Crisply Generated Fuzzy Concepts Radim B elohl avek Vladim r Sklen a r and Ji r Zacpal - PDF document

CrisplyGeneratedFuzzyConcepts269tofuzzysetting(generalizationofFCAfromthepointofviewoffuzzylogic).AgeneraldiscussionabouttherelationshipbetweenconceptualscalinginthesenseofFCAandmembershipfunctionsinthesenseoffuzzysetcanbefoundin[21].Inthepresentpaper,weareinterestedinFCAofdatawithfuzzyattributes(FCAf)intheframeworkoffuzzylogicandfuzzysettheory.Probablythe“rstpaperonthiswas[11].Lateron,FCAfwasdevelopedbyPollandt[18]and,inde-pendently,bythe“rstauthorofthispaper,e.g.[1,2,3,7].AnimportantaspectofFCAingeneralisthepossiblylargenumberofformalconceptsextractedfromdata.Inthispaper,weproposeandstudywhatwecallcrisplygeneratedformalfuzzyconcepts.TheseareparticularformalfuzzyconceptswhichcanbeconsideredmoreimportantŽthantheothers(non-crisplygenerated).Consider-ingonlycrisplygeneratedconcepts,themainpracticale
ectisthereductionofthenumberofformalconceptsextractedfromdata.Intherestofthissection,wepresentpreliminariesonfuzzylogicandFCAf.InSection2wepresentourapproachandtheoreticalresults.Section3containsexamplesandexperimentsstudyingmainlythereductionofthenumberofextractedconcepts.1.2PreliminariesFuzzySetsandFuzzyLogic.Weassumebasicfamiliaritywithfuzzylogicandfuzzysets[16,13,6].Anelementmaybelongtoafuzzysetinanintermedi-atedegreenotnecessarilybeing0or1.Formally,afuzzysetinauniverseisamappingassigningtoeachatruthdegreeissomepartiallyorderedsetoftruthdegreescontainingatleast0(fullfalsity)and1(fulltruth).needstobeequippedwithlogicalconnectives,e.g.(fuzzycon-(fuzzyimplication),etc.togetherwithlogicalconnectivesformsaoftruthdegrees.Weassumethatformsaso-calledcompleteresid-uatedlattice.Recallthatacompleteresiduatedlattice[6,13,14]isastructuresuchthat(1)isacompletelattice(withtheleastelement0,greatestelement1),i.e.apartiallyorderedsetinwhicharbi-traryin“ma()andsuprema()exist;(2)isacommutativemonoid,isabinaryoperationsatisfying;(3).Inwhatfollows,alwaysdenotesa“xedcompleteresiduatedlattice.Themostappliedsetoftruthdegreesistherealinterval[01];witha,b=max(a,b),andwiththreeimportantpairsoffuzzyconjunctionandfuzzyimplication:Lukasiewicz(=max(=min(11)),minimum(=min(a,b=1ifand=else),andproduct(=1ifand=else).Anotherpossibilityistotakea“nitechain)equippedwithLukasiewiczstructure(l,norminimum(k,lThesetofallfuzzysets(or-sets)inisdenoted.Forafuzzyset,the1-cutisanordinarysetiscalled 270R.Bavek,V.Sklen´r,andJ.Zacpalcrispif.Bywedenoteafuzzysetforwhich)=0for.ForfuzzysetsA,Bweputisasubsetof)ifforeachwehave).Moregenerally,thedegreeA,Btowhichisasubsetofisde“nedbyA,B)).Then,A,B)=1.FormalConceptAnalysisofDatawithFuzzyAttributes.besetsofobjectsandattributes,respectively,beafuzzyrelationbetween.Thatis,assignstoeachandeachtruthdegreex,ytowhichobjecthasattributeisasupportsetofsomecompleteresiduatedlattice).ThetripletX,Y,Iiscalledaformalfuzzycontext(correspondstoadatatablewithfuzzyattributes).Forfuzzysets,considerfuzzysets(denotedalso)de“nedbyx,yx,y))(2).Usingbasicrulesofpredicatefuzzylogic,onecanseethat)isthetruthdegreeofthepropositionissharedbyallobjectsfrom)isthetruthdegreeofhasallattributesfromŽ.PuttingX,Y,IA,BB,BX,Y,I)isthesetofallpairsA,Bsuchthat(a)isthecollectionofallobjectsthathavealltheattributesof(theintent)and(b)isthecollectionofallattributesthataresharedbyalltheobjectsof(theextent).ElementsofX,Y,I)arecalledformalconceptsofX,Y,I(formalfuzzyconcepts,formalX,Y,I)iscalledtheconceptlatticegivenbyX,Y,Iconceptlattice,-conceptlattice).BoththeextentandtheintentofaformalconceptA,Bareingeneralfuzzysets.Thiscorrespondstothefactthatingeneral,conceptsapplytoobjectsandattributestovariousintermediatedegrees,notonly0and1.)(3)X,Y,Imodelsthesubconcept-superconcepthi-erarchyinX,Y,IThefollowingisaversionofthemaintheoremforfuzzyconceptlattices(see[7,18]).Theorem1.ThesetX,Y,Iisunderacompletelatticewherein“maandsupremaaregivenby CrisplyGeneratedFuzzyConcepts271Moreover,anarbitrarycompletelatticeisisomorphictosomeX,Y,Ii
therearemappingssuchthatX,L-denseinV,Y,L-denseinV;x,yx,ay,bNotethatTheorem1canbeprovedbyreduction(see[18,4])tothemaintheoremofordinaryconceptlattices[20]ordirectlyintheframeworkoffuzzylogic[7].NotealsothatTheorem1isconcernedwithbivalentorder,thereisstillamoregeneralversion[7]dealingwithmany-valued(fuzzy)order.Taking(twotruthdegrees;bivalentcase),thenotionsofformalfuzzycontext,formalfuzzyconcept,andfuzzyconceptlatticecoincidewiththeordinarynotions[12].InthefollowingwedenoteExt(A,BX,Y,I)forsome(extentsofconcepts)andInt(A,BX,Y,I)forsome(intentsofconcepts).Recall[3]that.Asa2CrisplyGeneratedFormalConcepts2.1MotivationandDe“nitionAformalconceptA,Bconsistsofafuzzysetandafuzzysetsuchthat.Dueto(1)and(2),andthebasicrulesofpredicatefuzzylogic,thisdirectlycapturestheverbalde“nitionofaformalconceptinspiredbyPort-Royallogic.Nevertheless,thisde“nitionactuallyallowsforformalfuzzyA,Bsuchthat,forexample,foranywehave2and2.Averbaldescriptionofsuchaconceptisaconcepttowhicheachattributebelongstodegree1/2Ž.Suchaconcept,althoughsatisfyingtheverballydescribedcondition,willprobablybeconsiderednottheimportantoneŽ.ThisisbecausepeopleexpectconceptstobedeterminedbysomeattributesŽ,i.e.byanordinarysetofattributes.Thisleadstothefollowingde“nition.De“nition1.AformalfuzzyconceptA,BX,Y,Iiscalledcrisplygen-ifthereisacrispsetsuchthat(andthusWesaythatcrisplygeneratesA,B.LetX,Y,I)denotethecollectionofallcrisplygeneratedformalconceptsinX,Y,I,i.e.X,Y,IA,BX,Y,IthereisA,Bisacrisplygeneratedconceptwith,itmightbeactuallymoreinformativetowriteA,BinsteadofA,B.Doingso,noinformationislostsincethecorrespondingfuzzyconceptA,BcanbeobtainedfromA,B.Ingeneral,theremaybeseveralcrispswith.Toremovethisambiguity,wecanalwaystakethegreatestLemma1.ForacrisplygeneratedformalconceptA,Bisthelargestcrispforwhich 272R.Bavek,V.Sklen´r,andJ.ZacpalProof.A,Bbecrisplygeneratedbysome,i.e..Since,wehave.Thatis,containsanycrispwhichgen-A,B.Moreover,itselfisacrispsetwhichgeneratesA,B.Indeed,takesomecrispwhichgeneratesA,B.Weknowthat,fromwhichweget.Ontheotherhand,gives(whichshows(Crisplygeneratedformalconceptscanbealternativelyde“nedasmaximalA,Bcontainedinforwhichistheprojectionof.Callafuzzyrectangularrelationiftherearesuchthatx,y),written(callthenA,BrectangleA,Bissaidtobecontained.Weputifforeachwehave)and).Byan-projectionofasubsetwemeanafuzzysetde“nedbyx,yLemma2.A,Bisacrisplygeneratedconcepti
