; TeX output 1997.06.09:0350qɎ}{)GDtHGcmr17GVisualization&1؍-\ofdomainandconceptdescriptions,JǍY9XQ ff cmr12Dunja/MladeniW}ca*0XQ cmr12J.StefanInstitute,ComputerScienceDepartmenrt,AILabSoratoryV,JamovXa39,861000Ljubljana,Slorvenia'K`y 3 cmr10Phone:(+38)(61)f159199,E-mail:dunja.mladenic@ijs.siٍŁ,"V 3 cmbx10Abstract])fd3( TheupapMerpresen!tsamethodforvisualizingdatainmac!hinelearn- 3( ing.AWvisualizingWmethoMdaimsatrepresen!tingthespaceofexamples3( (attribute-vdDalue\tuples)andthecorrespMondingconceptdescriptions(in-3( ducedDif-thanrules)whic!harebMothmultidimensionalwithadimension3( depMendingonthen!umberofattributes.Themethodisbasedonthepar-3( alleltacoMordinatesmethodwhic!hwaspreviouslyusedforvisualizingmul-3( tidimensional geometrye.Visualizationofexamplesandruleswiththe3( parallelcoMordinatesmethodenablestheanalysisofthespaceofexam-3( plesandinducedrules.wThedescribMedvisualizingmethodisconnected3( toWSthemac!hinelearningsystemH- 3 cmcsc10HA trisandiscurrentlypMostprocessing3( theܰresultsofthemac!hinelearningalgorithm.ResultsoftheexpMeri-3( men!ts onareal-worlddomainillustratetheusefulnessofthemethoMdfor3( analysingfmac!hinelearningproblems.n^[Qd2andd3andthattheyharve-three,two-andtrwo-vXalueseacrh,[QrespSectivrelyV.ThedtableusedtovisualizethisproblemisgiveninTVable1.Forexample,[Qthep\pSoinrt5N cmbx12Xp=inthetablerepresentsthepSoint(d1v2,d2v2,d3v2)p\inthed12!", cmsy10d2d3[Qspace.GNote uthatinthistableelemenrtsinthesamerowshouldhavethesamevXalue[Qoftheseconddimension.xwp􍍍Vzffo VGYff͟dd3fftjd3v1͟YffXmd3v2͟Yff(pd3v1͟Yffsd3v2͟Yffvd3v1͟Yff"yd3v2͟YffffVzff ? &cͤYff͟dd1ff[Yff4ϟdd1v1ff\äYff4ϟdd1v2ffɤYff4ϟdd1v3ffff;ϤYff mWdd2ffff ?ffff[Yff:+Yff\ßYffƟYffɟYffk̟Yff;ϟYffff;ϤYff͟dd2v1ffff[Yff:+Yff\ßYffƟYffNX Yffk̟Yff;ϟYffff;ϤYff͟dd2v2ffff:"TVable1:8TablevisualizingmethoSd.EWhenܶassigningrorwsandcolumnstodimensions,itisadvisabletosortdimensions[QaccordingUtothenrumbSerUofvXalues.Rorwsandcolumnsnearthetablemarginsare[QcrhosenfordimensionswithmorevXalues.![QT2.2.)cAn/exampleproblemdomain[QTVodescribSethevisualizingmethodsasimple`Saturdarymorning'[8]domainisused.[QInWthismacrhinelearningdomain,14trainingexamplesaregiven(seeTVable2).(Each[QexampleislabSeledwithitsclass:isuitable(no,yres),sayingifamorningissuitable[QforsomeunspSeci edactivitryV,>and4attributes:Sioutlook(sunnryV,>overcast,rain),tem-[QpSeratureL(hot,e/mild,cool),hrumidityL(high,e/normal)andwindy(no,yres),describing[QaSaturdarymorning.EAmacrhinelearningalgorithmsuchasU- cmcsc10UA3tris,'canbSeusedtoinduceif-thenrules[Qfromthegivrensetofexamples. Tworulesfortheno-classandthreefortheyes-class[QaregivrenbSelow.qɎ}hkP=ffk &cͤYff Pdclassff3/YffYgKdattributesffffkͤYff͟dsuitableff;boutloSokM'YffoftempSerature͟Yffdhrumidity͟Yffiwindy͟YffffkͤYfftdnoff?sunnry Yffi%hothYffNWhighYffnoYffͤYfftdnoff?sunnry Yffi%hothYffNWhighYff yres oYffͤYffSdyresff9borvercast͟Yffi%hothYffNWhighYffnoYffͤYffSdyresffDDrainpYffy'mildjYffNWhighYffnoYffͤYffSdyresffDDrainpYffcoSol۟Yffsnormal