DSpace at VNU: Generalized picture distance measure and applications to picture fuzzy clustering

DSpace at VNU: Generalized picture distance measure and applications to picture fuzzy clustering tài liệu, giáo án, bài...

jo u r n al hom e p a g e :w w w e l s e v i e r c o m / l o c a t e / a s o c

Generalized picture distance measure and applications to picture

fuzzy clustering

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a r t i c l e i n f o

Keywords:

a b s t r a c t

Picturefuzzyset(PFS),whichisageneralizationoftraditionalfuzzysetandintuitionisticfuzzyset, showsgreatpromisesofbetteradaptationtomanypracticalproblemsinpatternrecognition,artificial life,robotic,expertandknowledge-basedsystemsthanexistingtypesoffuzzysets.Anemergingresearch trendinPFSisdevelopmentofclusteringalgorithmswhichcanexploitandinvestigatehiddenknowledge fromamassofdatasets.Distancemeasureisoneofthemostimportanttoolsinclusteringthatdetermine thedegreeofrelationshipbetweentwoobjects.Inthispaper,weproposeageneralizedpicturedistance measureandintegrateittoanovelhierarchicalpicturefuzzyclusteringmethodcalledHierarchicalPicture Clustering(HPC).Experimentalresultsshowthattheclusteringqualityoftheproposedalgorithmisbetter thanthoseoftherelevantones

©2016ElsevierB.V.Allrightsreserved

1 Introduction

Sincefuzzyset(FS)[49]wasfirstlyintroducedbyZadehin1965,

manyextensionsofFShavebeenproposedintheliteraturesuchas

thetype-2fuzzyset(T2FS)[18],roughset(RS)[24],softset,rough

softsetandfuzzysoftset[15],intuitionisticfuzzyset(IFS)[3],

intu-itionisticfuzzyroughset(IFRS)[51],softroughfuzzyset&softfuzzy

roughset[19],interval-valuedintuitionisticfuzzyset(IVIFS)[38]

andhesitantfuzzyset(HFS)[32].Theaimofthoseextensionsis

toovercomethelimitationsofFSregardingthedegreeof

fuzzi-ness,theuncertainty ofmembershipdegrees,and theexistence

ofneutrality.Recently,anewgeneralizedfuzzysetcalledpicture

fuzzyset(PFS)hasbeenproposedbyCuongandKreinovichinRef

[6].Theword“picture”inPFSreferstogeneralityasthissetisthe

directextensionofFSandIFS.Intheotherwords,PFSintegrates

informationofneutralandnegativeintoitsdefinitionsothatwhen

thevalue(s)ofone(both)ofthosedegreesis(are)equaltozero,it

returnstoIFS(FS)set.ComparingwithIFS,PFSdividesthehesitancy

degreeintotwoparts,i.e.,refusaldegreeandneutraldegree(see

Definition1andExamples1and2fordetails).Thissetshowsgreat

promisesofbetteradaptationtomanypracticalproblemsin

pat-ternrecognition,artificiallife,robotic,expertandknowledge-based

systemsthansomeexistingtypesoffuzzysets

Definition 1. Apicturefuzzyset(PFS)[6]inanon-emptysetXis,

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A=

x,A(x) ,A(x) ,A(x)|x∈X

, whereA(x) isthepositivedegreeofeachelementx∈X,A(x) is theneutraldegreeandA(x) isthenegativedegreesatisfyingthe constraints,

A(x) ,A(x) ,A(x) ∈ [0,1] ,∀x ∈X,

0≤A(x)+A(x)+A(x)≤1,∀x ∈X

TherefusaldegreeofanelementiscalculatedasA(x)=1− (A(x)+A(x)+A(x)),∀x∈X.InthecaseA(x)=0PFSreturns

totheIFSset,andwhenbothA(x)=A(x)=0,PFSreturnstothe

FSset.SomepropertiesofPFSoperations,theconvexcombination

ofPFS,etc.accompaniedwithproofscanbereferencedinRef.[6]

Example 1. Inademocraticelectionstation,thecouncilissues

500votingpapersforacandidate.Thevotingresultsaredivided intofourgroupsaccompaniedwiththenumberofpapersnamely

“votefor”(300),“abstain”(64),“voteagainst”(115)and“refusalof voting”(21).Group“abstain”meansthatthevotingpaperisawhite paperrejectingboth“agree”and“disagree”forthecandidatebut stilltakesthevote.Group“refusalofvoting”iseitherinvalidvoting papersorbypassingthevote.Thisexamplewashappenedinreality andIFScouldnothandleitsincetheneutralmembership(group