A,Bisamaximal(w.r.t.)rectanglecontainedinsuchthatistheprojectionofProof.Theassertionfollowsfrom[6…Theorem5.7],thefactthatA,Biscrisplygeneratedi
(seeLemma1),andfrom(x,y2.2IndependenceoftheChoiceofFuzzyLogicalConnectivesThenextstepistoobservethatrestrictingourselvestocrisplygeneratedcon-cepts,oneisnomoredependent(almost)onthelogicalconnectivesde“nedonthescaleoftruthdegrees.Toformulatethisprecisely,letusdenotethecon-ceptlatticeoverthestructureoftruthdegreesbyX,Y,I)anddenoteX,Y,I)thesetofallcrisplygeneratedconceptsofX,Y,I).Supposewehavetwostructureswithacommonsetoftruthdegrees,i.e.,andadatatable(formalfuzzycontext)X,Y,Iwhichis“lledwithtruthdegreesfromLemma3.Lethaveacommonsetoftruthdegrees,letX,Y,Iaformalfuzzycontextwithtruthdegreesfrom.ThenthereisanisomorphismbetweenX,Y,IX,Y,IsuchthatforthecorrespondingformalconceptsX,Y,IX,Y,IwehaveProof.Denotebytheoperatorsgeneratedby2).Recallthatforeachresiduatedimplicationconnectivewehave1.Therefore,foreachcrispwehavex,y))=x,y))=x,y).Thatis,doesnotdependon.Therefore,ifA,BX,Y,I)iscrisplygeneratedthenfromLemma1wehave(andsinceiscrisp,also(.ThisshowsthatA,D,for,isacrisplygeneratedformalconceptfromX,Y,I).Clearly,.If,i.e.islargerthan,thenLemma1gives(andso( CrisplyGeneratedFuzzyConcepts273whichisimpossiblesincebyLemma1,isthelargestonewith(InasimilarwayoneshowsthatifA,BX,Y,I)iscrisplygeneratedA,AX,Y,I)iscrisplygeneratedaswelland.Theassertionthenimmediatelyfollows.Notethatingeneral,X,Y,I)andX,Y,I)mayhavedi
erentnum-berofformalconcepts,i.e.thechoiceoffuzzylogicalconnectivesmatters.Lemma3showsthattheircrisplygeneratedpartsX,Y,I)andX,Y,Iareisomorphic.Thatis,ifweconsideronlycrisplygeneratedconcepts,thechoiceoffuzzylogicalconnectives,inasense,doesnotmatter.2.3ComputingAllCrisplyGeneratedFormalConceptsWenowpresentanalgorithmforgeneratingX,Y,I).Goingdirectlybydef-inition,i.e.creatingforeachcrisp,hasexponentialtimecomplexityandthus,cannotbeused.OuralgorithmisinspiredbyGantersNextClosurealgorithm[12…p.67]forgeneratinganordinaryconceptlattice,i.e.generatingallformalconceptsinlexicographicorder.Thisideacanbeadoptedtofuzzysettingtogenerateallcrisplygeneratedformalfuzzyconcepts.TheideaofouralgorithmistointroducealinearorderingX,Y,IsuchthatforagivenA,BX,Y,I),wecancomputeitsimmediatesucces-sorw.r.t.to.SinceaformalconceptA,Bisuniquelygivenbyitsintent,itissucienttogenerateallintents.ByInt)wedenoteallintentsofcrisplygeneratedfuzzyconcepts,i.e.IntA,BX,Y,I)forsome.Wesupposethatsuchthatif(thatis,theorderingofelementsofbyindicesextendstheirorderingin;suchanindexingisalwayspossibleandisautomaticallysatis“edifislinearlyorderedandweindextheelementsinusingthisorderfromtheleasttothegreatestelement,i.e.···).Forintroducearelationi
(and()forji.Furthermore,weputforsomeThatis,i
the“rstelementofonwhichdi
er,belongs;i.e.meansthatislexicographicallysmallerthanLemma4.isastricttotalorderonIntwhichextendsProof.EasytoseesinceeveryInt)withcommon1-cuts(i.e.with)areequal.Indeed,ByLemma1,ifFurthermore,for,weput:=(( 274R.Bavek,V.Sklen´r,andJ.ZacpalThatis,weobtainbytakingthe1-cutof,cuttingo