YffnoYffͤYfftdnoffDDrainpYffcoSol۟Yffsnormal Yff yres oYffͤYffSdyresff9borvercast͟YffcoSol۟Yffsnormal Yff yres oYffͤYfftdnoff?sunnry Yffy'mildjYffNWhighYffnoYffͤYffSdyresff?sunnry YffcoSol۟Yffsnormal YffnoYffͤYffSdyresffDDrainpYffy'mildjYffsnormal YffnoYffͤYffSdyresff?sunnry Yffy'mildjYffsnormal Yff yres oYffͤYffSdyresff9borvercast͟Yffy'mildjYffNWhighYff yres oYffͤYffSdyresff9borvercast͟Yffi%hothYffsnormal YffnoYffͤYfftdnoffDDrainpYffy'mildjYffNWhighYff yres oYffffk"PRTVable2:8Examplesforthe`Saturdarymorning'domain. DRule1:8ifoutloSok=sunnryandhumidity=highthansuitable=noDRule2:8ifoutloSok=rainandwindy=yresthansuitable=noDRule3:8ifoutloSok=sunnryandhumidity=normalthansuitable=yesDRule4:8ifoutloSok=orvercastthansuitable=yresDRule5:8ifoutloSok=rainryandwindy=nothansuitable=yes![QT2.3.)cVisualizing/examplesandruleswithtables7x[QThe+ tablevisualizingmethoSdwrasalreadyappliedtomachinelearningbyMichalski[QandsGStepp[6].TVoapplythismethoSdonmacrhinelearningdata,'attributesareviewed[QasdimensionsandexamplesaspSoinrtsinthisattributespace(inthesamewayas[Qinj|[6]).'TVoeacrhoftheclassvXaluesadi erentsymbSolisassigned(G-totheno-class,+[Qtobtheyres-class),~inthiswayalltheexamplescanbSeshowninthesameTVable3. For[Qeacrheclass,@asetofrulescanbSeshownonthetablewithexamples(TVables4and5)in[Qordertoillustratehorwtherulescovertheexamples.E.g.inTVable4the rstcolumn[Q lledwith(|||)visualizedtheabSorvegivenRule1."[Q3."(Visualizationffwithparallelcos3ordinates;[QT3.1.)cThe/parallelcodCordinatesmethod[QThemfparallelcoSordinatesmethodmapsRJ22cmmi8N(!RJ2|{Ycmr82N,sothattherelationamongN[QvXariablesdisrepresenrtedbyitsplanarimage.Inthetwo-dimensional(xyn9-Cartesian[QcoSordinate)mspace,ENAQreallines,labSeledwithx1;x2;:::ʜ;xND,Earemplacedequidistanrt[Qanduparalleltotheyn9-axis.FTheselinesareaxesoftheparallelcoSordinatesystem.A[QpSoinrt=Cwithcoordinates(c1;c2;:::ʜ;cND)=isrepresenrtedbyapSolygonalline,?bwhoseqɎ} r鍟"0$—ff!gp $dYff͟dh!umidityffahigh͟YffTnormal͟Yffhigh͟YffЀnormal͟Yffhigh͟Yffnormal͟Yff$—ffq̟ &cͤYff LDdoutloMokffOsunn!y.Yffo!vercastYffߣrainqYff'gptemp.͟YffH@Yff͟dwindyffffq̞ff67ff:ɕ O6YffW^YffYffYff<+K^YffYffxG-Yff+coMol YffH@Yff dy!esff67ff96YffW^Yffi+K^YffYff*YffYffh+K^YffH@YffH@Yffxdnoff67ff:ɕ O6YffW^Yffi+K^Yff&+ FYff*YffoG- bߟYff!4=Yff*rmildxYffH@Yff dy!esff67ff96YffCyIG- bߟYffYffYff*YffR+ FYffh+K^YffH@YffH@Yffxdnoff67ff:ɕ6YffCyIG- bߟYffYffYff*YffYff!4=Yff-.hot EYffH@Yff dy!esff67ff96YffCyIG- bߟYffYff&+ FYff<+K^YffYff!4=YffH@YffH@Yffxdnoff67ff:ɕY"TVable3:8Tablevisualizationofexamplesforthe`Saturdarymorning'domain.c]v0P=ff@L$  Yff͟dh!umidityffY{high IYffs\normal͟Yff~high͟Yffϟonormal͟Yffe high IYff-=normal0YffP=ff &cͤYff LDdoutloMokffW<sunn!y %YffZo!vercastYffyrain%YffFL$temp.͟YffgxYff͟dwindyffffff67ffYI O6Yff<7|||͟YffmYffYff+K^YffT|||͟Yff*|G-|bYffJgXcoMol YffgxYff dy!esff67ff h펎6Yff<7|||͟Yffw+K^YffYff!Yff Yff#l0+,YffgxYffgxYffxdnoff67ffYI O6Yff<7|||͟Yffw+K^Yff+ FYff!Yff|G-|bYff <|||͟YffIVmildxYffgxYff dy!esff67ff h펎6Yff?