“abstain”)doesnotexist

Example 2. Personnelselectionis averyimportantactivityin thehumanresourcemanagementofanorganization.Theprocess

ofselection followsa methodologytocollectinformation about http://dx.doi.org/10.1016/j.asoc.2016.05.009

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anindividualinordertodetermineifthat individualshouldbe

employed.Theselectionresultscouldbeclassifiedinto4classes:

truepositive,truenegative,falsenegative,andfalsepositivewhich

are somehow equivalent to the positive, neutral, negative and

refusaldegrees ofPFS.Eachcandidate isranked accordingto4

classesbyhisabilityandsuitabilityforthejob,andthefinaldecision

ismadebasedonresultsoftheclasses.Forexample,iftwo

candi-datesarerankedA-(50%,20%,20%,10%)andB-(40%,10%,30%,20%),

thefinaldecisioncanbemadethroughtheunionoperatorand

max-imumofthepositivedegreeinPFSwhichreturnsthevalueof50%

(Aisselected)

AnemergingtrendinPFSandotheradvancedfuzzysetsisthe

developmentofsoftcomputingmethodsespeciallyclustering

algo-rithmsonthesesets,whichcouldproducebetterqualityofresults

thanthat onFS For instance,clustering algorithmson interval

T2FSfocusingonuncertaintyassociatedwiththefuzzifierwere

investigatedinRefs.[14,52].RegardingtheIFSset,Pelekisetal

[23]proposedaclusteringapproachutilizingasimilarity-metric

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definedoverIFS.XuandWu[45]developedtheIFCMalgorithm

toclassifyIFSandinterval-valuedIFS.Sonetal.[26]proposedan

intuitionisticfuzzyclusteringalgorithmforgeo-demographic

anal-ysis.Xuandhisgroupdevelopedanumberofintuitionisticfuzzy

clusteringmethodsinvariouscontexts[36,37,39,42].Fuzzy

clus-teringalgorithmsonothersetsnamelyHFSandPFSwerefound

inRefs.[4,27].Itisclearfromtheliteraturethatdistancemeasure

isthemostimportantfactorforanefficientclusteringalgorithm

ThemostwidelyuseddistancemeasuresfortwoFSsAandBon

X=

X1, ,XN



istheHamming,EuclideanandHausdorffmetrics [6].BecauseoftheFS’sdrawbacks,distancemeasuresonothersets

mostlyIFShavebeenproposed.Atanassov[3],Chen[5],Dengfeng

andChuntian[7],Grzegorzewski[10],Hatzimichailidisetal.[11],

HungandYang[12,13],Lietal.[16],LiangandShi[17],Mitchell

[21],Papakostasetal.[22],SzmidtandKacprzyk[28–30],Wangand

Xin[35],XuandChen[41],XuandXia[46],YangandChiclana[47]

andXu[44]presentedsomedistancemeasuresinIFSnamelythe

(normalized)intuitionisticHammingandEuclideandistances,and

the(normalized)HausdorffintuitionisticHammingandEuclidean

distances.AbasicdistancemeasureonPFShasbeengivenbyCuong

andKreinovich[6]asfollows

=



1

N

N



i=1



.

WerecognizethatdP(A,B) isageneralizationofthoseinIFS

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andFSwhenA(x)=0andbothA(x)=A(x)=0,respectively

Asexplainedabove,theintegrationofneutraldegreeA(x) would

measureinformationofobjectsmoreaccuratelyandincrease

qual-ityandaccuracyofachievedresults.Yetagain,tohelpimproving

theperformanceasmotivatedbythepreviousresearchesonIFS

thattendedtocombinesomebasicdistancemeasuresintoa

com-plexoneto improvethegenerality and accuracy,in this paper

weproposeanovelgeneralizedpicturedistancemeasureanduse

itinanewclusteringmethodonPFScalledHierarchicalPicture

Clustering(HPC).Thereasonfordesigninganewmeasurecanbe

illustratedbyanexampleasfollows.Considerthatwewouldlike

tomeasurethetruth-valueofthepropositionG=“throughapoint

exteriortoalineonecandrawonlyoneparalleltothegivenline”

