=$65,000τ=$72,500
τ=$55,000τ=$230,000
GainRatioOnepotentiallimitationofimpuritymeasuressuchasentropyandGiniindexisthattheytendtofavorqualitativeattributeswithlargenumberofdistinctvalues.Figure3.12 showsthreecandidateattributesforpartitioningthedatasetgiveninTable3.3 .Aspreviouslymentioned,theattribute
isabetterchoicethantheattribute ,becauseitprovidesalargerinformationgain.However,ifwecomparethemagainst ,thelatterproducesthepurestpartitionswiththemaximuminformationgain,sincetheweightedentropyandGiniindexisequaltozeroforitschildren.Yet,
isnotagoodattributeforsplittingbecauseithasauniquevalueforeachinstance.Eventhoughatestconditioninvolving willaccuratelyclassifyeveryinstanceinthetrainingdata,wecannotusesuchatestconditiononnewtestinstanceswith valuesthathaven'tbeenseenbeforeduringtraining.Thisexamplesuggestshavingalowimpurityvaluealoneisinsufficienttofindagoodattributetestconditionforanode.AswewillseelaterinSection3.4 ,havingmorenumberofchildnodescanmakeadecisiontreemorecomplexandconsequentlymoresusceptibletooverfitting.Hence,thenumberofchildrenproducedbythesplittingattributeshouldalsobetakenintoconsiderationwhiledecidingthebestattributetestcondition.
Therearetwowaystoovercomethisproblem.Onewayistogenerateonlybinarydecisiontrees,thusavoidingthedifficultyofhandlingattributeswithvaryingnumberofpartitions.ThisstrategyisemployedbydecisiontreeclassifierssuchasCART.Anotherwayistomodifythesplittingcriteriontotakeintoaccountthenumberofpartitionsproducedbytheattribute.Forexample,intheC4.5decisiontreealgorithm,ameasureknownasgainratioisusedtocompensateforattributesthatproducealargenumberofchildnodes.Thismeasureiscomputedasfollows:
where isthenumberofinstancesassignedtonode andkisthetotalnumberofsplits.Thesplitinformationmeasurestheentropyofsplittinganodeintoitschildnodesandevaluatesifthesplitresultsinalargernumberofequally-sizedchildnodesornot.Forexample,ifeverypartitionhasthesamenumberofinstances,then andthesplitinformationwouldbeequaltolog k.Thus,ifanattributeproducesalargenumberofsplits,itssplitinformationisalsolarge,whichinturn,reducesthegainratio.
3.4.ExampleGainRatioConsiderthedatasetgiveninExercise2onpage185.Wewanttoselectthebestattributetestconditionamongthefollowingthreeattributes:
, ,and .Theentropybeforesplittingis
If isusedasattributetestcondition:
If isusedasattributetestcondition:
Finally,if isusedasattributetestcondition:
Gainratio=ΔinfoSplitInfo=Entropy(Parent)−∑i=1kN(vi)NEntropy(vi)−∑i=1kN(vi)Nlog2N(vi)N
(3.9)
N(vi) vi
∀i:N(vi)/N=1/k2
Entropy(parent)=−1020log21020−1020log21020=1.
Entropy(children)=1020[−610log2610−410log2410]×2=0.971GainRatio=1−0.971−1020log21020−1020log21020=0.0291=0.029
Entropy(children)=420[−14log214−34log234]+820×0+820[−18log218−78log278]=0.380GainRatio=1−0.380−420log2420−820log2820−820log2820=0.6201.52
Thus,eventhough hasthehighestinformationgain,itsgainratioislowerthan sinceitproducesalargernumberofsplits.
3.3.4AlgorithmforDecisionTreeInduction
Algorithm3.1 presentsapseudocodefordecisiontreeinductionalgorithm.TheinputtothisalgorithmisasetoftraininginstancesEalongwiththeattributesetF.Thealgorithmworksbyrecursivelyselectingthebestattributetosplitthedata(Step7)andexpandingthenodesofthetree(Steps11and12)untilthestoppingcriterionismet(Step1).Thedetailsofthisalgorithmareexplainedbelow.
1. The functionextendsthedecisiontreebycreatinganewnode.Anodeinthedecisiontreeeitherhasatestcondition,denotedasnode.testcond,oraclasslabel,denotedasnode.label.
2. The functiondeterminestheattributetestconditionforpartitioningthetraininginstancesassociatedwithanode.Thesplittingattributechosendependsontheimpuritymeasureused.ThepopularmeasuresincludeentropyandtheGiniindex.
3. The functiondeterminestheclasslabeltobeassignedtoaleafnode.Foreachleafnodet,let denotethefractionoftraininginstancesfromclassiassociatedwiththenodet.Thelabelassignedto
Entropy(children)=120[−11log211−01log201]×20=0GainRatio=1−0−120log2120×20=14.32=0.23
p(i|t)
theleafnodeistypicallytheonethatoccursmostfrequentlyinthetraininginstancesthatareassociatedwiththisnode.
Algorithm3.1Askeletondecisiontreeinductionalgorithm.
∈
∈
wheretheargmaxoperatorreturnstheclassithatmaximizes .Besidesprovidingtheinformationneededtodeterminetheclasslabel
leaf.label=argmaxip(i|t), (3.10)
p(i|t)
ofaleafnode, canalsobeusedasaroughestimateoftheprobabilitythataninstanceassignedtotheleafnodetbelongstoclassi.Sections4.11.2 and4.11.4 inthenextchapterdescribehowsuchprobabilityestimatescanbeusedtodeterminetheperformanceofadecisiontreeunderdifferentcostfunctions.
4. The functionisusedtoterminatethetree-growingprocessbycheckingwhetheralltheinstanceshaveidenticalclasslabelorattributevalues.Sincedecisiontreeclassifiersemployatop-down,recursivepartitioningapproachforbuildingamodel,thenumberoftraininginstancesassociatedwithanodedecreasesasthedepthofthetreeincreases.Asaresult,aleafnodemaycontaintoofewtraininginstancestomakeastatisticallysignificantdecisionaboutitsclasslabel.Thisisknownasthedatafragmentationproblem.Onewaytoavoidthisproblemistodisallowsplittingofanodewhenthenumberofinstancesassociatedwiththenodefallbelowacertainthreshold.Amoresystematicwaytocontrolthesizeofadecisiontree(numberofleafnodes)willbediscussedinSection3.5.4 .
3.3.5ExampleApplication:WebRobotDetection
Considerthetaskofdistinguishingtheaccesspatternsofwebrobotsfromthosegeneratedbyhumanusers.Awebrobot(alsoknownasawebcrawler)isasoftwareprogramthatautomaticallyretrievesfilesfromoneormorewebsitesbyfollowingthehyperlinksextractedfromaninitialsetofseedURLs.Theseprogramshavebeendeployedforvariouspurposes,fromgatheringwebpagesonbehalfofsearchenginestomoremaliciousactivitiessuchasspammingandcommittingclickfraudsinonlineadvertisements.
p(i|t)
Figure3.15.Inputdataforwebrobotdetection.
Thewebrobotdetectionproblemcanbecastasabinaryclassificationtask.Theinputdatafortheclassificationtaskisawebserverlog,asampleofwhichisshowninFigure3.15(a) .Eachlineinthelogfilecorrespondstoarequestmadebyaclient(i.e.,ahumanuserorawebrobot)tothewebserver.Thefieldsrecordedintheweblogincludetheclient'sIPaddress,timestampoftherequest,URLoftherequestedfile,sizeofthefile,anduseragent,whichisafieldthatcontainsidentifyinginformationabouttheclient.
Forhumanusers,theuseragentfieldspecifiesthetypeofwebbrowserormobiledeviceusedtofetchthefiles,whereasforwebrobots,itshouldtechnicallycontainthenameofthecrawlerprogram.However,webrobotsmayconcealtheirtrueidentitiesbydeclaringtheiruseragentfieldstobeidenticaltoknownbrowsers.Therefore,useragentisnotareliablefieldtodetectwebrobots.
Thefirststeptowardbuildingaclassificationmodelistopreciselydefineadatainstanceandassociatedattributes.Asimpleapproachistoconsidereachlogentryasadatainstanceandusetheappropriatefieldsinthelogfileasitsattributeset.Thisapproach,however,isinadequateforseveralreasons.First,manyoftheattributesarenominal-valuedandhaveawiderangeofdomainvalues.Forexample,thenumberofuniqueclientIPaddresses,URLs,andreferrersinalogfilecanbeverylarge.Theseattributesareundesirableforbuildingadecisiontreebecausetheirsplitinformationisextremelyhigh(seeEquation(3.9) ).Inaddition,itmightnotbepossibletoclassifytestinstancescontainingIPaddresses,URLs,orreferrersthatarenotpresentinthetrainingdata.Finally,byconsideringeachlogentryasaseparatedatainstance,wedisregardthesequenceofwebpagesretrievedbytheclient—acriticalpieceofinformationthatcanhelpdistinguishwebrobotaccessesfromthoseofahumanuser.
Abetteralternativeistoconsidereachwebsessionasadatainstance.Awebsessionisasequenceofrequestsmadebyaclientduringagivenvisittothewebsite.Eachwebsessioncanbemodeledasadirectedgraph,inwhichthenodescorrespondtowebpagesandtheedgescorrespondtohyperlinksconnectingonewebpagetoanother.Figure3.15(b) showsagraphicalrepresentationofthefirstwebsessiongiveninthelogfile.Everywebsessioncanbecharacterizedusingsomemeaningfulattributesaboutthegraphthatcontaindiscriminatoryinformation.Figure3.15(c) showssomeoftheattributesextractedfromthegraph,includingthedepthandbreadthofits
correspondingtreerootedattheentrypointtothewebsite.Forexample,thedepthandbreadthofthetreeshowninFigure3.15(b) arebothequaltotwo.