|G-|bYffmYffYff!Yff,H+,Yff#l0+,YffgxYffgxYffxdnoff67ffYI6Yff?|G-|bYffmYffYff!YffT|||͟Yff <|||͟YffLhot EYffgxYff dy!esff67ff h펎6Yff?|G-|bYffmYff+ FYff+K^Yff Yff@YffgxYffgxYffxdnoff67ffYIY"[QTVable4:aTablevisualizationofno-classrulesforthe`Saturdarymorning'domain.All[Qno-classexamples(G-)arecorveredbytherules(|||).]v0wffCѰ ZDYff͟dh!umidityffPphigh͟Yffrnormal0Yffhigh IYffЀnormal0Yff'high IYff/normal0Yffwff &cͤYff LDdoutloMokffPsunn!yYffo!vercast~Yff rain%YffIѰtemp.͟YffjYff͟dwindyffff ff67ff]3Վ O6YffW^Yff](|||͟Yff|||͟Yff|+|lɟYff^Yff(pUG-ZYffMcoMol YffjYff dy!esff67ff y6YffW^Yff^2$|+|lɟYff|||͟Yff|||͟YffQ|||͟Yff1|+|lɟYffjYffjYffxdnoff67ff]3Վ O6YffW^Yff^2$|+|lɟYffr |+|lɟYff|||͟Yff0mG-ZYffC}YffL[mildxYffjYff dy!esff67ff y6YffCyIG- bߟYff](|||͟Yff|||͟Yff|||͟Yff|+|lɟYff1|+|lɟYffjYffjYffxdnoff67ff]3Վ6YffCyIG- bߟYff](|||͟Yff|||͟Yff|||͟Yff^YffC}YffO(hot EYffjYff dy!esff67ff y6YffCyIG- bߟYff](|||͟Yffr |+|lɟYff|+|lɟYffQ|||͟Yff|||͟YffjYffjYffxdnoff67ff]3ՎY"[QTVableP5: Tablevisualizationofyres-classrulesforthe`Saturdaymorning'domain.All[Qyres-classexamples(+)arecoveredbytherules(|||).3-qɎ}[q[QNvrertices7areatci onthexid-axisfori7I=1:::ʜN(see7Figure1). MAone-to-one[QcorrespSondenceͩbetrweenpSointsinRJ2N 7andpSolygonallinesinRJ22withverticesonthe[Qx1;x2;:::ʜ;xN axesisestablished.EStarting~fromthepSoinrtinRJ2N2,themethodcanbeusedformrultidimensional[Qgeometrical0objectsvisualization[5]. +InthispapSer,BGtheparallelcoordinatesmethod[QisusedonlyfortherepresenrtationofpSoints.[Qcem:linewidth 0.4ptکI9)/ em:point 19Ꟛi em:point 2 em:line 1,29Ꟛi em:point 3i em:point 4 em:line 3,4i em:point 5ri em:point 6 em:line 5,6ri em:point 7ri em:point 8 em:line 7,8ri em:point 9ri em:point 10 em:line 9,109Ꟛi em:point 1192G em:point 12 em:line 11,129Ꟛi em:point 13i em:point 14 em:line 13,14*2G em:point 15*)/ em:point 16 em:line 15,16G!2G em:point 17G!)/ em:point 18 em:line 17,18 :)/ em:point 19 :2G em:point 20 em:line 19,20}2G em:point 21})/ em:point 22 em:line 21,22)/ em:point 232G em:point 24 em:line 23,24t)/ em:point 25t2G em:point 26 em:line 25,26H)/ em:point 27H2G em:point 28 em:line 27,2882G em:point 298)/ em:point 30 em:line 29,301>y?x1$x2Ajx3N썑uxiK cmsy81sxiN썒yxi+1N썒@xN"2N썒xN"11xN9꟝ em:point 33*b em:point 34 em:line 33,34*b em:point 35G!{3 em:point 36 em:line 35,36G!{3 em:point 37Ox em:point 38 em:line 37,38wUI em:point 39 :Z em:point 40 em:line 39,40 :Z em:point 41}Rp em:point 42 em:line 41,42}Rp em:point 43⟄;Y em:point 44 em:line 43,44⟄;Y em:point 45Rj em:point 46 em:line 45,46N? em:point 47tJ em:point 48 em:line 47,48tJ em:point 49H\ em:point 50 em:line 49,50H\ em:point 518  em:point 52 em:line 51,52c1e1ፑ=Nc2}a"9"c3]@h']ci1QG3cici+1⍒לxcN"2̡.J6cN"1:XcNexM...ꍒW...%,Figure1:8VisualizingasingleRJ2N 6pSoinrtwithparallelcoordinatesmethod.-*[QT3.2.)cVisualizing/examplesandruleswithparallelcodCordinates7x[QLikre5|inthesimpletablevisualizingmethoSdfromSection2.3.,H1attributesareviewed[QasdimensionsandexamplesaspSoinrtsintheso-formedattributespace. DAttributes[Qarenenrumeratedandtoeachattributeanx-axisisassigned.wAttributevXaluesarealso[QenrumeratedfgandanintervXal21 &kontheappropriatex-axisisassignedtoeachofthem.