Thepropositionisincomplete,sinceitdoesnotspecifythetypeof

geometricalspaceitbelongsto.InanEuclideangeometricspace

theproposition Gistrue;in aRiemanniangeometricspacethe

propositionGisfalse(sincethereisnoparallelpassingthrough

anexteriorpointtoagivenline);inageometricspacecovering

thePFSset(constructedfrommixedspaces,forexamplefroma

partofEuclideansubspacetogetherwithanotherpartof Riemann-ianspace)thepropositionGisindeterminate(trueandfalseinthe sametime)[48].Itisobviousthatobjects,notions,ideas,etc.can

bebettermeasuredinPFSthaninothertypesoffuzzysets

Themaindifferencesoftheproposeddistancemeasurewith

dP(A,B) and thoseonIFSsuchasinXu[44] arehighlightedas follows

Firstly,asbeingshownabove,dP(A,B) isanaturalexpansionof thewell-knownMinkowskidistanceoforderp≥1betweentwo pointsunderfuzzyenvironments.Whenp=1orp=2,wehavethe Manhattanand Euclideandistances,respectively.Inthelimiting caseofpreachinginfinity,weobtaintheChebyshevdistance.The Minkowskidistancehasthebestperformancefornumericaldata but works ineffectivelywithasymmetric binaryvariables, non-metricvectorobjects,etc.[20].Forexample,thesimilaritybetween twovectorscanbedenotedasacosinemeasurewhichisfurther usedtodefinea distance[48].Forasymmetricbinaryvariables, thecontingencytable,whichreflectsthematchingstatesbetween twoobjects,isusedtocomputethedistancebetweenasymmetric binaryvariables[25].Itisveryoftenthatanon-linearfunctionis adoptedasthedistancemetricforprocessingnon-sphericaldata [9].Oneofthemostcommonwaystocreatesuchthefunctionis combiningthebasicdistancemeasuresintoacomplexonesothat thedeficienciesofthestandalonemetricsaresettled.This intu-itionleadstodebutoftheproposedmeasurewhichmayenhance performanceandaccuracyofresults

Secondly,theproposedmeasureisacombinationofthe Ham-ming,EuclideanandHausdorffdistances.ItisdifferenttodP(A,B) whichinessenceisthenormalizedformofwell-knownMinkowski distanceoforderp≥1.Inthenextsection,wewillexplainwhy thehybridizationshouldbemadeandemphasizeonthe advan-tagesanddisadvantagesofusingtheproposedmeasure.However,

itisnotedthattheproposeddistancemeasureisageneralization versionofdP(A,B)

Thirdly,theproposeddistancemeasureisdifferenttothoseon IFSsuchasinXu[44]inmanyaspects.Letustakesomeexamples.In Ref.[44],XugeneralizedtheintuitionisticHammingandEuclidean distancesofSzmidtandKacprzyk[28]asbelow

=



1 2N

N

 i=1



.

Hethendefinedseveralsimilaritymeasuresfromtheabove dis-tancefunction,forinstance:



1 2N

N

 i=1



,

⎛

⎜

⎜

⎝

N

 i=1



N

 i=1



⎞

⎟

⎟

⎠ 1/˛

Even though d (A,B) is quite similar to dP(A,B), we recog-nize that d (A,B) is designed on thebasis of IFS which means

A(x)+A(x)+A(x)=1whiledP(A,B) isthedistanceonPFS sat-isfying0≤A(x)+A(x)+A(x)≤1.Indeed,itisnotintuitiveand

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logicalwhentakingthedifferencebetweenA(x) andB(x) since

thesevaluescanbecalculatedthroughotherdegrees.Intheother

word,althoughd(A,B)isexpressedasafunctionofthree

compo-nents,itturnsoutthatd(A,B)isdependentontwovariables.Thisis

differenttodP(A,B) whichismeasuredbythreeseparatedegrees

Thus,werealizethatd(A,B)isdifferenttodP(A,B) andcertainly

muchdifferenttotheproposed(hybrid)measure.Again,inRef

[39]Xuetal.proposedtwointuitionisticfuzzysimilaritymeasures

forspectralclusteringbasedontheminimumoperatorbetween

themembershipandnon-membershipdegreesofIFS.Those

sim-ilaritymeasuresaredefinedbasedonthestandard intuitionistic

Hamming,EuclideanandHausdorffdistances.Anoverviewof

dis-tanceandsimilaritymeasuresofIFSgiven byXuandChen[41]

affirmedthatmostoftherelevantworksinIFSpaymuchattention

tothesimilaritydegreesbasedonthreebasicdistancefunctions

namelytheintuitionisticHamming,EuclideanandHausdorff.The

analysisclearlypointoutthedifferenceandnoveltyoftheproposed

distancemeasurewiththoseonIFS

Oncedefining the generalized picture distancemeasure, we

applyit toa newclusteringmethod calledHierarchicalPicture

Clustering(HPC).Itusesasimplerstrategyandeasierfor

imple-mentationthantheintuitionisticfuzzyclustering[36–38,39,42]