ThederivedattributesshowninFigure3.15(c) aremoreinformativethantheoriginalattributesgiveninthelogfilebecausetheycharacterizethebehavioroftheclientatthewebsite.Usingthisapproach,adatasetcontaining2916instanceswascreated,withequalnumbersofsessionsduetowebrobots(class1)andhumanusers(class0).10%ofthedatawerereservedfortrainingwhiletheremaining90%wereusedfortesting.TheinduceddecisiontreeisshowninFigure3.16 ,whichhasanerrorrateequalto3.8%onthetrainingsetand5.3%onthetestset.Inadditiontoitslowerrorrate,thetreealsorevealssomeinterestingpropertiesthatcanhelpdiscriminatewebrobotsfromhumanusers:
1. Accessesbywebrobotstendtobebroadbutshallow,whereasaccessesbyhumanuserstendtobemorefocused(narrowbutdeep).
2. Webrobotsseldomretrievetheimagepagesassociatedwithawebpage.
3. Sessionsduetowebrobotstendtobelongandcontainalargenumberofrequestedpages.
4. Webrobotsaremorelikelytomakerepeatedrequestsforthesamewebpagethanhumanuserssincethewebpagesretrievedbyhumanusersareoftencachedbythebrowser.
3.3.6CharacteristicsofDecisionTreeClassifiers
Thefollowingisasummaryoftheimportantcharacteristicsofdecisiontreeinductionalgorithms.
1. Applicability:Decisiontreesareanonparametricapproachforbuildingclassificationmodels.Thisapproachdoesnotrequireanypriorassumptionabouttheprobabilitydistributiongoverningtheclassandattributesofthedata,andthus,isapplicabletoawidevarietyofdatasets.Itisalsoapplicabletobothcategoricalandcontinuousdatawithoutrequiringtheattributestobetransformedintoacommonrepresentationviabinarization,normalization,orstandardization.UnlikesomebinaryclassifiersdescribedinChapter4 ,itcanalsodealwithmulticlassproblemswithouttheneedtodecomposethemintomultiplebinaryclassificationtasks.Anotherappealingfeatureofdecisiontreeclassifiersisthattheinducedtrees,especiallytheshorterones,arerelativelyeasytointerpret.Theaccuraciesofthetreesarealsoquitecomparabletootherclassificationtechniquesformanysimpledatasets.
2. Expressiveness:Adecisiontreeprovidesauniversalrepresentationfordiscrete-valuedfunctions.Inotherwords,itcanencodeanyfunctionofdiscrete-valuedattributes.Thisisbecauseeverydiscrete-valuedfunctioncanberepresentedasanassignmenttable,whereeveryuniquecombinationofdiscreteattributesisassignedaclasslabel.Sinceeverycombinationofattributescanberepresentedasaleafinthedecisiontree,wecanalwaysfindadecisiontreewhoselabelassignmentsattheleafnodesmatcheswiththeassignmenttableoftheoriginalfunction.Decisiontreescanalsohelpinprovidingcompactrepresentationsoffunctionswhensomeoftheuniquecombinationsofattributescanberepresentedbythesameleafnode.Forexample,Figure3.17 showstheassignmenttableoftheBooleanfunction
involvingfourbinaryattributes,resultinginatotalofpossibleassignments.ThetreeshowninFigure3.17 shows
(A∧B)∨(C∧D)24=16
acompressedencodingofthisassignmenttable.Insteadofrequiringafully-growntreewith16leafnodes,itispossibletoencodethefunctionusingasimplertreewithonly7leafnodes.Nevertheless,notalldecisiontreesfordiscrete-valuedattributescanbesimplified.Onenotableexampleistheparityfunction,whosevalueis1whenthereisanevennumberoftruevaluesamongitsBooleanattributes,and0otherwise.Accuratemodelingofsuchafunctionrequiresafulldecisiontreewith nodes,wheredisthenumberofBooleanattributes(seeExercise1onpage185).
2d
Figure3.16.Decisiontreemodelforwebrobotdetection.
Figure3.17.DecisiontreefortheBooleanfunction .
3. ComputationalEfficiency:Sincethenumberofpossibledecisiontreescanbeverylarge,manydecisiontreealgorithmsemployaheuristic-basedapproachtoguidetheirsearchinthevasthypothesisspace.Forexample,thealgorithmpresentedinSection3.3.4 usesagreedy,top-down,recursivepartitioningstrategyforgrowingadecisiontree.Formanydatasets,suchtechniquesquicklyconstructareasonablygooddecisiontreeevenwhenthetrainingsetsizeisverylarge.Furthermore,onceadecisiontreehasbeenbuilt,classifyingatestrecordisextremelyfast,withaworst-casecomplexityofO(w),wherewisthemaximumdepthofthetree.
4. HandlingMissingValues:Adecisiontreeclassifiercanhandlemissingattributevaluesinanumberofways,bothinthetrainingandthetestsets.Whentherearemissingvaluesinthetestset,theclassifiermustdecidewhichbranchtofollowifthevalueofasplitting
(A∧B)∨(C∧D)
nodeattributeismissingforagiventestinstance.Oneapproach,knownastheprobabilisticsplitmethod,whichisemployedbytheC4.5decisiontreeclassifier,distributesthedatainstancetoeverychildofthesplittingnodeaccordingtotheprobabilitythatthemissingattributehasaparticularvalue.Incontrast,theCARTalgorithmusesthesurrogatesplitmethod,wheretheinstancewhosesplittingattributevalueismissingisassignedtooneofthechildnodesbasedonthevalueofanothernon-missingsurrogateattributewhosesplitsmostresemblethepartitionsmadebythemissingattribute.Anotherapproach,knownastheseparateclassmethodisusedbytheCHAIDalgorithm,wherethemissingvalueistreatedasaseparatecategoricalvaluedistinctfromothervaluesofthesplittingattribute.Figure3.18showsanexampleofthethreedifferentwaysforhandlingmissingvaluesinadecisiontreeclassifier.Otherstrategiesfordealingwithmissingvaluesarebasedondatapreprocessing,wheretheinstancewithmissingvalueiseitherimputedwiththemode(forcategoricalattribute)ormean(forcontinuousattribute)valueordiscardedbeforetheclassifieristrained.
Figure3.18.Methodsforhandlingmissingattributevaluesindecisiontreeclassifier.
Duringtraining,ifanattributevhasmissingvaluesinsomeofthetraininginstancesassociatedwithanode,weneedawaytomeasurethegaininpurityifvisusedforsplitting.Onesimplewayistoexcludeinstanceswithmissingvaluesofvinthecountingofinstancesassociatedwitheverychildnode,generatedforeverypossibleoutcomeofv.Further,ifvischosenastheattributetestconditionatanode,traininginstanceswithmissingvaluesofvcanbepropagatedtothechildnodesusinganyofthemethodsdescribedaboveforhandlingmissingvaluesintestinstances.
5. HandlingInteractionsamongAttributes:Attributesareconsideredinteractingiftheyareabletodistinguishbetweenclasseswhenusedtogether,butindividuallytheyprovidelittleornoinformation.Duetothegreedynatureofthesplittingcriteriaindecisiontrees,suchattributescouldbepassedoverinfavorofotherattributesthatarenotasuseful.Thiscouldresultinmorecomplexdecisiontreesthannecessary.Hence,decisiontreescanperformpoorlywhenthereareinteractionsamongattributes.Toillustratethispoint,considerthethree-dimensionaldatashowninFigure3.19(a) ,whichcontains2000datapointsfromoneoftwoclasses,denotedas and inthediagram.Figure3.19(b) showsthedistributionofthetwoclassesinthetwo-dimensionalspaceinvolvingattributesXandY,whichisanoisyversionoftheXORBooleanfunction.Wecanseethateventhoughthetwoclassesarewell-separatedinthistwo-dimensionalspace,neitherofthetwoattributescontainsufficientinformationtodistinguishbetweenthetwoclasseswhenusedalone.Forexample,theentropiesofthefollowingattributetestconditions: and ,arecloseto1,indicatingthatneitherXnorYprovideanyreductionintheimpuritymeasurewhenusedindividually.XandYthusrepresentacaseofinteractionamongattributes.Thedatasetalsocontainsathirdattribute,Z,inwhichbothclassesaredistributeduniformly,asshowninFigures3.19(c) and
+ ∘
X≤10 Y≤10
3.19(d) ,andhence,theentropyofanysplitinvolvingZiscloseto1.Asaresult,Zisaslikelytobechosenforsplittingastheinteractingbutusefulattributes,XandY.Forfurtherillustrationofthisissue,readersarereferredtoExample3.7 inSection3.4.1 andExercise7attheendofthischapter.
Figure3.19.ExampleofaXORdatainvolvingXandY,alongwithanirrelevantattributeZ.
6. HandlingIrrelevantAttributes:Anattributeisirrelevantifitisnotusefulfortheclassificationtask.Sinceirrelevantattributesarepoorlyassociatedwiththetargetclasslabels,theywillprovidelittleornogaininpurityandthuswillbepassedoverbyothermorerelevantfeatures.Hence,thepresenceofasmallnumberofirrelevantattributeswillnotimpactthedecisiontreeconstructionprocess.However,notallattributesthatprovidelittletonogainareirrelevant(seeFigure3.19 ).Hence,iftheclassificationproblemiscomplex(e.g.,involvinginteractionsamongattributes)andtherearealargenumberofirrelevantattributes,thensomeoftheseattributesmaybeaccidentallychosenduringthetree-growingprocess,sincetheymayprovideabettergainthanarelevantattributejustbyrandomchance.Featureselectiontechniquescanhelptoimprovetheaccuracyofdecisiontreesbyeliminatingtheirrelevantattributesduringpreprocessing.WewillinvestigatetheissueoftoomanyirrelevantattributesinSection3.4.1 .
7. HandlingRedundantAttributes:Anattributeisredundantifitisstronglycorrelatedwithanotherattributeinthedata.Sinceredundantattributesshowsimilargainsinpurityiftheyareselectedforsplitting,onlyoneofthemwillbeselectedasanattributetestconditioninthedecisiontreealgorithm.Decisiontreescanthushandlethepresenceofredundantattributes.