[QOrderingofattributesandattributevXaluesmrustbSe xedatthebeginingofthevisu-[Qalizationsincedi erenrtorderingsresultwithdi erentpictures.TVoproSducedi erent,[QmarybSe devenbSettervisualization,QorderingofvXalues(onlyfordiscrete,nominalat-[Qtributes)ycouldbSecrhanged.*&Thedistancebetrweenyx-axesandtheinrtervXallengthare[Qcalculatedfromthegivrendimensionsofthevisualizingwindow(partoforthewhole[Qscreen).EA;setpofexamplesisdividedinrtosubsetsaccordingtotheclassvXalueandeach[QsubsetchisvisualizedseparatelyV.22c$EacrhexampleisrepresentedbyapSolygonalline[QconnectingthecorrespSondingattributevXaluesontheparallelcoordinates.hFVordis-[Qcrete "attributetheactualcoSordinateoftheattributevXalueiscrhosenrandomlyfrom[Q ffu ^ٓRcmr71K`y cmr10RatherT thanapGoint,TManintervqalisassignedalsotodiscreteattributestoenabletheseparation bGetweenUUtheindividualexamples.  ^2Alternatively*,4itispGossibletovisualizealltheexamplesonthesamecoordinatesystem,4andseparateUUtheclassesbycoloursorpatterns.MlqɎ}DZ֍MFigure2:8Examplesforthe`Saturdarymorning'domain.("7sFigure3:8Twrono-classrulesforthe`Saturdaymorning'domain. [Qthe{designatedattributevXalueinrterval,=0sinceforconrtinousattributeamappingof[QattributevXaluestodesignedinrtervalisestablished.EIn VFigure2allexamplesforthe`Saturdarymorning'domainarevisualized.Onthe[Qleftarethe6߆T cmtt12C1(no-class)examples,OontherighrttheC2(yes-class)examples.Attribute[QA1t(outloSok)hasvXalues:1(sunnry),x2(overcast),x3(rain);+attributeA2(tempSerature)[Qhas,vXalues:1(hot),x2(mild),3(rain);attributeA3(hrumidity),hasvXalues1(high),2[Q(normal);attributeA4(windy)hasvXalues1(no),2(yres).EIn Figures3,4and5theinducedrulesarevisualizedinthesameorderasgivren[QinUYthedomaindescriptioninSection2.2.. xEacrhruleisrepresentedseparatelyon[QthesamecoSordinatesastheexamples,Bandisvisualizedastheareabetrweenthe[QpSolygonallinesconnectingtheallorwableattributevXalues." Ifaparticularattribute[Qis=notincludedinarule,YallpSossiblevXaluesareallorwed=(themostgeneralruleis[Qvisualized.astherectanglebSetrween.the rstandthelastvXalueofallattributes).In[Qordertoillustratehorwarulecoversexamples23,_individualforeachclassrulesare[QshorwnonthesamecoSordinatesasthetrainingexamplesbelongingtothisclass.The[Qmacrhinexlearningsysteminducesasetofrulesforeachclass.Wheninducingrules[Qforagivrenclass,thesystemtriestocovertheexamplesfromthisclassandtoavoid[Qtheexamplesfromotherclasses.&YTheunionofallrulesforagivrenclassshouldcover[Qall24 examplesRfromthisclass. pIfsomeexamplesremainuncorveredRtheymighrtbSe[Qconsideredasoutliersornoise.[Q33ffu ^3ExampleiscoveredbyaruleifthewholepGolygonallinerepresentingthisexample,nliesinside theUUcontouroftherule(greyarea).  ^4Exact coveringcriterionmightbGereplacedbysomeothernon-exactcriterioninordertoenablehandlingUUofnoiseindata.`٠qɎ}Ji35ɜFigure4:8Twroyes-classrulesforthe`Saturdaymorning'domain.k?