theelementsi,...,n,joiningwithelementandapplyingtheclosureLemma5.Thefollowingassertionsaretrue.(1)If,and(2)if(3)ifProof.(1)followsdirectlyfromde“nition.(2)Fromwehave.Putting,wethushave,whence(.(3)Fromwehave.Using(2)weget(andso.Ontheotherhand,.Therefore,.Finally,by(2),1=(),i.e.provingTheorem2.For,theleastcrisplygeneratedintentIntwhichisgreaterthanisgivenbywhereisthegreatestelementwithProof.betherequiredsuccessorof.WehaveBBforsome.ByLemma5(3),.ByLemma5(2),andthusandso.Sinceisthesuccessorof,wehave.Itremainstoshowthatisthegreatestelementwith).If)forthenLemma5(1)yields),i.e.whichisacontradictiontoistheimmediateofTheorem2leadstothefollowingalgorithm.X,Y,I,OUTPUT:IntThetimecomplexityofcomputingfromthenextcrisplygeneratedintent).Therefore,ouralgorithmhaspolynomialtimedelaycomplex-ity[15](generatingcrisplygeneratedintents,onegeneratesthesuccessorinpolynomialtime)).ThetimecomplexityofthealgorithmisthusInt CrisplyGeneratedFuzzyConcepts275Remark1.Notethatin[8]wepresentedanalgorithmforgeneratingallformalfuzzyconceptsofX,Y,I).ThisalgorithmisinspiredbyGantersNextClosurewhichisitsparticularcase.Usingthisalgorithm,wecangenerateX,Y,Iinthefollowingway:GenerateallA,BX,Y,I)andforeachsuchA,BtestbyLemma1whetherA,Biscrisplygenerated.Compratedtothis,thealgorithmpresentedheregeneratesX,Y,I)directly,goingfromonecrisplygeneratedconcepttothenextone.Wedemonstratethespeed-upinSection3.2.4CrisplyGeneratedFuzzyConceptsasFixedPointsofFuzzyGalois-LikeMappingsItiswell-knownthatordinaryformalconceptsX,Y,Iareexactlythe“xedpointsofaGaloisconnectionformedby(theconceptderivationoperators)inducedby[19,12].Moreover,eachGaloisconnectionbetweenisinducedbysomerelation.In[2],thisfactwasgeneralizedtothesettingoffuzzylogic:CallafuzzyGaloisconnectionbetweenanypairofmappings)(6))(7)foreachA,AB,B.Itwasprovedin[2]thatgivenX,Y,I,thepairde“nedby(1)and(2)isafuzzyGaloisconnectionand,conversely,eachfuzzyGaloisconnectionisinducedbysomeX,Y,Iby(1)and(2).TherelationshipbetweenfuzzyGaloisconnectionsandfuzzyrelationsbetweenisone-to-one.Anaturalquestionarisesastowhethercrisplygeneratedfuzzyconceptscanbethoughtofas“xedpointsofsuitablemappings,possiblyaxiomaticallyde“nable.Inthefollowing,wepresentapositiveanswer.Infact,whatwearegoingtopresentisaspecialcaseofamoregeneralcaseofso-called(fuzzy)Galoisconnectionswithhedges[10].However,tokeepourdiscussionintheframeworkofcrisplygeneratedconcepts,wedonotgotothemoregeneralnotionsof[10]andpresenttheresultswithproofsforourspecialcase.ConsidermappingsresultingfromX,Y,Ix,y))(10)x,yNotethatwehavearede“nedby(1)and(2).Now,denotebyY,Ithesetofall“xedpointsof,i.e.Y,IA,BB,B 276R.Bavek,V.Sklen´r,andJ.ZacpalTheorem3.Y,IX,Y,I,i.e.crisplygeneratedfuzzyconceptsareexactlythe“xedpointsofProof.Ž:IfA,BY,I,i.e.and(.Therefore,A,BX,Y,I),byde“nition.Ž:LetA,BX,Y,I),i.e.,andsomecrisp.Weneedtoverify.ByLemma1,itclearlysucestocheck(,i.e..Asweneedtoverify.But.Indeed,the“rstequalityfollowsfromthefactthatiscrispandthus.Forthesecondequality,(followsfromand(followsfrom,from,andfromthefactthatif(justput).Hence,A,BY,INow,weturntotheinvestigationofthepropertiesofandtheproblemofaxiomatizationoftheseproperties.Lemma6.de“nedby(10)and(11)satisfyA,BB,A)(12)foreveryA,AProof.WehaveA,Bx,y)))=x,y)))=x,y)))=B,Aproving(12).(13)isaconsequenceofpropertiesoffuzzyGaloisconnections[2].De“nition2.Apairofmappingssatisfying(12)and(13)iscalledac-GaloisconnectionbetweenThefollowingaresomeconsequencesof(12).Lemma7.satisfy(12)then)(16))(17)foranyB,B CrisplyGeneratedFuzzyConcepts277Proof.(14):Weshowthat)=1i