For instance, Xu et al [42] proposed intuitionistic clustering

using associationcoefficients of IFS toconstruct anassociation

matrix,whichisthentransformedintoanequivalentassociation

matrix.Basedonthe-cuttingmatrixoftheequivalentassociation

matrix,clustersofIFSsarethendetermined.Xuetal.[39]defined

two intuitionisticfuzzysimilarity measuresfor constructingan

intuitionisticfuzzysimilaritymeasurematrixusedbyaspectral

algorithmtoclusterintuitionisticfuzzydata.Theun-normalized

graphLaplacianandeigenvectorswereoptedtoclusterthesamples

inspectralclustering.Wangetal.[36]presentedanettingmethod

tomakeclusteringanalysisofIFSsviatheintuitionisticfuzzy

sim-ilaritymatrix.Wangetal.[37]proposedtheintuitionisticfuzzy

squareproductwhichistransformedtotheintuitionisticfuzzy

sim-ilaritymatrixfordirectintuitionisticfuzzyclusteringbasedona

confidencelevel.Thosealgorithmsaremostlycomplexand

time-consumingsincetheyfirstlyconstructedtheintuitionisticfuzzy

similaritymatrixandtheneitherusedanexhaustediterative

strat-egyto gettheequivalentassociation matrix [42]or a complex

calculationthroughgraphLaplacian[39],nettingmethod[36],etc

Meanwhile,HPCreliessolelyonthegeneralizedpicturedistance

measureandhierarchicalclusteringschemefortheclassification

ofPFSs Itis indeedrecognizedthat HPChastheadvantages of

simpleprocessingandintuitivemanners.Butmorethanthat,HPC

providesthewaytodealwithPFSdatawhichwerenot

investi-gatedbytheexistingintuitionisticfuzzyclusteringalgorithms.As

mentionedearly,therearemanyeventsandphenomenathatare

representedbythePFSset.Whenfacingwiththosedata,

cluster-ingalgorithmsonIFSworkineffectivelysincetheydonottakeinto

accounttherefusal/neutralinformation.Combiningtherefusaland

neutraldegreesinIFSwouldmakelostinformation;letussayfor

example:aPFS-A={(x,0.3,0,0.1);(y,0.4,0.1,0.1)}andaIFS-B={(x,

0.3,0.1);(y,0.4,0.1)}.ItisobviousthatIFSregardsneutralvalues

ofxandybeing0.6and0.5,respectively.Yet,infactthemost

dom-inantpartintheneutralvaluesofIFSistherefusaldegree.The

observationinArevealsthatthe“real”neutralandrefusaldegrees

ofxare0and0.6whilethoseofyare0.1and0.4,respectively

Thus,it ismisleadingifweuseclusteringalgorithmsonIFSfor

dealingwithPFSdata.Inshort,weclearlyrecognizetheroleand

advantagesofHPCincomparisonwiththerelevantclustering

algo-rithmsonIFS.Wedonotmentionthe(comparisonof)clustering

qualitiesofthosealgorithmssincetheyaredesignedondifferent

basesets.However,wewouldliketoemphasizeonthesimplicity

andfirstdebutofaclusteringalgorithmonPFSwhichisthemain

contributioninthispaper

Therestofthepaperisorganizedasfollows.Section2presents thegeneralizedpicturedistancemeasureandtheHPCalgorithm Section3validatestheproposedalgorithmbyexperiments Sec-tion4drawstheconclusionsand delineatesthefuture research directions

2 The proposed methodology

Inthissection,wefirstlyintroduce thedefinitionof general-izedpicturedistancemeasureandthenpresentanovelhierarchical picturefuzzyclustering(HPC)

Definition 2. Afunctiond (A,B) withA,B ∈PFS(X)iscalled pic-turedistancemeasureifitsatisfies:

0≤d (A,B)≤1,

d (A,B)=0⇔A=B,

d (A,B)=d (B,A) ,

AB×d (A,B)+AC×d (A,C)≥BC×d (B,C)∀A,B,C ∈PFS(X) wherethesymbol“×”isthearithmeticalproduct.AB,BCandAC arecompositionoperationsofA,B,C ∈PFS(X).Asanexample,the followingmin-maxcompositionformulaeareusedtocalculatethe triple (AB,AC,BC) fromthemembershipfunctionsofA,B,C ∈ PFS(X)

AB=min

i

 max

A(xi) ,B(xi)

BC=min

i

 max

B(xi) ,C(xi)

Ac=min

i

 max

A(xi) ,c(xi)