8. UsingRectilinearSplits:Thetestconditionsdescribedsofarinthischapterinvolveusingonlyasingleattributeatatime.Asaconsequence,thetree-growingprocedurecanbeviewedastheprocessofpartitioningtheattributespaceintodisjointregionsuntileachregioncontainsrecordsofthesameclass.Theborderbetweentwoneighboringregionsofdifferentclassesisknownasadecisionboundary.Figure3.20 showsthedecisiontreeaswellasthedecisionboundaryforabinaryclassificationproblem.Sincethetestconditioninvolvesonlyasingleattribute,thedecisionboundariesare
rectilinear;i.e.,paralleltothecoordinateaxes.Thislimitstheexpressivenessofdecisiontreesinrepresentingdecisionboundariesofdatasetswithcontinuousattributes.Figure3.21 showsatwo-dimensionaldatasetinvolvingbinaryclassesthatcannotbeperfectlyclassifiedbyadecisiontreewhoseattributetestconditionsaredefinedbasedonsingleattributes.ThebinaryclassesinthedatasetaregeneratedfromtwoskewedGaussiandistributions,centeredat(8,8)and(12,12),respectively.Thetruedecisionboundaryisrepresentedbythediagonaldashedline,whereastherectilineardecisionboundaryproducedbythedecisiontreeclassifierisshownbythethicksolidline.Incontrast,anobliquedecisiontreemayovercomethislimitationbyallowingthetestconditiontobespecifiedusingmorethanoneattribute.Forexample,thebinaryclassificationdatashowninFigure3.21 canbeeasilyrepresentedbyanobliquedecisiontreewithasinglerootnodewithtestcondition
Figure3.20.
x+y<20.
Exampleofadecisiontreeanditsdecisionboundariesforatwo-dimensionaldataset.
Figure3.21.Exampleofdatasetthatcannotbepartitionedoptimallyusingadecisiontreewithsingleattributetestconditions.Thetruedecisionboundaryisshownbythedashedline.
Althoughanobliquedecisiontreeismoreexpressiveandcanproducemorecompacttrees,findingtheoptimaltestconditioniscomputationallymoreexpensive.
9. ChoiceofImpurityMeasure:Itshouldbenotedthatthechoiceofimpuritymeasureoftenhaslittleeffectontheperformanceofdecisiontreeclassifierssincemanyoftheimpuritymeasuresarequiteconsistentwitheachother,asshowninFigure3.11 onpage129.Instead,thestrategyusedtoprunethetreehasagreaterimpactonthefinaltreethanthechoiceofimpuritymeasure.
3.4ModelOverfittingMethodspresentedsofartrytolearnclassificationmodelsthatshowthelowesterroronthetrainingset.However,aswewillshowinthefollowingexample,evenifamodelfitswelloverthetrainingdata,itcanstillshowpoorgeneralizationperformance,aphenomenonknownasmodeloverfitting.
Figure3.22.Examplesoftrainingandtestsetsofatwo-dimensionalclassificationproblem.
Figure3.23.Effectofvaryingtreesize(numberofleafnodes)ontrainingandtesterrors.
3.5.ExampleOverfittingandUnderfittingofDecisionTreesConsiderthetwo-dimensionaldatasetshowninFigure3.22(a) .Thedatasetcontainsinstancesthatbelongtotwoseparateclasses,representedas and ,respectively,whereeachclasshas5400instances.Allinstancesbelongingtothe classweregeneratedfromauniformdistribution.Forthe class,5000instancesweregeneratedfromaGaussiandistributioncenteredat(10,10)withunitvariance,whiletheremaining400instancesweresampledfromthesameuniformdistributionasthe class.WecanseefromFigure3.22(a) thatthe classcanbelargelydistinguishedfromthe classbydrawingacircleofappropriatesizecenteredat(10,10).Tolearnaclassifierusingthistwo-dimensionaldataset,werandomlysampled10%ofthedatafortrainingandusedtheremaining90%fortesting.Thetrainingset,showninFigure3.22(b) ,looksquiterepresentativeoftheoveralldata.WeusedGiniindexasthe
+ ∘∘
+
∘ +∘
impuritymeasuretoconstructdecisiontreesofincreasingsizes(numberofleafnodes),byrecursivelyexpandinganodeintochildnodestilleveryleafnodewaspure,asdescribedinSection3.3.4 .
Figure3.23(a) showschangesinthetrainingandtesterrorratesasthesizeofthetreevariesfrom1to8.Botherrorratesareinitiallylargewhenthetreehasonlyoneortwoleafnodes.Thissituationisknownasmodelunderfitting.Underfittingoccurswhenthelearneddecisiontreeistoosimplistic,andthus,incapableoffullyrepresentingthetruerelationshipbetweentheattributesandtheclasslabels.Asweincreasethetreesizefrom1to8,wecanobservetwoeffects.First,boththeerrorratesdecreasesincelargertreesareabletorepresentmorecomplexdecisionboundaries.Second,thetrainingandtesterrorratesarequiteclosetoeachother,whichindicatesthattheperformanceonthetrainingsetisfairlyrepresentativeofthegeneralizationperformance.Aswefurtherincreasethesizeofthetreefrom8to150,thetrainingerrorcontinuestosteadilydecreasetilliteventuallyreacheszero,asshowninFigure3.23(b) .However,inastrikingcontrast,thetesterrorrateceasestodecreaseanyfurtherbeyondacertaintreesize,andthenitbeginstoincrease.Thetrainingerrorratethusgrosslyunder-estimatesthetesterrorrateoncethetreebecomestoolarge.Further,thegapbetweenthetrainingandtesterrorrateskeepsonwideningasweincreasethetreesize.Thisbehavior,whichmayseemcounter-intuitiveatfirst,canbeattributedtothephenomenaofmodeloverfitting.
3.4.1ReasonsforModelOverfitting
Modeloverfittingisthephenomenawhere,inthepursuitofminimizingthetrainingerrorrate,anoverlycomplexmodelisselectedthatcapturesspecific
patternsinthetrainingdatabutfailstolearnthetruenatureofrelationshipsbetweenattributesandclasslabelsintheoveralldata.Toillustratethis,Figure3.24 showsdecisiontreesandtheircorrespondingdecisionboundaries(shadedrectanglesrepresentregionsassignedtothe class)fortwotreesofsizes5and50.Wecanseethatthedecisiontreeofsize5appearsquitesimpleanditsdecisionboundariesprovideareasonableapproximationtotheidealdecisionboundary,whichinthiscasecorrespondstoacirclecenteredaroundtheGaussiandistributionat(10,10).Althoughitstrainingandtesterrorratesarenon-zero,theyareveryclosetoeachother,whichindicatesthatthepatternslearnedinthetrainingsetshouldgeneralizewelloverthetestset.Ontheotherhand,thedecisiontreeofsize50appearsmuchmorecomplexthanthetreeofsize5,withcomplicateddecisionboundaries.Forexample,someofitsshadedrectangles(assignedtheclass)attempttocovernarrowregionsintheinputspacethatcontainonlyoneortwo traininginstances.Notethattheprevalenceof instancesinsuchregionsishighlyspecifictothetrainingset,astheseregionsaremostlydominatedby-instancesintheoveralldata.Hence,inanattempttoperfectlyfitthetrainingdata,thedecisiontreeofsize50startsfinetuningitselftospecificpatternsinthetrainingdata,leadingtopoorperformanceonanindependentlychosentestset.
+
+
+ +
Figure3.24.Decisiontreeswithdifferentmodelcomplexities.
Figure3.25.Performanceofdecisiontreesusing20%datafortraining(twicetheoriginaltrainingsize).
Thereareanumberoffactorsthatinfluencemodeloverfitting.Inthefollowing,weprovidebriefdescriptionsoftwoofthemajorfactors:limitedtrainingsizeandhighmodelcomplexity.Thoughtheyarenotexhaustive,theinterplaybetweenthemcanhelpexplainmostofthecommonmodeloverfittingphenomenainreal-worldapplications.
LimitedTrainingSizeNotethatatrainingsetconsistingofafinitenumberofinstancescanonlyprovidealimitedrepresentationoftheoveralldata.Hence,itispossiblethatthepatternslearnedfromatrainingsetdonotfullyrepresentthetruepatternsintheoveralldata,leadingtomodeloverfitting.Ingeneral,asweincreasethesizeofatrainingset(numberoftraininginstances),thepatternslearnedfromthetrainingsetstartresemblingthetruepatternsintheoveralldata.Hence,
theeffectofoverfittingcanbereducedbyincreasingthetrainingsize,asillustratedinthefollowingexample.
3.6ExampleEffectofTrainingSizeSupposethatweusetwicethenumberoftraininginstancesthanwhatwehadusedintheexperimentsconductedinExample3.5 .Specifically,weuse20%datafortrainingandusetheremainderfortesting.Figure3.25(b) showsthetrainingandtesterrorratesasthesizeofthetreeisvariedfrom1to150.TherearetwomajordifferencesinthetrendsshowninthisfigureandthoseshowninFigure3.23(b) (usingonly10%ofthedatafortraining).First,eventhoughthetrainingerrorratedecreaseswithincreasingtreesizeinbothfigures,itsrateofdecreaseismuchsmallerwhenweusetwicethetrainingsize.Second,foragiventreesize,thegapbetweenthetrainingandtesterrorratesismuchsmallerwhenweusetwicethetrainingsize.Thesedifferencessuggestthatthepatternslearnedusing20%ofdatafortrainingaremoregeneralizablethanthoselearnedusing10%ofdatafortraining.
Figure3.25(a) showsthedecisionboundariesforthetreeofsize50,learnedusing20%ofdatafortraining.Incontrasttothetreeofthesamesizelearnedusing10%datafortraining(seeFigure3.24(d) ),wecanseethatthedecisiontreeisnotcapturingspecificpatternsofnoisyinstancesinthetrainingset.Instead,thehighmodelcomplexityof50leafnodesisbeingeffectivelyusedtolearntheboundariesofthe instancescenteredat(10,10).
HighModelComplexityGenerally,amorecomplexmodelhasabetterabilitytorepresentcomplexpatternsinthedata.Forexample,decisiontreeswithlargernumberofleaf
+
+
nodescanrepresentmorecomplexdecisionboundariesthandecisiontreeswithfewerleafnodes.However,anoverlycomplexmodelalsohasatendencytolearnspecificpatternsinthetrainingsetthatdonotgeneralizewelloverunseeninstances.Modelswithhighcomplexityshouldthusbejudiciouslyusedtoavoidoverfitting.