)=1foreach.Using(12)wehaveAsaresult,wehave)=1i
1i
foreachwehavei
foreachwehave)i
foreachwehaveA,B)i
(15)isjust(14)for=1.(16)and(17)followfrom)and)whichwenowverify.First,)(here=1for=1and=0otherwise).For)justputintheforegoingequality.Lemma8.Letbeac-Galoisconnection.Thenthereisafuzzyrelationsuchthatwhereareinducedby(10)and(11).Proof.bede“nedbyx,y).Thenusing(16),itisstraightforwardtoshow.Furthermore,using(14)and(15),and(17)wegetx,yNext,wehavethedesiredone-to-onecorrespondencebetweenfuzzyrelationsandc-Galoisconnections.Theorem4.Letbeafuzzyrelation,letbede“nedby(10)and(11).Letbeac-Galoisconnection.Thensatisfy(12)and(13).de“nedasintheproofofLemma8isafuzzyrelationandwehaveProof.Duetothepreviousresults,itremainstocheck.Wehavex,yz,yx,y),completingtheproof.Comingbacktoconditions(6)…(9),onecaneasilyseethattheyareingeneralnotsatis“edbyac-Galoisconnection.Thenextlemmashowspropertiesofc-Galoisconnectionswhichareanalogousto(6)…(9). 278R.Bavek,V.Sklen´r,andJ.ZacpalLemma9.Forac-Galoisconnection,wehave)(18))(19)Proof.Bydirectveri“cation.2.5MainTheoremforX,Y,INowwepresentaversionofmaintheoremofconceptlatticesforX,Y,IDuetothelimitedscopeofthepaper,wepresentonlysketchofproof.Note(see[6])thatforafuzzysetisasubsetofu,a.Conversely,forafuzzysetinde“nedbyu,a.Now,forafuzzy,de“neanordinaryrelationbetweenx,ax,yTheorem5.ThesetX,Y,Iequippedwithisacompletelatticewherein“maandsupremaaregivenbyMoreover,anarbitrarycompletelatticeisisomorphictosomeX,Y,Ii
therearemappingssuchthatX,L-denseinV,-denseinV,andx,yx,aProof.Sketch:Analogouslyasin[4],wecan“ndabijectionbetweenc-Galoisconnectionsbetween,andordinaryGaloisconnectionsbetween.Underthisbijection,(fuzzyrelationcorrespondinganc-Galoisconnec-tion)correspondsto(ordinaryrelationcorrespondingtoaGaloisconnection)andthecorrespondingc-GaloisconnectionandGaloisconnectionhaveisomor-phiclatticesof“xedpoints.OneofthemisourX,Y,I),theotheroneisL,Y,I).Now,L,Y,I)isanordinaryconceptlattice,andthusobeysWillesMainTheorem[20].TranslatingtheMainTheoremtoX,Y,Ithengivesourtheorem.Corollary1.X,Y,Iisa-subsemilatticeofX,Y,IRemark2.X,Y,I)neednotbea-subsemilatticeofX,Y,I).Onecanverifybytaking,and CrisplyGeneratedFuzzyConcepts2792.6CrisplyGeneratedConceptsandOne-SidedFuzzyConceptsIn[22],theauthorsdealwiththefollowing.LetX,Y,Ibeafuzzycontext(with=[01]).De“nemappings(assigningafuzzysetattributestoaofobjects)and(assigningaofobjectstoafuzzysetofattributes)byx,y)(23)foreachx,yThesamede“nitionwaslaterrediscoveredŽbyKrajci[17].PairsA,Barecalledone-sidedfuzzyconcepts(isaset,isafuzzyset)in[17].BydirectcomputationonecanverifythatA,Bisaonesidedfuzzyconcepti