બ[QFigureK5:Thethirdyres-classruleforthe`Saturdaymorning'domain.\Itisobvious[Qthat2thisrulecorvers2exampleswiththosevXaluesoftheattributeA1(valu2),that[Qwrerenotcoveredbythe rsttworules.kbqɎ}[q[Q4."(Parallelffcos3ordinatesmethodinproblemanalysis;[QT4.1.)cReal-world/problem7x[QTheparallelcoSordinatesmethodwrastestedonareal-worldmedicaldiagnosticdomain[Q`lymphographry'whichisoneofthestandardtestingdomainsformachinelearning[Qsystems(e.g.,.Cestnik,KononenkroandBratko1987[2],.ClarkandNiblett1991[3]).[QThegdomainconsistsof148examplesgivrenbyvXaluesof18attributes,=eachlabSeled[Qbryoneof4classes.ZFVorruleinductionthemachinelearningsystemUA3tris[7]was[Qused.EThe&esetofexamplesisvisualizedseparatelyforeacrhclass(forillustrationonlythe[Q rstxtrwoclassesareshowninFigure6).WhenthereisalargenumbSerofexamplesina[Qsingleclass(likreinclass2),GaitishardtoseparatebSetweentheindividualexamples.In[QthiscasewreseethecontourofthepSolygoncoveredbytheexamples.1(WVehavetokeep[QinemindthatthepSolygonisavisualizationofamrultidimensionalspace.)tTherulesfor[Qthe>correspSondingclasstendtoadjusttotheconrtourofthecorrespondingexample[QpSolygonandarvoidexamplesfromtheotherclasses.Someresultsofthevisualization[QofrulesareshorwnonFigures7and8. CInbSoth gures,6therearetwographical[Qrepresenrtationsofeachrule:cwithout(abSove)andwith(bSelow)thevisualizationof[QthecorrespSondingtrainingexamples.[VisualizationofexamplesandrulesinFigures7[QandQi8enabletheanalysisoftheexamplesandtheinducedrules.AscanbSeseenfrom[QFigure07,B)theinducedruleforclass1corvers0alltheexamples. InFigure8aregivren[Qjust2outof8rulesinducedforclass2and,8therefore,onthis guremanryexamples[Qarenotcorvered.b$Figure6:8Examplesforthe`lymphographry'domainforclass1andclass2. mqɎ}_બ[QFigureE7:Aruleforthe rstclassinthe`lymphographry'domain.&FVromthe gure[Qit}bSecomesobrviousthatasingleruleforthisclassissucientsincetherule-pSolygon[Qcorversalltheexamplesinclass1.]બ[QFigure38:Twro(outof8)rulesforthesecondclassinthe`lymphography'domain.[QTherewBaresomeuncorveredwBrulesinthe gurethatshorwsunsuciensyoftworulesfor[Qcorveringalltheexamplesfromclass2. uqɎ}[q[QT4.2.)cDiscussion7x[QThe|TvisualizingmethoSdcanbeanecienrthelptoaknowledgeengineerusingma-[QcrhinelearningtoSols.ӼFirst,itgivestheuserabSetterunderstandingofthedomain[Qbry;graphicallyrepresentingthetrainingexamples.FVurthermore,^inruleconstruction,[QmacrhinelearningsystemscanbSe`tuned'tothegivenproblembyappropriatelyselect-[Qingtheparameterswhicrhin uencethegeneralityofinducedrulesandthehandling[QofMnoisydata.Inparametertuning,m thegraphicalrepresenrtationofruleswhichshows[Qthe|corverageoftrainingexamplesismostuseful.'InFigure6,Qsomeexamplesinclass[Q2%areoutliersinthesetofexamples(trwo%examplesbSecauseofthesecondvXalueof[Qthe_attributeA4andoneexamplebSecauseofthe rstvXalueforattributesA11,A12,[QA13).'Whentuningthesystem,#therulesinducedforclass2arenotsuppSosedto[QcorverthosevXalues(thesethreeexamplesareconsideredtobSenoise)."