Onemeasureofmodelcomplexityisthenumberof“parameters”thatneedtobeinferredfromthetrainingset.Forexample,inthecaseofdecisiontreeinduction,theattributetestconditionsatinternalnodescorrespondtotheparametersofthemodelthatneedtobeinferredfromthetrainingset.Adecisiontreewithlargernumberofattributetestconditions(andconsequentlymoreleafnodes)thusinvolvesmore“parameters”andhenceismorecomplex.
Givenaclassofmodelswithacertainnumberofparameters,alearningalgorithmattemptstoselectthebestcombinationofparametervaluesthatmaximizesanevaluationmetric(e.g.,accuracy)overthetrainingset.Ifthenumberofparametervaluecombinations(andhencethecomplexity)islarge,thelearningalgorithmhastoselectthebestcombinationfromalargenumberofpossibilities,usingalimitedtrainingset.Insuchcases,thereisahighchanceforthelearningalgorithmtopickaspuriouscombinationofparametersthatmaximizestheevaluationmetricjustbyrandomchance.Thisissimilartothemultiplecomparisonsproblem(alsoreferredasmultipletestingproblem)instatistics.
Asanillustrationofthemultiplecomparisonsproblem,considerthetaskofpredictingwhetherthestockmarketwillriseorfallinthenexttentradingdays.Ifastockanalystsimplymakesrandomguesses,theprobabilitythatherpredictioniscorrectonanytradingdayis0.5.However,theprobabilitythatshewillpredictcorrectlyatleastnineoutoftentimesis
whichisextremelylow.
Supposeweareinterestedinchoosinganinvestmentadvisorfromapoolof200stockanalysts.Ourstrategyistoselecttheanalystwhomakesthemostnumberofcorrectpredictionsinthenexttentradingdays.Theflawinthisstrategyisthatevenifalltheanalystsmaketheirpredictionsinarandomfashion,theprobabilitythatatleastoneofthemmakesatleastninecorrectpredictionsis
whichisveryhigh.Althougheachanalysthasalowprobabilityofpredictingatleastninetimescorrectly,consideredtogether,wehaveahighprobabilityoffindingatleastoneanalystwhocandoso.However,thereisnoguaranteeinthefuturethatsuchananalystwillcontinuetomakeaccuratepredictionsbyrandomguessing.
Howdoesthemultiplecomparisonsproblemrelatetomodeloverfitting?Inthecontextoflearningaclassificationmodel,eachcombinationofparametervaluescorrespondstoananalyst,whilethenumberoftraininginstancescorrespondstothenumberofdays.Analogoustothetaskofselectingthebestanalystwhomakesthemostaccuratepredictionsonconsecutivedays,thetaskofalearningalgorithmistoselectthebestcombinationofparametersthatresultsinthehighestaccuracyonthetrainingset.Ifthenumberofparametercombinationsislargebutthetrainingsizeissmall,itishighlylikelyforthelearningalgorithmtochooseaspuriousparametercombinationthatprovideshightrainingaccuracyjustbyrandomchance.Inthefollowingexample,weillustratethephenomenaofoverfittingduetomultiplecomparisonsinthecontextofdecisiontreeinduction.
(109)+(1010)210=0.0107,
1−(1−0.0107)200=0.8847,
Figure3.26.Exampleofatwo-dimensional(X-Y)dataset.
Figure3.27.
Trainingandtesterrorratesillustratingtheeffectofmultiplecomparisonsproblemonmodeloverfitting.
3.7.ExampleMultipleComparisonsandOverfittingConsiderthetwo-dimensionaldatasetshowninFigure3.26 containing500 and500 instances,whichissimilartothedatashowninFigure3.19 .Inthisdataset,thedistributionsofbothclassesarewell-separatedinthetwo-dimensional(XY)attributespace,butnoneofthetwoattributes(XorY)aresufficientlyinformativetobeusedaloneforseparatingthetwoclasses.Hence,splittingthedatasetbasedonanyvalueofanXorYattributewillprovideclosetozeroreductioninanimpuritymeasure.However,ifXandYattributesareusedtogetherinthesplittingcriterion(e.g.,splittingXat10andYat10),thetwoclassescanbeeffectivelyseparated.
+ ∘
Figure3.28.Decisiontreewith6leafnodesusingXandYasattributes.Splitshavebeennumberedfrom1to5inorderofotheroccurrenceinthetree.
Figure3.27(a) showsthetrainingandtesterrorratesforlearningdecisiontreesofvaryingsizes,when30%ofthedataisusedfortrainingandtheremainderofthedatafortesting.Wecanseethatthetwoclassescanbeseparatedusingasmallnumberofleafnodes.Figure3.28showsthedecisionboundariesforthetreewithsixleafnodes,wherethesplitshavebeennumberedaccordingtotheirorderofappearanceinthetree.Notethattheeventhoughsplits1and3providetrivialgains,theirconsequentsplits(2,4,and5)providelargegains,resultingineffectivediscriminationofthetwoclasses.
Assumeweadd100irrelevantattributestothetwo-dimensionalX-Ydata.Learningadecisiontreefromthisresultantdatawillbechallengingbecausethenumberofcandidateattributestochooseforsplittingateveryinternalnodewillincreasefromtwoto102.Withsuchalargenumberofcandidateattributetestconditionstochoosefrom,itisquitelikelythatspuriousattributetestconditionswillbeselectedatinternalnodesbecauseofthemultiplecomparisonsproblem.Figure3.27(b) showsthetrainingandtesterrorratesafteradding100irrelevantattributestothetrainingset.Wecanseethatthetesterrorrateremainscloseto0.5evenafterusing50leafnodes,whilethetrainingerrorratekeepsondecliningandeventuallybecomes0.
3.5ModelSelectionTherearemanypossibleclassificationmodelswithvaryinglevelsofmodelcomplexitythatcanbeusedtocapturepatternsinthetrainingdata.Amongthesepossibilities,wewanttoselectthemodelthatshowslowestgeneralizationerrorrate.Theprocessofselectingamodelwiththerightlevelofcomplexity,whichisexpectedtogeneralizewelloverunseentestinstances,isknownasmodelselection.Asdescribedintheprevioussection,thetrainingerrorratecannotbereliablyusedasthesolecriterionformodelselection.Inthefollowing,wepresentthreegenericapproachestoestimatethegeneralizationperformanceofamodelthatcanbeusedformodelselection.Weconcludethissectionbypresentingspecificstrategiesforusingtheseapproachesinthecontextofdecisiontreeinduction.
3.5.1UsingaValidationSet
Notethatwecanalwaysestimatethegeneralizationerrorrateofamodelbyusing“out-of-sample”estimates,i.e.byevaluatingthemodelonaseparatevalidationsetthatisnotusedfortrainingthemodel.Theerrorrateonthevalidationset,termedasthevalidationerrorrate,isabetterindicatorofgeneralizationperformancethanthetrainingerrorrate,sincethevalidationsethasnotbeenusedfortrainingthemodel.Thevalidationerrorratecanbeusedformodelselectionasfollows.
GivenatrainingsetD.train,wecanpartitionD.trainintotwosmallersubsets,D.trandD.val,suchthatD.trisusedfortrainingwhileD.valisusedasthevalidationset.Forexample,two-thirdsofD.traincanbereservedasD.trfor
training,whiletheremainingone-thirdisusedasD.valforcomputingvalidationerrorrate.ForanychoiceofclassificationmodelmthatistrainedonD.tr,wecanestimateitsvalidationerrorrateonD.val, .Themodelthatshowsthelowestvalueof canthenbeselectedasthepreferredchoiceofmodel.
Theuseofvalidationsetprovidesagenericapproachformodelselection.However,onelimitationofthisapproachisthatitissensitivetothesizesofD.trandD.val,obtainedbypartitioningD.train.IfthesizeofD.tristoosmall,itmayresultinthelearningofapoorclassificationmodelwithsub-standardperformance,sinceasmallertrainingsetwillbelessrepresentativeoftheoveralldata.Ontheotherhand,ifthesizeofD.valistoosmall,thevalidationerrorratemightnotbereliableforselectingmodels,asitwouldbecomputedoverasmallnumberofinstances.
Figure3.29.
errval(m)errval(m)
ClassdistributionofvalidationdataforthetwodecisiontreesshowninFigure3.30 .
3.8.ExampleValidationErrorInthefollowingexample,weillustrateonepossibleapproachforusingavalidationsetindecisiontreeinduction.Figure3.29 showsthepredictedlabelsattheleafnodesofthedecisiontreesgeneratedinFigure3.30 .Thecountsgivenbeneaththeleafnodesrepresenttheproportionofdatainstancesinthevalidationsetthatreacheachofthenodes.Basedonthepredictedlabelsofthenodes,thevalidationerrorrateforthelefttreeis ,whilethevalidationerrorratefortherighttreeis .Basedontheirvalidationerrorrates,therighttreeispreferredovertheleftone.
3.5.2IncorporatingModelComplexity
Sincethechanceformodeloverfittingincreasesasthemodelbecomesmorecomplex,amodelselectionapproachshouldnotonlyconsiderthetrainingerrorratebutalsothemodelcomplexity.Thisstrategyisinspiredbyawell-knownprincipleknownasOccam'srazorortheprincipleofparsimony,whichsuggeststhatgiventwomodelswiththesameerrors,thesimplermodelispreferredoverthemorecomplexmodel.Agenericapproachtoaccountformodelcomplexitywhileestimatinggeneralizationperformanceisformallydescribedasfollows.