itisoftheformA,BforsomefuzzyconceptX,Y,IwhichiscrisplygeneratedbyextentsŽ,i.e.suchthatforsomewehave)forsomeset.Therefore,uptoexchangingrolesofextentsandintents,[22,17]infactdealwithparticularformalfuzzyconcepts(crisplygeneratedbyextents)fromX,Y,I),onlythatinsteadofA,BA,B.Asaconsequence,ourresultpresentedinthispaperapplyinanappropriatemodi“cationtoone-sidedfuzzyconceptsof[22,17].3ExamplesandExperimentsTab.1describeseconomicindexesofselectedcountries,transformedto[0togetaformalfuzzycontext.Usingminimum-basedfuzzylogicaloperations,thecorrespondingconceptlatticeX,Y,I)contains304formalconceptsandisdepictedinFig.1.ThecorrespondingsetX,Y,I)ofallcrisplygeneratedfuzzyconceptscontains27formalconceptsandisdepictedinFig.2.Asweareinterestedonlyinthereductionofthesizeoftheconceptlattice,weomitthedescriptionsofformalconcepts. Fig.1.ConceptlatticecorrespondingtodatafromTab.1Nextweshowresultsofexperimentsdemonstratingthefactorofreduction.Thatis,weareinterestedintheratioX,Y,IX,Y,I(thesmaller,thelargerthereduction).Tab.2showsthevaluesoffor10experiments CrisplyGeneratedFuzzyConcepts281Table2.Behaviorof(averageAv,dispersionVar)independenceonthesizeofinputdatatable(rows);columnscorrespondtoexperiments 12345678910AvVar 0.580.40.380.530.480.380.430.410.480.330.4410.0733 0.310.310.380.430.380.320.430.420.360.380.3720.0443 0.460.370.310.480.450.270.410.430.40.370.3950.0635 .................................... 0.090.110.10.10.10.10.10.090.10.110.0990.0066 0.10.090.080.10.090.10.090.090.090.080.0900.0079 0.080.070.090.080.090.090.080.070.090.080.0810.0074Table3.Dependenceof(averageAv,dispersionVar)onthenumberof1sinobjectattributes(rows);columnscorrespondtoexperiments 12345678910AvVar 0.10.080.070.080.080.090.10.080.080.080.0840.0100 0.090.10.110.110.080.090.10.080.090.090.0940.0086 0.120.110.110.10.110.110.120.10.10.110.1070.0084 0.120.130.110.140.140.130.130.120.140.140.1300.0093 0.180.150.150.160.160.160.160.170.160.170.1620.0095 0.180.210.170.170.20.20.20.180.20.220.1930.0151 0.240.240.260.260.280.260.220.270.250.280.2560.0193 0.360.330.340.360.350.350.330.340.370.350.3470.0121 0.540.590.550.560.530.480.510.560.520.550.5390.0298 1,0000.0000Table4.Dependenceof(averageAv,dispersionVar)onthenumberofnonzerovaluesinattributes(rows);columnscorrespondtoexperiments 12345678910AvVar 0.590.590.640.560.620.560.580.630.610.630.6000.0273 0.550.550.470.490.590.430.510.470.550.440.5040.0503 0.620.430.570.50.480.540.470.480.610.450.5140.0642 .................................... 0.060.070.080.070.080.070.070.080.060.050.0670.0089 0.030.050.040.060.040.070.040.040.040.030.0440.0113 0.040.050.060.070.050.040.060.080.050.070.0580.01170(thenumbervariesfrom1to15,rows);columnsrepresentexperi-ments;weconsideraverageanddispersionof,seeTab.4.Next,werandomlygeneratedinputdatatableswith20objectsand20attributeswithvarying31(rows),seeTab.5;columnsrepresentexperiments;weconsideraverageanddispersionofInthelastexperiment,weobservedthespeed-upofthealgorithmdescribedinSection2.3comparedtojustusing[8]andtestingwhichconceptsarecrisplygenerated,seeRemark1.Werandomlygeneratedseveraldatatableswithdi- CrisplyGeneratedFuzzyConcepts2836.BavekR.:FuzzyRelationalSystems:FoundationsandPrinciples.Kluwer,Academic/PlenumPublishers,NewYork,2002.7.BavekR.:Conceptlatticesandorderinfuzzylogic.Ann.PureAppl.Logic(2004),277…298.8.BavekR.:Proc.FourthInt.Conf.onRecentAdvancesinSoftComputingNottingham,UnitedKingdom,12…13December,2002,pp.200…205.9.BavekR.