The aim of those formulae is to specify fuzzy coefficients

AB,AC,BC ∈ [0,1] forthefuzzytriangularinequalityasinthe 4thpropertyofthisdefinition.Besidesthemin-max,some typi-calcompositionssuchasmax-prod,Lukasiewiczt-norm,etc.can

beusedaccordingly.Ageometricalrepresentationofthe4th prop-ertyisgiveninFig.1.Itisclearfromthefigurethatanewfuzzy representationofABisA’B’whichisboundedinafuzzydomain calledArea1satisfyingd(A’B’)=AB×d (A,B).Then,there exists fuzzyrepresentationsofACandBCnamelyA’C’andB’C’thatbelong

toequivalentfuzzydomains—Area2andArea3respectivelysothat

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(AB,AC,BC) sothatthe4thpropertyholdsthend (A,B) isa

pic-turedistancemeasure.Thisimpliesthatthedistancemeasureis

constructedonafuzzyspace.Intheequivalentarticlesonfuzzysets

and topology, Zadeh and coworkers [5,15,18,19,24,32,38,49,51]

suggestedthat the triangularinequality for a metric shouldbe

fuzzifiedbymembershipdegreessothatconditionsand

proper-tiesoffuzzytopologyhold.Thiscanberegardedassoftversionof

themetricdefinitioninahardspace

Definition 3. Thefunctionbelowisageneralizedpicturedistance

measurebetweenA,B∈PFS(X)

dG(A,B)=



1 N N



i =1



pi+ p

i + p i

pi,pi,ip 1/p



1 N N



i =1



pi+ p

i + p i

pi,pi,ip 1/p

+

 max

i



˚A

i,˚B i

 +1 N N



i =1

|˚A

i −˚B

i|p

1/p

+1

where

i=|A(xi)−B(xi)|,(i=1, ,N)

i=|A(xi)−B(xi)|,(i=1, ,N)

i=|A(xi)−B(xi)|,(i=1, ,N)

˚A

i =|A(xi)+A(xi)+A(xi)|,(i=1, ,N)

˚B

i =|B(xi)+B(xi)+B(xi)|, (i=1, ,N)

Remarks.

1)dG(A,B) is a hybrid measure of the well-known Hamming,

EuclideanandHausdorffdistances.Specifically,whenp=1,we

have thehybridbetweenHausdorffand Hammingmeasures

Whenp=2,a hybridofHausdorffandEuclideandistances is

recognized

2)dG(A,B) isnotatrivialhybridizationofsuchtheexisting

meas-uresinthesensethatitdoesnotphysicallymixthosemeasures

togetherwithouttakingcareoftheirmeaningandcontexts.In

fact,dG(A,B) hasbeendesignedonthebasisofthepicturefuzzy

setrepresentedintheformofmembershipvaluesi,i,i,

˚A

i and˚B

i.ItisregardedasageneralizationofdP(A,B),whichis

abasicpicturedistancemeasureofCuongandKreinovich[6],by

employingtheintegrationofothermeasuressuchasHamming,

EuclideanandHausdorffdistances

3)ThereasonsforthehybridizationindG(A,B) canbeexplainedas

follows.NotethatthebasicpicturedistancemeasureofCuong

and Kreinovich relies ontheHamming(p=1)and Euclidean

(p=2)distanceswhichwereshowntohavelimitationsin

deal-ingwithnon-sphericaldatasets[5,7,8,12,13,28–30].Sincethey

assumethatsamplepointsaredistributedaroundsamplemean

inasphericalmanner,theprobabilityofatestpointbelongingto

thesetdependsnotonlyonthedistancefromthesamplemean

butalsoonthedirectionsoastoavoidnon-spherical

distribu-tions[35,41,47].Meanwhile,Hausdorffmetricmeasureshowfar

twosubsetsofametricspacearefromeachother.Itturnstheset

ofnon-emptycompactsubsetsofametricspaceintoametric

spaceinitsownright.Thus,Hausdorffdistancehasthe

advan-tageofbeingsensitivetoposition[40,41,45].Anotherimportant

advantageofHausdorffdistanceisthepossibilityofusing

sep-aratelydissimilaritymeasuresbetweenoneobjectandapart

ofanother[46].Therefore,combiningHausdorffdistancewith

HammingandEuclideanmeasuresinageneralizedpicture

dis-tancemeasureasindG(A,B) wouldachievetheadvantagesof

eachmeasureaswellasincreasetheperformance

4)dG(A,B) is applicableto a large class of problems Asbeing

demonstratedinDefinition3,dG(A,B) iscomputedthroughthe

degreesofPFS(i,i,i,˚A

iand˚B

i)whichareappropriate

forPFSdata.Nonetheless,othertypesofcrispdata,e.g., numer-ical,categoricaldataandimagescanalsobeusedwithinthis measurewiththesupportofafuzzificationprocess.Forinstance,

animageisfirstlyextractedintofeaturerecordswhicharethen fuzzifiedbytheGaussianmembershipfunctiontomakefuzzy data.NotethateachdegreeinPFSwouldhavedifferent mem-bershipfunctionssothatwewillobtainvaluesofdegreesfor eachrecord.Thenextprocessisthendonewithintheachieved PFSdataasinDefinition3.Othertypesofdatacanbehandled analogously.Thisremarkshowsthegeneralityoftheproposed measure