GivenatrainingsetD.train,letusconsiderlearningaclassificationmodelmthatbelongstoacertainclassofmodels, .Forexample,if representsthesetofallpossibledecisiontrees,thenmcancorrespondtoaspecificdecision
errval(TL)=6/16=0.375errval(TR)=4/16=0.25
M M
treelearnedfromthetrainingset.Weareinterestedinestimatingthegeneralizationerrorrateofm,gen.error(m).Asdiscussedpreviously,thetrainingerrorrateofm,train.error(m,D.train),canunder-estimategen.error(m)whenthemodelcomplexityishigh.Hence,werepresentgen.error(m)asafunctionofnotjustthetrainingerrorratebutalsothemodelcomplexityof asfollows:
where isahyper-parameterthatstrikesabalancebetweenminimizingtrainingerrorandreducingmodelcomplexity.Ahighervalueof givesmoreemphasistothemodelcomplexityintheestimationofgeneralizationperformance.Tochoosetherightvalueof ,wecanmakeuseofthevalidationsetinasimilarwayasdescribedin3.5.1 .Forexample,wecaniteratethrougharangeofvaluesof andforeverypossiblevalue,wecanlearnamodelonasubsetofthetrainingset,D.tr,andcomputeitsvalidationerrorrateonaseparatesubset,D.val.Wecanthenselectthevalueof thatprovidesthelowestvalidationerrorrate.
Equation3.11 providesonepossibleapproachforincorporatingmodelcomplexityintotheestimateofgeneralizationperformance.Thisapproachisattheheartofanumberoftechniquesforestimatinggeneralizationperformance,suchasthestructuralriskminimizationprinciple,theAkaike'sInformationCriterion(AIC),andtheBayesianInformationCriterion(BIC).Thestructuralriskminimizationprincipleservesasthebuildingblockforlearningsupportvectormachines,whichwillbediscussedlaterinChapter4 .FormoredetailsonAICandBIC,seetheBibliographicNotes.
Inthefollowing,wepresenttwodifferentapproachesforestimatingthecomplexityofamodel, .Whiletheformerisspecifictodecisiontrees,thelatterismoregenericandcanbeusedwithanyclassofmodels.
M,complexity(M),
gen.error(m)=train.error(m,D.train)+α×complexity(M), (3.11)
αα
α
α
α
complexity(M)
EstimatingtheComplexityofDecisionTreesInthecontextofdecisiontrees,thecomplexityofadecisiontreecanbeestimatedastheratioofthenumberofleafnodestothenumberoftraininginstances.Letkbethenumberofleafnodesand bethenumberoftraininginstances.Thecomplexityofadecisiontreecanthenbedescribedas
.Thisreflectstheintuitionthatforalargertrainingsize,wecanlearnadecisiontreewithlargernumberofleafnodeswithoutitbecomingoverlycomplex.ThegeneralizationerrorrateofadecisiontreeTcanthenbecomputedusingEquation3.11 asfollows:
whereerr(T)isthetrainingerrorofthedecisiontreeand isahyper-parameterthatmakesatrade-offbetweenreducingtrainingerrorandminimizingmodelcomplexity,similartotheuseof inEquation3.11 .canbeviewedastherelativecostofaddingaleafnoderelativetoincurringatrainingerror.Intheliteratureondecisiontreeinduction,theaboveapproachforestimatinggeneralizationerrorrateisalsotermedasthepessimisticerrorestimate.Itiscalledpessimisticasitassumesthegeneralizationerrorratetobeworsethanthetrainingerrorrate(byaddingapenaltytermformodelcomplexity).Ontheotherhand,simplyusingthetrainingerrorrateasanestimateofthegeneralizationerrorrateiscalledtheoptimisticerrorestimateortheresubstitutionestimate.
3.9.ExampleGeneralizationErrorEstimatesConsiderthetwobinarydecisiontrees, and ,showninFigure3.30 .Bothtreesaregeneratedfromthesametrainingdataand isgeneratedbyexpandingthreeleafnodesof .Thecountsshownintheleafnodesofthetreesrepresenttheclassdistributionofthetraining
Ntrain
k/Ntrain
errgen(T)=err(T)+Ω×kNtrain,
Ω
α Ω
TL TRTL
TR
instances.Ifeachleafnodeislabeledaccordingtothemajorityclassoftraininginstancesthatreachthenode,thetrainingerrorrateforthelefttreewillbe ,whilethetrainingerrorratefortherighttreewillbe .Basedontheirtrainingerrorratesalone,wouldpreferredover ,eventhough ismorecomplex(contains
largernumberofleafnodes)than .
Figure3.30.Exampleoftwodecisiontreesgeneratedfromthesametrainingdata.
Now,assumethatthecostassociatedwitheachleafnodeis .Then,thegeneralizationerrorestimatefor willbe
andthegeneralizationerrorestimatefor willbe
err(TL)=4/24=0.167err(TR)=6/24=0.25
TL TR TLTR
Ω=0.5TL
errgen(TL)=424+0.5×724=7.524=0.3125
TR
errgen(TR)=624+0.5×424=824=0.3333.
Since hasalowergeneralizationerrorrate,itwillstillbepreferredover.Notethat impliesthatanodeshouldalwaysbeexpandedinto
itstwochildnodesifitimprovesthepredictionofatleastonetraininginstance,sinceexpandinganodeislesscostlythanmisclassifyingatraininginstance.Ontheotherhand,if ,thenthegeneralizationerrorratefor is andfor is
.Inthiscase, willbepreferredoverbecauseithasalowergeneralizationerrorrate.Thisexampleillustratesthatdifferentchoicesof canchangeourpreferenceofdecisiontreesbasedontheirgeneralizationerrorestimates.However,foragivenchoiceof ,thepessimisticerrorestimateprovidesanapproachformodelingthegeneralizationperformanceonunseentestinstances.Thevalueof canbeselectedwiththehelpofavalidationset.
MinimumDescriptionLengthPrincipleAnotherwaytoincorporatemodelcomplexityisbasedonaninformation-theoreticapproachknownastheminimumdescriptionlengthorMDLprinciple.Toillustratethisapproach,considertheexampleshowninFigure3.31 .Inthisexample,bothperson andperson aregivenasetofinstanceswithknownattributevalues .AssumepersonAknowstheclasslabelyforeveryinstance,whileperson hasnosuchinformation. wouldliketosharetheclassinformationwith bysendingamessagecontainingthelabels.Themessagewouldcontain bitsofinformation,whereNisthenumberofinstances.
TLTR Ω=0.5
Ω=1TL errgen(TL)=11/24=0.458 TR
errgen(TR)=10/24=0.417 TR TL
Ω
ΩΩ
Θ(N)
Figure3.31.Anillustrationoftheminimumdescriptionlengthprinciple.
Alternatively,insteadofsendingtheclasslabelsexplicitly, canbuildaclassificationmodelfromtheinstancesandtransmititto . canthenapplythemodeltodeterminetheclasslabelsoftheinstances.Ifthemodelis100%accurate,thenthecostfortransmissionisequaltothenumberofbitsrequiredtoencodethemodel.Otherwise, mustalsotransmitinformationaboutwhichinstancesaremisclassifiedbythemodelsothat canreproducethesameclasslabels.Thus,theoveralltransmissioncost,whichisequaltothetotaldescriptionlengthofthemessage,is
wherethefirsttermontheright-handsideisthenumberofbitsneededtoencodethemisclassifiedinstances,whilethesecondtermisthenumberofbitsrequiredtoencodethemodel.Thereisalsoahyper-parameter thattrades-offtherelativecostsofthemisclassifiedinstancesandthemodel.
Cost(model,data)=Cost(data|model)+α×Cost(model), (3.12)
α
NoticethefamiliaritybetweenthisequationandthegenericequationforgeneralizationerrorratepresentedinEquation3.11 .Agoodmodelmusthaveatotaldescriptionlengthlessthanthenumberofbitsrequiredtoencodetheentiresequenceofclasslabels.Furthermore,giventwocompetingmodels,themodelwithlowertotaldescriptionlengthispreferred.AnexampleshowinghowtocomputethetotaldescriptionlengthofadecisiontreeisgiveninExercise10onpage189.
3.5.3EstimatingStatisticalBounds
InsteadofusingEquation3.11 toestimatethegeneralizationerrorrateofamodel,analternativewayistoapplyastatisticalcorrectiontothetrainingerrorrateofthemodelthatisindicativeofitsmodelcomplexity.Thiscanbedoneiftheprobabilitydistributionoftrainingerrorisavailableorcanbeassumed.Forexample,thenumberoferrorscommittedbyaleafnodeinadecisiontreecanbeassumedtofollowabinomialdistribution.Wecanthuscomputeanupperboundlimittotheobservedtrainingerrorratethatcanbeusedformodelselection,asillustratedinthefollowingexample.
3.10.ExampleStatisticalBoundsonTrainingErrorConsidertheleft-mostbranchofthebinarydecisiontreesshowninFigure3.30 .Observethattheleft-mostleafnodeof hasbeenexpandedintotwochildnodesin .Beforesplitting,thetrainingerrorrateofthenodeis .Byapproximatingabinomialdistributionwithanormaldistribution,thefollowingupperboundofthetrainingerrorrateecanbederived:
TRTL
2/7=0.286
where istheconfidencelevel, isthestandardizedvaluefromastandardnormaldistribution,andNisthetotalnumberoftraininginstancesusedtocomputee.Byreplacing and ,theupperboundfortheerrorrateis ,whichcorrespondsto errors.Ifweexpandthenodeintoitschildnodesasshownin ,thetrainingerrorratesforthechildnodesare
and ,respectively.UsingEquation(3.13) ,theupperboundsoftheseerrorratesare and
,respectively.Theoveralltrainingerrorofthechildnodesis ,whichislargerthantheestimatederrorforthecorrespondingnodein ,suggestingthatitshouldnotbesplit.
3.5.4ModelSelectionforDecisionTrees
Buildingonthegenericapproachespresentedabove,wepresenttwocommonlyusedmodelselectionstrategiesfordecisiontreeinduction.