:Whatisafuzzyconceptlattice(inpreparation).10.BavekR.,Funiokov´aT.,VychodilV.:Galoisconnectionswithhedges(submit-ted).PreliminaryversiontoappearinProc.8-thFuzzyDays,Dortmund,Septem-ber2004.11.BuruscoA.,Fuentes-Gonz´alesR.:ThestudyoftheL-fuzzyconceptlattice.ware&SoftComputing,3:209…218,1994.12.GanterB.,WilleR.:FormalConceptAnalysis.MathematicalFoundations.Springer,Berlin,1999.13.H´ajekP.:MetamathematicsofFuzzyLogic.Kluwer,Dordrecht,1998.14.H¨ohleU.:Onthefundamentalsoffuzzysettheory.J.Math.Anal.Appl.15.JohnsonD.S.,YannakakisM.,PapadimitrouC.H.:Ongeneratingallmaximalindependentsets.Inf.ProcessingLetters(1988),129…133.16.KlirG.J.,YuanB.:FuzzySetsandFuzzyLogic.TheoryandApplications.PrenticeHall,UpperSaddleRiver,NJ,1995.17.KrajciS.:Clusterbasede
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CrisplyGeneratedFuzzyConceptsRadimBavek,Vladim´šrSklen´r,andJišZacpalDept.ComputerScience,Palack´yUniversity,Tomkova40,CZ-77900Olomouc,CzechRepublicradim.belohlavek,vladimir.sklenar,jiri.zacpalInformalconceptanalysisofdatawithfuzzyattributes,boththeextentandtheintentofaformal(fuzzy)conceptmaybefuzzysets.Inthispaperwefocusonso-calledcrisplygeneratedformalconcepts.AconceptA,BX,Y,I)iscrisplygeneratedif(andso)forsomecrisp(i.e.,ordinary)setofattributes(gen-erator).Consideringonlycrisplygeneratedconceptshastwopracticalconsequences.First,thenumberofcrisplygeneratedformalconceptsisconsiderablylessthanthenumberofallformalfuzzyconcepts.Second,sincecrisplygeneratedconceptsmaybeidenti“edwitha(ordinary,notfuzzy)setofattributes(thelargestgenerator),theymightbeconsideredtheimportantonesŽamongallformalfuzzyconcepts.Wepresentba-sicpropertiesofthesetofallcrisplygeneratedconcepts,analgorithmforlistingallcrisplygeneratedconcepts,aversionofthemaintheoremofconceptlatticesforcrisplygeneratedconcepts,andshowthatcrisplygeneratedconceptsarejustthe“xedpointsofpairsofmappingsre-semblingGaloisconnections.Furthermore,weshowconnectionstootherpapersonformalconceptanalysisofdatawithfuzzyattributes.Also,wepresentexamplesdemonstratingthereductionofthenumberofformalconceptsandthespeed-upofouralgorithm(comparedtolistingofallformalconceptsandtestingwhetheraconceptiscrisplygenerated).1ProblemSettingandPreliminaries1.1ProblemSettingFormalconceptanalysis(FCA)[12]dealswithobject-attributedatatables(ob-jectsandattributescorrespondingtotablerowsandcolumns,respectively).Inthebasicsetting,attributesareassumedtobebinary,i.e.tableentriesare1or0accordingtowhetheranattributeappliestoanobjectornot.Iftheattributesunderconsiderationarefuzzy(likecheapŽ,expensiveŽ),eachtableentrycon-tainsatruthdegreetowhichanattributeappliestoanobject.Thedegreescanbetakenfromsomeappropriatescalecontaining0(doesnotapplyatall)and1(fullyapplies)asbounds.Themostpopularchoiceissomesubintervalof[01],butingeneral,degreesneednotbenumbers.Adatatablewithtruthdegreescanbeconsideredamany-valuedcontextandcanbetransformedtoabinarydatatableviaso-calledconceptualscaling[12].Alternatively,thetablewithtruthdegreescanbeapproachedusingtheapparatusofFCAgeneralizedB.GanterandR.Godin(Eds.):ICFCA2005,LNCS3403,pp.268…283,2005.Springer-VerlagBerlinHeidelberg2005 