Theorem 1. dG(A,B) isapicturedistancemeasure

Proof. FromDefinition2,itisobviousthatthegeneralized pic-turedistancessatisfythreefirstconditions.Forthelastcondition regardingtriangularinequalityinPFS,wehavetoprovethe exist-enceofatriple (AB,AC,BC) sothattheconditionholds.Since workingonPFSwhose dataelementshaveassociated member-shipvalues,it isclearthateachdistancemeasurebetweentwo setsinPFSshouldbeaccompaniedwithacompositionfunction

ofthosemembershipvalues.Assuch,thelastconditionisoften namedasthe(picture)fuzzytriangularinequalityorsoft triangu-larinequality.Intheextentofthisproof,wewillshowthatthere existsdiscretevaluesforthetriple (AB,AC,BC).Considerp=1 ForA,B∈PFS(X),letusdenote:

AB1=

N



i =1

i+i+i

AB2=max

i,i,i

,

AB3=max

i



˚A

i,˚B i

 ,

AB4=

N



i=1

|˚A

i −˚Bi|

Thefollowinginequalityisneededtoprove:

AB1+AB2

AB1+AB2+AB3+AB4+1+ AC1+AC2

AC1+AC2+AC3+AC4+1

≥ BC1+BC2

3 (BC1+BC2+BC3+BC4+1). ThefactsbelowcomefromthedefinitionofPFS

|A(xi)−B(xi)|+|A(xi)−C(xi)|≥|B(xi)−C(xi)|,

|A(xi)−B(xi)|+|A(xi)−C(xi)|≥|B(xi)−C(xi)|,

|A(xi)−B(xi)|+|A(xi)−C(xi)|≥|B(xi)−C(xi)|

Itfollowsthat,

AB1+AC1≥BC1,

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Table 1

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AB2+AC2≥BC2

Assume:

max{|B(x)−C(x)|,|B(x)−C(x)|,|B(x)−C(x)|}

=|B(x)−C(x)|

Then,

|B(x)−C(x)|≤|A(x)−B(x)|+|A(x)−C(x)|

≤max{|A(x)−B(x)|,|A(x)−B(x)|,|A(x)−B(x)|}+

max{|A(x)−C(x)|,|A(x)−C(x)|,|A(x)−C(x)|}

BC1+BC2

BC1+BC2+BC3+BC4+1

AB1+AB2+AC1+AC2+BC3+BC4+1

AB1+AB2+AC1+AC2+BC3+BC4+1(∗)

Ifoneofthefactsbelowhappen,

max

i



˚Ai,˚Bi,˚Ci

=˚Bj,

max

i



˚Ai,˚Bi,˚Ci

=˚Cj, ThenBC3≥AB3

Againifmax

i



˚A

i,˚B

i,˚C i



=˚A

j,

i



˚A

i,˚B i



−max

i



˚B

i,˚C i



≤ max

i



˚A

i −˚B

i,˚B

i −˚C i



≤max

i



˚A

i −˚B

i +˚B

i −˚C i



= max

i



˚A

i −˚C

i



≤3AC2,

AB3−BC3≤3AC2

Analogously,weachieve

BC4≥AB4,

Or3AC2+BC4≥AB4

Thus,

Combine(*,**,***),theinequalityisproven.Thus,dG(A,B) isa

picturedistancemeasure

Definition 4. The average picture set of Ai ∈PFS (X) (index

i=1, ,N)isdenotedasAVG (Ai),

AVG (Ai)=



x,N1

N



i=1

i(x) ,1 N

N



i=1

i(x) ,1 N

N



i=1

i(x)|x∈X



,

where i(x), i(x), i(x) are thepositive, neutral and negative membershipdegreesofAi,respectively

Definition 5. PictureDistanceMatrix(PDM)ofAi∈PFS (X) (index

i=1, ,N)isasimilaritymatrixsizedN×Nwhereeachelementis computedbyDefinition3

TheHPCAlgorithm:

Step1:GivenacollectionofAi∈PFS (X) (indexi=1, ,N) Con-sidereachAiisauniquecluster

Step2:CalculatePictureDistanceMatrix(PDM)

Step3:MergetwoconsecutivePFSsetsbasedonPDMand cal-culatenewcentersbyDefinition4.Noticethatonlytwoclusters arejointedineachstage

Step4:RepeatStep2withAibeingreplacedwiththenew cen-tersuntilthedesirablenumberofclustersisachieved

3 Evaluation

In this section, we aimtovalidate whetherthenew metric canaccurately measure dataelementsof thepicturefuzzyset:

Ai∈PFS (X).Eventhoughthereexistmanyextensionsofthe classi-calFuzzyC-Means(FCM)intheliteraturethatusedtheEuclideanor HammingorMahalanobisdistancesforobtainingclustersof spher-icalorellipticgeometricalform,theywerenotdesignedtoworkin thePFSsetwhichcontainstheinformation ofpositive,negative andneutralasinDefinition1.Therefore,inordertoclassifyPFS elements:Ai ∈PFS (X),weshouldusethebasicandgeneralized picture distancemeasures-dP(A,B) and dG(A,B) ina hierarchi-cal clusteringalgorithm like HPCrespectively Thissection will compare thosemeasures interms of performance ofclustering algorithms.Therefore,wehaveimplementedtheHPCalgorithm withdG(A,B) inadditiontoavariantofHPCusingdP(A,B) called

CK.TheIntuitionisticHierarchicalClustering(IHC)algorithm[43] hasbeenimplementedtoevaluateclusteringqualityofHPCand CK

The experimental data consists of 4 datasets The first one, Guangzhoucar[50]describedinTable1,isasmalldatasetconsists

of5newcarsintheGuangzhoumarketevaluatedby6criteria:Fuel (G1),Aerod(G2),Price(G3),Comfort(G4),Design(G5)andSafety (G6).Dataofeachcarforagivencriterionconsistofthree com-ponentsrepresentingforthepositive,theneutralandthenegative degrees.SumoftheneutralandthenegativedegreesinTable1

isthenon-membershipvalueinRef.[50].Thesecondone, Build-ingmaterials[40]showninTable2,isanothersmalldatasethas5 buildingmaterialsnamelySealant,Floorvarnish,Wallpaint, Car-petandChlorideflooringcharacterizedby8attributes.Sumofthe neutralandthenegativedegreesinTable2isthenon-membership valueinRef.[40].Theaimofthisdatasetistovalidatethe algo-rithmsonadatasethavinglargernumberofattributesthanthat

of theGuangzhoucar dataset.Thethirdone,Heart Disease[34] whoseapartisexpressedinTable3,isareallargedatasetfrom UCIMachineLearningRepositoryconsistsof270patients acquir-ingheartdiseasecategorizedby3attributessuchasAge(3#Age), Bloodpressure(mmHg/patient,10#Trestbps)andheartrate(#32 Thalach).Thepositive,theneutraland thenegativedegreesare fuzzifiedfromcrispdatausingGaussian,triangularandtrapezoid

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427

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Table 2

Table 3

Q8

membershipfunctions.Theaimofthisdatasetistovalidatethe

algorithmsonadatasethavinglargernumberofobjectsthanthat

oftheGuangzhoucardataset.Lastly,ForestCoverType[33]isalarge

datasetextractedfromUCIMachineLearningRepositoryincludes

1000instancesin10dimensionsshowingtheactualforestcover

typeforagivenobservation(30×30mcell)determinedfromUS

ForestService(USFS)Region2ResourceInformationSystem(RIS)

Thepositive,theneutralandthenegativedegreesarefuzzifiedfrom

crispdatausingGaussian,triangularandtrapezoidmembership

functions.Theaimofthisdatasetistovalidatethealgorithmsona

largedatasethavingboththenumberofobjectsandthenumberof

attributesgreaterthanthoseoftheGuangzhoucardataset

Inordertoevaluateclusteringqualitiesofthealgorithms,we

useNMI(NormalizedMutualInformation),F-MeasureandPurity

Theseevaluationindicesarethe-larger-the-better

NMI=

k



j =1

r



i =1

nijlogn×nij

ni×n j







 r



i =1

nilogni

n

 ⎛

⎝ k

j =1

njlognj

n

⎞

⎠ ,

Precisioni= 1

ni

k

max

j =1



nij ,(i=1, ,r),

Recalli= 1

nj∗

k

max

j =1



nij ,(i=1, ,r),

j∗=argmaxk

j =1



nij ,(j∗∈ [1,k] ),

Fi=2×Precisioni×Recalli Precisioni+Recalli ,

F-Measure=1

r

r



i =1

Fi,

Purity=1n

r



i=1

k

max

j=1



nij ,

where

• T=

T1, ,Tk

andC=

C1, ,Cr

arekcorrectandrpredicted clusters,respectively

• nistotalnumberofdatapoints

• nij=|Ci∩Tj|:commonnumberofdatapointsbetweenCiandTj (i=1, ,r;j=1, ,k)

• ni=

k



j=1

nij:numberofdatapointsofCi(i=1, ,r)

• nj=

r



i=1

nij:numberofdatapointsofTj(j=1, ,k)

Firstly,weillustratetheactivitiesoftheHPCalgorithmtoclassify theGuangzhoucardatasetinTable1.Inthefirstphase,eachcarin thedatasetisauniquecluster

 Car1 , Car2 , Car3 , Car4 , Car5

Table 4

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499

PDM1=

⎛

⎜

⎜

⎝

0 0.2908 0.3275 0.2560 0.3406

0.2908 0 0.3036 0.2912 0.3286

0.3275 0.3036 0 0.3581 0.3062

0.2560 0.2912 0.3581 0 0.3767

0.3406 0.3286 0.3062 0.3767 0

⎞

⎟

⎟

⎠.