Prepruning(EarlyStoppingRule)
Inthisapproach,thetree-growingalgorithmishaltedbeforegeneratingafullygrowntreethatperfectlyfitstheentiretrainingdata.Todothis,amorerestrictivestoppingconditionmustbeused;e.g.,stopexpandingaleafnodewhentheobservedgaininthegeneralizationerrorestimatefallsbelowacertainthreshold.Thisestimateofthegeneralizationerrorratecanbe
eupper(N,e,α)=e+zα/222N+zα/2e(1−e)N+zα/224N21+zα/22N, (3.13)
α zα/2
α=25%,N=7, e=2/7eupper(7,2/7,0.25)=0.503
7×0.503=3.521TL
1/4=0.250 1/3=0.333eupper(4,1/4,0.25)=0.537
eupper(3,1/3,0.25)=0.6504×0.537+3×0.650=4.098
TR
computedusinganyoftheapproachespresentedintheprecedingthreesubsections,e.g.,byusingpessimisticerrorestimates,byusingvalidationerrorestimates,orbyusingstatisticalbounds.Theadvantageofprepruningisthatitavoidsthecomputationsassociatedwithgeneratingoverlycomplexsubtreesthatoverfitthetrainingdata.However,onemajordrawbackofthismethodisthat,evenifnosignificantgainisobtainedusingoneoftheexistingsplittingcriterion,subsequentsplittingmayresultinbettersubtrees.Suchsubtreeswouldnotbereachedifprepruningisusedbecauseofthegreedynatureofdecisiontreeinduction.
Post-pruning
Inthisapproach,thedecisiontreeisinitiallygrowntoitsmaximumsize.Thisisfollowedbyatree-pruningstep,whichproceedstotrimthefullygrowntreeinabottom-upfashion.Trimmingcanbedonebyreplacingasubtreewith(1)anewleafnodewhoseclasslabelisdeterminedfromthemajorityclassofinstancesaffiliatedwiththesubtree(approachknownassubtreereplacement),or(2)themostfrequentlyusedbranchofthesubtree(approachknownassubtreeraising).Thetree-pruningstepterminateswhennofurtherimprovementinthegeneralizationerrorestimateisobservedbeyondacertainthreshold.Again,theestimatesofgeneralizationerrorratecanbecomputedusinganyoftheapproachespresentedinthepreviousthreesubsections.Post-pruningtendstogivebetterresultsthanprepruningbecauseitmakespruningdecisionsbasedonafullygrowntree,unlikeprepruning,whichcansufferfromprematureterminationofthetree-growingprocess.However,forpost-pruning,theadditionalcomputationsneededtogrowthefulltreemaybewastedwhenthesubtreeispruned.
Figure3.32 illustratesthesimplifieddecisiontreemodelforthewebrobotdetectionexamplegiveninSection3.3.5 .Noticethatthesubtreerootedat
hasbeenreplacedbyoneofitsbranchescorrespondingtodepth=1
,and ,usingsubtreeraising.Ontheotherhand,thesubtreecorrespondingto and hasbeenreplacedbyaleafnodeassignedtoclass0,usingsubtreereplacement.Thesubtreefor
and remainsintact.
Figure3.32.Post-pruningofthedecisiontreeforwebrobotdetection.
breadth<=7,width>3 MultiP=1depth>1 MultiAgent=0
depth>1 MultiAgent=1
3.6ModelEvaluationTheprevioussectiondiscussedseveralapproachesformodelselectionthatcanbeusedtolearnaclassificationmodelfromatrainingsetD.train.Herewediscussmethodsforestimatingitsgeneralizationperformance,i.e.itsperformanceonunseeninstancesoutsideofD.train.Thisprocessisknownasmodelevaluation.
NotethatmodelselectionapproachesdiscussedinSection3.5 alsocomputeanestimateofthegeneralizationperformanceusingthetrainingsetD.train.However,theseestimatesarebiasedindicatorsoftheperformanceonunseeninstances,sincetheywereusedtoguidetheselectionofclassificationmodel.Forexample,ifweusethevalidationerrorrateformodelselection(asdescribedinSection3.5.1 ),theresultingmodelwouldbedeliberatelychosentominimizetheerrorsonthevalidationset.Thevalidationerrorratemaythusunder-estimatethetruegeneralizationerrorrate,andhencecannotbereliablyusedformodelevaluation.
Acorrectapproachformodelevaluationwouldbetoassesstheperformanceofalearnedmodelonalabeledtestsethasnotbeenusedatanystageofmodelselection.ThiscanbeachievedbypartitioningtheentiresetoflabeledinstancesD,intotwodisjointsubsets,D.train,whichisusedformodelselectionandD.test,whichisusedforcomputingthetesterrorrate, .Inthefollowing,wepresenttwodifferentapproachesforpartitioningDintoD.trainandD.test,andcomputingthetesterrorrate, .
3.6.1HoldoutMethod
errtest
errtest
Themostbasictechniqueforpartitioningalabeleddatasetistheholdoutmethod,wherethelabeledsetDisrandomlypartitionedintotwodisjointsets,calledthetrainingsetD.trainandthetestsetD.test.AclassificationmodelistheninducedfromD.trainusingthemodelselectionapproachespresentedinSection3.5 ,anditserrorrateonD.test, ,isusedasanestimateofthegeneralizationerrorrate.Theproportionofdatareservedfortrainingandfortestingistypicallyatthediscretionoftheanalysts,e.g.,two-thirdsfortrainingandone-thirdfortesting.
Similartothetrade-offfacedwhilepartitioningD.trainintoD.trandD.valinSection3.5.1 ,choosingtherightfractionoflabeleddatatobeusedfortrainingandtestingisnon-trivial.IfthesizeofD.trainissmall,thelearnedclassificationmodelmaybeimproperlylearnedusinganinsufficientnumberoftraininginstances,resultinginabiasedestimationofgeneralizationperformance.Ontheotherhand,ifthesizeofD.testissmall, maybelessreliableasitwouldbecomputedoverasmallnumberoftestinstances.Moreover, canhaveahighvarianceaswechangetherandompartitioningofDintoD.trainandD.test.
Theholdoutmethodcanberepeatedseveraltimestoobtainadistributionofthetesterrorrates,anapproachknownasrandomsubsamplingorrepeatedholdoutmethod.Thismethodproducesadistributionoftheerrorratesthatcanbeusedtounderstandthevarianceof .
3.6.2Cross-Validation
Cross-validationisawidely-usedmodelevaluationmethodthataimstomakeeffectiveuseofalllabeledinstancesinDforbothtrainingandtesting.Toillustratethismethod,supposethatwearegivenalabeledsetthatwehave
errtest
errtest
errtest
errtest
randomlypartitionedintothreeequal-sizedsubsets, ,and ,asshowninFigure3.33 .Forthefirstrun,wetrainamodelusingsubsetsandS (shownasemptyblocks)andtestthemodelonsubset .Thetesterrorrateon ,denotedas ,isthuscomputedinthefirstrun.Similarly,forthesecondrun,weuse and asthetrainingsetand asthetestset,tocomputethetesterrorrate, ,on .Finally,weuseand fortraininginthethirdrun,while isusedfortesting,thusresultinginthetesterrorrate for .Theoveralltesterrorrateisobtainedbysummingupthenumberoferrorscommittedineachtestsubsetacrossallrunsanddividingitbythetotalnumberofinstances.Thisapproachiscalledthree-foldcross-validation.
Figure3.33.Exampledemonstratingthetechniqueof3-foldcross-validation.
Thek-foldcross-validationmethodgeneralizesthisapproachbysegmentingthelabeleddataD(ofsizeN)intokequal-sizedpartitions(orfolds).Duringthei run,oneofthepartitionsofDischosenasD.test(i)fortesting,whiletherestofthepartitionsareusedasD.train(i)fortraining.Amodelm(i)islearnedusingD.train(i)andappliedonD.test(i)toobtainthesumoftesterrors,
S1,S2 S3S2
3 S1S1 err(S1)
S1 S3 S2err(S2) S2 S1
S3 S3err(S3) S3
th
.Thisprocedureisrepeatedktimes.Thetotaltesterrorrate, ,isthencomputedas
Everyinstanceinthedataisthususedfortestingexactlyonceandfortrainingexactly times.Also,everyrunuses fractionofthedatafortrainingand1/kfractionfortesting.
Therightchoiceofkink-foldcross-validationdependsonanumberofcharacteristicsoftheproblem.Asmallvalueofkwillresultinasmallertrainingsetateveryrun,whichwillresultinalargerestimateofgeneralizationerrorratethanwhatisexpectedofamodeltrainedovertheentirelabeledset.Ontheotherhand,ahighvalueofkresultsinalargertrainingsetateveryrun,whichreducesthebiasintheestimateofgeneralizationerrorrate.Intheextremecase,when ,everyrunusesexactlyonedatainstancefortestingandtheremainderofthedatafortesting.Thisspecialcaseofk-foldcross-validationiscalledtheleave-one-outapproach.Thisapproachhastheadvantageofutilizingasmuchdataaspossiblefortraining.However,leave-one-outcanproducequitemisleadingresultsinsomespecialscenarios,asillustratedinExercise11.Furthermore,leave-one-outcanbecomputationallyexpensiveforlargedatasetsasthecross-validationprocedureneedstoberepeatedNtimes.Formostpracticalapplications,thechoiceofkbetween5and10providesareasonableapproachforestimatingthegeneralizationerrorrate,becauseeachfoldisabletomakeuseof80%to90%ofthelabeleddatafortraining.
Thek-foldcross-validationmethod,asdescribedabove,producesasingleestimateofthegeneralizationerrorrate,withoutprovidinganyinformationaboutthevarianceoftheestimate.Toobtainthisinformation,wecanrunk-foldcross-validationforeverypossiblepartitioningofthedataintokpartitions,
errsum(i) errtest
errtest=∑i=1kerrsum(i)N. (3.14)
(k−1) (k−1)/k
k=N
andobtainadistributionoftesterrorratescomputedforeverysuchpartitioning.Theaveragetesterrorrateacrossallpossiblepartitioningsservesasamorerobustestimateofgeneralizationerrorrate.Thisapproachofestimatingthegeneralizationerrorrateanditsvarianceisknownasthecompletecross-validationapproach.Eventhoughsuchanestimateisquiterobust,itisusuallytooexpensivetoconsiderallpossiblepartitioningsofalargedatasetintokpartitions.Amorepracticalsolutionistorepeatthecross-validationapproachmultipletimes,usingadifferentrandompartitioningofthedataintokpartitionsateverytime,andusetheaveragetesterrorrateastheestimateofgeneralizationerrorrate.Notethatsincethereisonlyonepossiblepartitioningfortheleave-one-outapproach,itisnotpossibletoestimatethevarianceofgeneralizationerrorrate,whichisanotherlimitationofthismethod.