280R.Bavek,V.Sklen´r,andJ.ZacpalTable1.Economicindexes:datatablewithfuzzyattributes 1234567 1Czech 0.40.40.60.20.20.40.22Hungary 0.41.00.40.00.00.40.23Poland 0.21.01.00.00.00.00.04Slovakia 0.20.61.00.00.20.20.25Austria 1.00.00.20.20.21.01.06France 1.00.00.60.40.40.60.67Italy 1.00.20.60.00.20.60.48Germany 1.00.00.60.20.21.00.69UK 1.00.20.40.00.20.60.610Japan 1.00.00.40.20.20.40.211Canada 1.00.20.41.01.01.01.012USA 1.00.20.41.01.00.20.4:1-highgrossdomesticproductpercapita(USD),2-highconsumerpriceindex(1995=100highunemploymentrate(percent-ILO),4-highelectricityproductionpercapita(kWh),5-highenergyconsumptionpercapita(GJ),6-highexportpercapita(USD),7-highimportpercapita(USD) 25 24 20 19 12 23 11 10 22 21 2 1 18 17 9 8 7 16 14 6 4 15 5 13 3 0 Fig.2.CrisplygeneratedformalconceptscorrespondingtodatafromTab.1(columns)runoverrandomlygeneratedformalcontexts(rows)withthenumberofobjectsequaltothenumberofattributes(from5to25objects/attributes)andwith=11(11truthdegrees).Moreover,weshowaverageanddispersion.Wecanseethatthedispersionislowandthatdecreaseswithgrowingsizeofdata.Furtherexperimentsneedtoberuntoshowinmoredetailthebehavior.Inthesecondexperiment,werandomlygeneratedtableswith20objectsand20attributes,=11withminimum-basedfuzzyconjunction,eachobjectwith10attributeswithadegree0(and10attributeswithadegree=0);ofthetenattributewithnonzerodegrees,wevariedthenumberofattributes,from1to10(rows),withdegree=1;columnsrepresentexperiments;weconsideraverageanddispersionof,seeTab.3.Inthethirdexperiment,werandomlygeneratedtableswith20objectsand20attributes,=11withminimum-basedfuzzyconjunction,eachobjectwithvaryingnumberofattributeswitha 282R.Bavek,V.Sklen´r,andJ.ZacpalTable5.Dependenceof(averageAv,dispersionVar)onthenumberoftruthdegrees(rows);columnscorrespondtoexperiments 12345678910AvVar 3 0.360.340.340.340.330.30.340.420.330.360.3460.0286 0.180.210.190.260.230.210.240.170.180.180.2050.0312 0.170.230.160.210.170.180.20.170.190.220.1870.0244 0.130.150.220.140.180.170.170.150.160.160.1630.0220 0.140.180.150.150.150.150.160.130.130.160.1500.0151 0.140.20.140.140.130.160.160.110.110.180.1470.0280 0.170.140.130.10.150.140.130.150.160.210.1470.0276mensions50objects50attributesto70objects70attributeswithunderaconstraintthateachobjecthas10attributeswithanonzerodegreeand4oftheseequal1.ThegraphinFig.3demonstratesthespeed-upindependenceonthesizeofinputdata(50to70),i.e.theratioT/TisthetimeneededforcomputingthewholeX,Y,I)andusingLemma1totestwhethereachconceptiscrisplygenerated,andisthetimeneededbythealgorithmfromSection2.3. Fig.3.Speed-upofalgorithmfromSection2.3,seeRemark1Acknowledgment.TheauthorsacknowledgesupportbygrantNo.1ET101370417oftheGAAVCR.RadimBavekgratefullyacknowledgessupportbygrantNo.201/02/P076ofGACR.1.BavekR.:Fuzzyconceptsandconceptualstructures:inducedsimilarities.InProc.JointConf.Inf.Sci.98,Vol.I,pages179…182,Durham,NC,1998.2.BavekR.:FuzzyGaloisconnections.Math.Log.Quart.(4)(1999),497…504.3.BavekR.:Similarityrelationsinconceptlattices.J.LogicComput.845,2000.4.BavekR.:ReductionandasimpleproofofcharacterizationoffuzzyconceptFundamentaInformaticae(4)(2001),277…285.5.BavekR.:Fuzzyclosureoperators.J.Math.Anal.Appl.(2001),473…489.