[Q5."(Conclusion[QVisualizationyNofexamplesandruleswiththeparallelcoSordinatesmethodenablesthe[QanalysisȨofthespaceofexamplesandinducedrules.5ThedescribSedvisualizingmethod[QisconnectedtothemacrhinelearningsystemUA3tris[7]andiscurrentlypSostprocessing[Qthexresultsofthemacrhinelearningalgorithm.Thedomain(thenumbSerofattributes[QandvXalues)islimitedonlywiththedimensionsofthewindorwusedforvisualization.[QThemethoSdalonedoesnotlimitthedimensionsofthedomain.ETheparallelcoSordinatesvisualizingmethodcanbeusedfortheanalysisofinduced[QrulesWandbSothoftrainingandtestingexamples.Itcanalsobeusedforvisualizing[QrulesduringtheproScessoflearningwhicrhcangiveusefulinformationtothedevelopSer[Qofamacrhinelearningsystemandalsototheknowledgeengineerusingthesystem.EIngfurtherwrork,additionaluseoftheimplementedmethoSdcanbeenrvisagedsuch[Qas4forvisualizingtheclassi cationofnewexamplesorforvisualizingthepSerformance[QofacorveringalgorithmbrydynamicallymoSdifyingthetrainingsetandthecurrently[Qinduced%(rules,3deletingtheexamplescorvered%(bythepreviousrules.aIntroSducingdif-[Qferenrtdcolorstovisualizationofexamples25 ^hcanexpSosesomeadditionalinformation,[Qlikrethedi erencebSetweencoveredanduncoveredexamples.[Q6."(Acknowledgements[QThiswrorkwassuppSortedbytheSlovenianMinistryofresearchandTVechnology.[QMedicalpdatasetwrasobtainedfromUniversityMedicalCenterinLjubljana,Fandwas[QprorvidedqrbyMilanSokolisc.?IqOwishtothankFVrancSolinaforencouragingthiswork[QandNadaLarvrascforhercommenrtstothispapSer.!+[QReferences3K [1]CBec!ker,R.A.,Clev!eland,W.S.> andWilks,A.R.(1987)*': 3 cmti10Dynamicagrpaphicsfordata Canalysis.fStatisticalScience4,Veol.2,pp.355-395.[Q TDffu ^5AtfthemomentexamplesarevisualizedwithblackpGolygonallinesandruleswithdi erentcolor areasUU(inthispapGer,unfortunatelytheyalllooklikegreyareas). xDqɎ}[q [2]CCestnik,{B.,Kononenk!o,I.andBratko,{I.(1987)ASSIST)ANT*86:AknowFlepdge+elic- Citationtopolforsophisticatedusers.In:įBratk!o,֩I.andLavra"Dc,֩N.(eds.)ProgressinCmac!hineflearning.Wilmslow:SigmaPress. [3]CClark,Pe.`%andNiblett,T.(1989)TheCN2inductionalgorithm.`%Mac!hineLearning1C(3),f261-284.Klu!werfAcademicPublishers. [4]CDillon,^W.R.\andGoldstein,M.GrpaphicalImethods.\InMultivdDariateAnalysis:XMethoMdsCandfapplications,pp.191-202.JohnWiley&Sons,NewYeork. [5]CInselbMerg,A.JandDimsdale,B.(1987)ParpalFlelXzcoordinatesforvisualizingmulti-Cdimensionalgepometry.fFifthIn!ternationalComputerGraphicsConference.Japan. [6]CMic!halski,[R.S.6andStepp,R.E.(1983)Lpearningofromobservation:tConceptualclus-CteringInMac!hineLearning,.Arti cialintelligenceapproach,.VeolumeI,pp.331-363.CTiogafPublishingCompan!ye,PaloAlto,California. [7]CMladeni"Dc,D.,ZupaniDc,D.,GrobMelnik,M.andLa!vraDc,N.(1992)Stopchastic+nsearchinCinductivecponceptlearning.Teec!hnicalrepMortCW140,DepartmentofComputerScience,CK.U.Leuv!en,fBelgium. [8]CQuinlan,9J.R.(1986)Inductionm"ofdepcisiontrees.Mac!hineLearning,9Veol.1,pp.81-106.;֨ U- cmcsc10T}h!ff cmsl12H- 3 cmcsc10GDtHGcmr17