Thek-foldcross-validationdoesnotguaranteethatthefractionofpositiveandnegativeinstancesineverypartitionofthedataisequaltothefractionobservedintheoveralldata.Asimplesolutiontothisproblemistoperformastratifiedsamplingofthepositiveandnegativeinstancesintokpartitions,anapproachcalledstratifiedcross-validation.
Ink-foldcross-validation,adifferentmodelislearnedateveryrunandtheperformanceofeveryoneofthekmodelsontheirrespectivetestfoldsisthenaggregatedtocomputetheoveralltesterrorrate, .Hence, doesnotreflectthegeneralizationerrorrateofanyofthekmodels.Instead,itreflectstheexpectedgeneralizationerrorrateofthemodelselectionapproach,whenappliedonatrainingsetofthesamesizeasoneofthetrainingfolds .Thisisdifferentthanthe computedintheholdoutmethod,whichexactlycorrespondstothespecificmodellearnedoverD.train.Hence,althougheffectivelyutilizingeverydatainstanceinDfortrainingandtesting,the computedinthecross-validationmethoddoesnotrepresenttheperformanceofasinglemodellearnedoveraspecificD.train.
errtest errtest
(N(k−1)/k) errtest
errtest
Nonetheless,inpractice, istypicallyusedasanestimateofthegeneralizationerrorofamodelbuiltonD.Onemotivationforthisisthatwhenthesizeofthetrainingfoldsisclosertothesizeoftheoveralldata(whenkislarge),then resemblestheexpectedperformanceofamodellearnedoveradatasetofthesamesizeasD.Forexample,whenkis10,everytrainingfoldis90%oftheoveralldata.The thenshouldapproachtheexpectedperformanceofamodellearnedover90%oftheoveralldata,whichwillbeclosetotheexpectedperformanceofamodellearnedoverD.
errtest
errtest
errtest
3.7PresenceofHyper-parametersHyper-parametersareparametersoflearningalgorithmsthatneedtobedeterminedbeforelearningtheclassificationmodel.Forinstance,considerthehyper-parameter thatappearedinEquation3.11 ,whichisrepeatedhereforconvenience.Thisequationwasusedforestimatingthegeneralizationerrorforamodelselectionapproachthatusedanexplicitrepresentationsofmodelcomplexity.(SeeSection3.5.2 .)
Forotherexamplesofhyper-parameters,seeChapter4 .
Unlikeregularmodelparameters,suchasthetestconditionsintheinternalnodesofadecisiontree,hyper-parameterssuchas donotappearinthefinalclassificationmodelthatisusedtoclassifyunlabeledinstances.However,thevaluesofhyper-parametersneedtobedeterminedduringmodelselection—aprocessknownashyper-parameterselection—andmustbetakenintoaccountduringmodelevaluation.Fortunately,bothtaskscanbeeffectivelyaccomplishedviaslightmodificationsofthecross-validationapproachdescribedintheprevioussection.
3.7.1Hyper-parameterSelection
InSection3.5.2 ,avalidationsetwasusedtoselect andthisapproachisgenerallyapplicableforhyper-parametersection.Letpbethehyper-parameterthatneedstobeselectedfromafiniterangeofvalues,
α
gen.error(m)=train.error(m,D.train)+α×complexity(M)
α
α
P=
.PartitionD.trainintoD.trandD.val.Foreverychoiceofhyper-parametervalue ,wecanlearnamodel onD.tr,andapplythismodelonD.valtoobtainthevalidationerrorrate .Let bethehyper-parametervaluethatprovidesthelowestvalidationerrorrate.Wecanthenusethemodel correspondingto asthefinalchoiceofclassificationmodel.
Theaboveapproach,althoughuseful,usesonlyasubsetofthedata,D.train,fortrainingandasubset,D.val,forvalidation.Theframeworkofcross-validation,presentedinSection3.6.2 ,addressesbothofthoseissues,albeitinthecontextofmodelevaluation.Hereweindicatehowtouseacross-validationapproachforhyper-parameterselection.Toillustratethisapproach,letuspartitionD.trainintothreefoldsasshowninFigure3.34 .Ateveryrun,oneofthefoldsisusedasD.valforvalidation,andtheremainingtwofoldsareusedasD.trforlearningamodel,foreverychoiceofhyper-parametervalue .Theoverallvalidationerrorratecorrespondingtoeachiscomputedbysummingtheerrorsacrossallthethreefolds.Wethenselectthehyper-parametervalue thatprovidesthelowestvalidationerrorrate,anduseittolearnamodel ontheentiretrainingsetD.train.
Figure3.34.Exampledemonstratingthe3-foldcross-validationframeworkforhyper-parameterselectionusingD.train.
{p1,p2,…pn}pi mi
errval(pi) p*
m* p*
pi pi
p*m*
Algorithm3.2 generalizestheaboveapproachusingak-foldcross-validationframeworkforhyper-parameterselection.Atthei runofcross-validation,thedatainthei foldisusedasD.val(i)forvalidation(Step4),whiletheremainderofthedatainD.trainisusedasD.tr(i)fortraining(Step5).Thenforeverychoiceofhyper-parametervalue ,amodelislearnedonD.tr(i)(Step7),whichisappliedonD.val(i)tocomputeitsvalidationerror(Step8).Thisisusedtocomputethevalidationerrorratecorrespondingtomodelslearningusing overallthefolds(Step11).Thehyper-parametervalue thatprovidesthelowestvalidationerrorrate(Step12)isnowusedtolearnthefinalmodel ontheentiretrainingsetD.train(Step13).Hence,attheendofthisalgorithm,weobtainthebestchoiceofthehyper-parametervalueaswellasthefinalclassificationmodel(Step14),bothofwhichareobtainedbymakinganeffectiveuseofeverydatainstanceinD.train.
Algorithm3.2Proceduremodel-select(k, ,D.train)
∈
th
th
pi
pip*
m*
P
∑
3.7.2NestedCross-Validation
TheapproachoftheprevioussectionprovidesawaytoeffectivelyusealltheinstancesinD.traintolearnaclassificationmodelwhenhyper-parameterselectionisrequired.ThisapproachcanbeappliedovertheentiredatasetDtolearnthefinalclassificationmodel.However,applyingAlgorithm3.2 onDwouldonlyreturnthefinalclassificationmodel butnotanestimateofitsgeneralizationperformance, .RecallthatthevalidationerrorratesusedinAlgorithm3.2 cannotbeusedasestimatesofgeneralizationperformance,sincetheyareusedtoguidetheselectionofthefinalmodel .However,tocompute ,wecanagainuseacross-validationframeworkforevaluatingtheperformanceontheentiredatasetD,asdescribedoriginallyinSection3.6.2 .Inthisapproach,DispartitionedintoD.train(fortraining)andD.test(fortesting)ateveryrunofcross-validation.Whenhyper-parametersareinvolved,wecanuseAlgorithm3.2 totrainamodelusingD.trainateveryrun,thus“internally”usingcross-validationformodelselection.Thisapproachiscallednestedcross-validationordoublecross-validation.Algorithm3.3describesthecompleteapproachforestimating
usingnestedcross-validationinthepresenceofhyper-parameters.
Asanillustrationofthisapproach,seeFigure3.35 wherethelabeledsetDispartitionedintoD.trainandD.test,usinga3-foldcross-validationmethod.
m*errtest
m*errtest
errtest
Figure3.35.Exampledemonstrating3-foldnestedcross-validationforcomputing .
Atthei runofthismethod,oneofthefoldsisusedasthetestset,D.test(i),whiletheremainingtwofoldsareusedasthetrainingset,D.train(i).ThisisrepresentedinFigure3.35 asthei “outer”run.InordertoselectamodelusingD.train(i),weagainusean“inner”3-foldcross-validationframeworkthatpartitionsD.train(i)intoD.trandD.valateveryoneofthethreeinnerruns(iterations).AsdescribedinSection3.7 ,wecanusetheinnercross-validationframeworktoselectthebesthyper-parametervalue aswellasitsresultingclassificationmodel learnedoverD.train(i).Wecanthenapply onD.test(i)toobtainthetesterroratthei outerrun.Byrepeatingthisprocessforeveryouterrun,wecancomputetheaveragetesterrorrate,
,overtheentirelabeledsetD.Notethatintheaboveapproach,theinnercross-validationframeworkisbeingusedformodelselectionwhiletheoutercross-validationframeworkisbeingusedformodelevaluation.
Algorithm3.3Thenestedcross-validationapproachforcomputing .
errtest
th
th
p*(i)m*(i)
m*(i) th
errtest
errtest
∑
3.8PitfallsofModelSelectionandEvaluationModelselectionandevaluation,whenusedeffectively,serveasexcellenttoolsforlearningclassificationmodelsandassessingtheirgeneralizationperformance.However,whenusingthemeffectivelyinpracticalsettings,thereareseveralpitfallsthatcanresultinimproperandoftenmisleadingconclusions.Someofthesepitfallsaresimpletounderstandandeasytoavoid,whileothersarequitesubtleinnatureanddifficulttocatch.Inthefollowing,wepresenttwoofthesepitfallsanddiscussbestpracticestoavoidthem.
3.8.1OverlapbetweenTrainingandTestSets
Oneofthebasicrequirementsofacleanmodelselectionandevaluationsetupisthatthedatausedformodelselection(D.train)mustbekeptseparatefromthedatausedformodelevaluation(D.test).Ifthereisanyoverlapbetweenthetwo,thetesterrorrate computedoverD.testcannotbeconsideredrepresentativeoftheperformanceonunseeninstances.Comparingtheeffectivenessofclassificationmodelsusing canthenbequitemisleading,asanoverlycomplexmodelcanshowaninaccuratelylowvalueof duetomodeloverfitting(seeExercise12attheendofthischapter).
errtest
errtest
errtest
ToillustratetheimportanceofensuringnooverlapbetweenD.trainandD.test,consideralabeleddatasetwherealltheattributesareirrelevant,i.e.theyhavenorelationshipwiththeclasslabels.Usingsuchattributes,weshouldexpectnoclassificationmodeltoperformbetterthanrandomguessing.However,ifthetestsetinvolvesevenasmallnumberofdatainstancesthatwereusedfortraining,thereisapossibilityforanoverlycomplexmodeltoshowbetterperformancethanrandom,eventhoughtheattributesarecompletelyirrelevant.AswewillseelaterinChapter10 ,thisscenariocanactuallybeusedasacriteriontodetectoverfittingduetoimpropersetupofexperiment.Ifamodelshowsbetterperformancethanarandomclassifierevenwhentheattributesareirrelevant,itisanindicationofapotentialfeedbackbetweenthetrainingandtestsets.