Becaused (Car1,Car4)=0.2560 is theminimalvalue among

alldistances,Car1 andCar4are groupedintoacluster

Remov-ingalldistancevaluesrelatedtoCar1andCar4,werecognizethat

d (Car2,Car3)=0.3036istheminimalvalueamongall.Thus,Car2

andCar3aremergedintoanothercluster.Resultsofthesecond

phaseare:



Car1,Car4

, Car2,Car3

, Car5

Using Definition 4, the centers of 

Car1,Car4

and



Car2,Car3

are:

Fuel

Aerod

Price

Comfort

Design

Safety

0.3)

(0.5, 0.05, 0.05)

(0.65, 0.05, 0.1)

(0.8,0.05, 0.05)

(0.15,0.2, 0.35)

(0.6,0.165, 0.085)

0.25,

0.15)

(0.3,0.3, 0.2)

(0.35,0.1, 0.25)

(0.15,0.1, 0.25)

(0.3,0.35, 0.25)

(0.5,0.2, 0.15)

Next,wecalculatethePDMofPhase2

PDM2=

⎛

⎝00.28870.28870 0.34850.2958

0.3485 0.2958 0

⎞

⎠

Sinced

Car1,Car4

, Car2,Car3

=0.2887isthesmallest valueamongalldistancesinPDM2,wecombinethosecarsintoa

cluster.Resultsofthethirdphaseare:



Car1,Car4,Car2,Car3

, Car5 Thecenterofcluster

Car1,Car4,Car2,Car3

is:

Fuel

Aerod

Price

Comfort

Design

Safety

0.2,

0.225)

(0.4, 0.175, 0.125)

(0.5, 0075, 0.175)

(0.475, 0.075, 0.15)

(0.225, 0.275, 0.3)

(0.55, 0.1825, 0.1175)

ThePDMofPhase3is:

PDM3=0.3110

Lastly,inthefourthphase,allcarsaregroupedintoaunique

cluster.AhierarchicaltreefortheclassificationofGuangzhoucar

datasetusingHPCalgorithmisshowninFig.2.Ifwecomputethe

averagevaluesofthepositive,theneutralandthenegative

mem-bershipsofallcarsandgroupbyphasesthenwegettheresultsin

Table4

Inordertovisualizetheclusteringresults,weusePrincipal

Com-ponentAnalysis(PCA),whichisawell-knownmethodinstatistics,

toreducedimensionsofdatainTable4andgettheresultsinTable5

2Ddistributionsofdatapointsandcentersofallphasesarealso

depictedinFigs.3–6

Secondly,wecomparetheclusteringqualitiesofHPCand CK

through evaluation indices on the experimental datasets The

resultsontheGuangzhoucardatasetareshowninTables6–9

Table 5

Table 6

Table 7

Table 8

Table 9

TheresultshaveshownthatclusteringqualityofHPCisbetter thanthatofCK.Moreover,asillustratedinFigs.2and7andTable6,

weclearlyrecognizethatthehierarchicaltreeofIHCisidenticalto HPC.Thismeansthatusingthegeneralizedpicturedistance mea-sureinclusteringalgorithmsresultsinbetterqualitythanusingthe basicpicturedistance

Analogously, wemadethecomparisononotherdatasetsand achievedtheresultsinTables10–19.Thevaluesin thesetables affirmtheefficiencyofHPCeveninthecasesthatthenumberof attributesorthenumberofobjectsishigherthanthatofGuangzhou car.Thisclearlyshowsthefactthatusingthegeneralizedpicture distancemeasurewouldresultinaccuratecalculationsofsimilarity betweenobjects

In Figs 8 and 9, we illustrate the hierarchical tree of HPC for buildingmaterialsand aclusteringtool calledHPCS—akind

ofknowledge-basedsystemstoassistclusteringonPFSdatasets, respectively

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Fig 2. The hierarchical tree of HPC for Guangzhou car.

Fig 5. The distributions of data and centers in Phase 3.

Table 10

Carpet,

Carpet,

Carpet,

Carpet,

Sealant

Sealant

Table 11

Table 12

Table 13

Table 14

Table 15