3.8.2UseofValidationErrorasGeneralizationError
Thevalidationerrorrate servesanimportantroleduringmodelselection,asitprovides“out-of-sample”errorestimatesofmodelsonD.val,whichisnotusedfortrainingthemodels.Hence, servesasabettermetricthanthetrainingerrorrateforselectingmodelsandhyper-parametervalues,asdescribedinSections3.5.1 and3.7 ,respectively.However,oncethevalidationsethasbeenusedforselectingaclassificationmodel
nolongerreflectstheperformanceof onunseeninstances.
Torealizethepitfallinusingvalidationerrorrateasanestimateofgeneralizationperformance,considertheproblemofselectingahyper-parametervaluepfromarangeofvalues usingavalidationsetD.val.Ifthenumberofpossiblevaluesin isquitelargeandthesizeofD.valissmall,itis
errval
errval
m*,errval m*
P,P
possibletoselectahyper-parametervalue thatshowsfavorableperformanceonD.valjustbyrandomchance.NoticethesimilarityofthisproblemwiththemultiplecomparisonsproblemdiscussedinSection3.4.1 .Eventhoughtheclassificationmodel learnedusing wouldshowalowvalidationerrorrate,itwouldlackgeneralizabilityonunseentestinstances.
ThecorrectapproachforestimatingthegeneralizationerrorrateofamodelistouseanindependentlychosentestsetD.testthathasn'tbeenusedin
anywaytoinfluencetheselectionof .Asaruleofthumb,thetestsetshouldneverbeexaminedduringmodelselection,toensuretheabsenceofanyformofoverfitting.Iftheinsightsgainedfromanyportionofalabeleddatasethelpinimprovingtheclassificationmodeleveninanindirectway,thenthatportionofdatamustbediscardedduringtesting.
p*
m* p*
m*m*
3.9ModelComparisonOnedifficultywhencomparingtheperformanceofdifferentclassificationmodelsiswhethertheobserveddifferenceintheirperformanceisstatisticallysignificant.Forexample,considerapairofclassificationmodels, and .Suppose achieves85%accuracywhenevaluatedonatestsetcontaining30instances,while achieves75%accuracyonadifferenttestsetcontaining5000instances.Basedonthisinformation,is abettermodelthan ?Thisexampleraisestwokeyquestionsregardingthestatisticalsignificanceofaperformancemetric:
1. Although hasahigheraccuracythan ,itwastestedonasmallertestset.Howmuchconfidencedowehavethattheaccuracyfor isactually85%?
2. Isitpossibletoexplainthedifferenceinaccuraciesbetween andasaresultofvariationsinthecompositionoftheirtestsets?
Thefirstquestionrelatestotheissueofestimatingtheconfidenceintervalofmodelaccuracy.Thesecondquestionrelatestotheissueoftestingthestatisticalsignificanceoftheobserveddeviation.Theseissuesareinvestigatedintheremainderofthissection.
3.9.1EstimatingtheConfidenceIntervalforAccuracy
*
MA MBMA
MBMA
MB
MA MBMA
MAMB
Todetermineitsconfidenceinterval,weneedtoestablishtheprobabilitydistributionforsampleaccuracy.Thissectiondescribesanapproachforderivingtheconfidenceintervalbymodelingtheclassificationtaskasabinomialrandomexperiment.Thefollowingdescribescharacteristicsofsuchanexperiment:
1. TherandomexperimentconsistsofNindependenttrials,whereeachtrialhastwopossibleoutcomes:successorfailure.
2. Theprobabilityofsuccess,p,ineachtrialisconstant.
AnexampleofabinomialexperimentiscountingthenumberofheadsthatturnupwhenacoinisflippedNtimes.IfXisthenumberofsuccessesobservedinNtrials,thentheprobabilitythatXtakesaparticularvalueisgivenbyabinomialdistributionwithmean andvariance :
Forexample,ifthecoinisfair andisflippedfiftytimes,thentheprobabilitythattheheadshowsup20timesis
Iftheexperimentisrepeatedmanytimes,thentheaveragenumberofheadsexpectedtoshowupis whileitsvarianceis
Thetaskofpredictingtheclasslabelsoftestinstancescanalsobeconsideredasabinomialexperiment.GivenatestsetthatcontainsNinstances,letXbethenumberofinstancescorrectlypredictedbyamodelandpbethetrueaccuracyofthemodel.Ifthepredictiontaskismodeledasabinomialexperiment,thenXhasabinomialdistributionwithmean andvariance Itcanbeshownthattheempiricalaccuracy, also
Np Np(1−p)
P(X=υ)=(Nυ)pυ(1−p)N−υ.
(p=0.5)
P(X=20)=(5020)0.520(1−0.5)30=0.0419.
50×0.5=25, 50×0.5×0.5=12.5.
NpNp(1−p). acc=X/N,
hasabinomialdistributionwithmeanpandvariance (seeExercise14).ThebinomialdistributioncanbeapproximatedbyanormaldistributionwhenNissufficientlylarge.Basedonthenormaldistribution,theconfidenceintervalforacccanbederivedasfollows:
where and aretheupperandlowerboundsobtainedfromastandardnormaldistributionatconfidencelevel Sinceastandardnormaldistributionissymmetricaround itfollowsthatRearrangingthisinequalityleadstothefollowingconfidenceintervalforp:
Thefollowingtableshowsthevaluesof atdifferentconfidencelevels:
0.99 0.98 0.95 0.9 0.8 0.7 0.5
2.58 2.33 1.96 1.65 1.28 1.04 0.67
3.11.ExampleConfidenceIntervalforAccuracyConsideramodelthathasanaccuracyof80%whenevaluatedon100testinstances.Whatistheconfidenceintervalforitstrueaccuracyata95%confidencelevel?Theconfidencelevelof95%correspondsto
accordingtothetablegivenabove.InsertingthistermintoEquation3.16 yieldsaconfidenceintervalbetween71.1%and86.7%.Thefollowingtableshowstheconfidenceintervalwhenthenumberofinstances,N,increases:
N 20 50 100 500 1000 5000
p(1−p)/N
P(−Zα/2≤acc−pp(1−p)/N≤Z1−α/2)=1−α, (3.15)
Zα/2 Z1−α/2(1−α).
Z=0, Zα/2=Z1−α/2.
2×N×acc×Zα/22±Zα/2Zα/22+4Nacc−4Nacc22(N+Zα/22). (3.16)
Zα/2
1−α
Zα/2
Za/2=1.96
Confidence 0.584 0.670 0.711 0.763 0.774 0.789
Interval
NotethattheconfidenceintervalbecomestighterwhenNincreases.
3.9.2ComparingthePerformanceofTwoModels
Considerapairofmodels, and whichareevaluatedontwoindependenttestsets, and Let denotethenumberofinstancesin
and denotethenumberofinstancesin Inaddition,supposetheerrorratefor on is andtheerrorratefor on is Ourgoalistotestwhethertheobserveddifferencebetween and isstatisticallysignificant.
Assumingthat and aresufficientlylarge,theerrorrates and canbeapproximatedusingnormaldistributions.Iftheobserveddifferenceintheerrorrateisdenotedas thendisalsonormallydistributedwithmean ,itstruedifference,andvariance, Thevarianceofdcanbecomputedasfollows:
where and arethevariancesoftheerrorrates.Finally,atthe confidencelevel,itcanbeshownthattheconfidenceintervalforthetruedifferencedtisgivenbythefollowingequation:
−0.919 −0.888 −0.867 −0.833 −0.824 −0.811
M1 M2,D1 D2. n1
D1 n2 D2.M1 D1 e1 M2 D2 e2.
e1 e2
n1 n2 e1 e2
d=e1−e2,dt σd2.
σd2≃σ^d2=e1(1−e1)n1+e2(1−e2)n2, (3.17)
e1(1−e1)/n1 e2(1−e1)/n2(1−α)%
3.12.ExampleSignificanceTestingConsidertheproblemdescribedatthebeginningofthissection.Modelhasanerrorrateof whenappliedto testinstances,whilemodel hasanerrorrateof whenappliedto testinstances.Theobserveddifferenceintheirerrorratesis
.Inthisexample,weareperformingatwo-sidedtesttocheckwhether or .Theestimatedvarianceoftheobserveddifferenceinerrorratescanbecomputedasfollows:
or .InsertingthisvalueintoEquation3.18 ,weobtainthefollowingconfidenceintervalfor at95%confidencelevel:
Astheintervalspansthevaluezero,wecanconcludethattheobserveddifferenceisnotstatisticallysignificantata95%confidencelevel.
Atwhatconfidencelevelcanwerejectthehypothesisthat ?Todothis,weneedtodeterminethevalueof suchthattheconfidenceintervalfordoesnotspanthevaluezero.Wecanreversetheprecedingcomputationandlookforthevalue suchthat .Replacingthevaluesofdand
gives .Thisvaluefirstoccurswhen (foratwo-sidedtest).Theresultsuggeststhatthenullhypothesiscanberejectedatconfidencelevelof93.6%orlower.
dt=d±zα/2σ^d. (3.18)
MAe1=0.15 N1=30
MB e2=0.25 N2=5000
d=|0.15−0.25|=0.1dt=0 dt≠0
σ^d2=0.15(1−0.15)30+0.25(1−0.25)5000=0.0043
σ^d=0.0655dt
dt=0.1±1.96×0.0655=0.1±0.128.
dt=0Zα/2 dt
Zα/2 d>Zσ/2σ^dσ^d Zσ/2<1.527 (1−α)<~0.936