Annals of Fuzzy Mathematics and Informatics
ISSN: 2093–9310 (print version)
ISSN: 2287–6235 (electronic version) http://www.afmi.or.kr
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c KyungMoonSaCo. http://www.kyungmoon.com
Supervisedpatternrecognitionusingsimilarity measurebetweentwointervalvaluedneutrosophic softsets
AnjanMukherjee,SadhanSarkar
Receivedddmm201y; Acceptedddmm201y
Abstract. F.Smarandacheintroducedtheconceptofneutrosophicset in1995andP.K.Majiintroducedthenotionofneutrosophicsoftsetin 2013,whichisahybridizationofneutrosophicsetandsoftset.IrfanDeli introducedtheconceptofintervalvaluedneutrosophicsoftsets.Interval valuedneutrosophicsoftsetsaremostefficienttoolstodealswithproblemsthatcontainuncertaintysuchasprobleminsocial,economicsystem, medicaldiagnosis,patternrecognition,gametheory,codingtheoryandso on.Inthisarticleweintroducesimilaritymeasurebetweentwointerval valuedneutrosophicsoftsetsandstudysomebasicpropertiesofsimilaritymeasure.Analgorithmisdevelopedinintervalvaluedneutrosophic softsetsettingusingsimilaritymeasure.Usingthisalgorithmamodel isconstructedforsupervisedpatternrecognitionproblemusingsimilarity measure.
2010AMSClassification: 03E72
Keywords: Fuzzyset,neutrosophicsoftset,intervalvaluedneutrosophicsoftset, patternrecognition.
CorrespondingAuthor: AnjanMukherjee(anjan2002 m@yahoo.co.in )
1. Introduction
In1999,Molodtsov[8]introducedtheconceptofsoftsettheorywhichiscompletelynewapproachformodelinguncertainty.Inthispaper[8]Molodtsovestablishedthefundamentalresultsofthisnewtheoryandsuccessfullyappliedthesoft settheoryintoseveraldirections.Majietal.[6]definedandstudiedseveralbasic notionsofsoftsettheoryin2003.PieandMiao[11],AktasandCagman[1]andAli etal.[2]improvedtheworkofMajietal.[6].TheintuitionisticfuzzysetisintroducedbyAtanaasov[3]asageneralizationoffuzzyset[17]whereheaddeddegreeof non-membershipwithdegreeofmembership.Smarandache[12, 13, 14]introduced
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theconceptofneutrosophicsetwhichisamathematicaltoolforhandlingproblems involvingimprecise,indeterminacyandinconsistantdata.Thewords“neutrosophy” and“neutrosophic”wereintroducedbyF.Smarandacheinhis1998book.Etymologically,“neutro-sophy”(noun)[Frenchneutre<f Latinneuter,neutralandGreek sophia,skill/wisdom]meansknowledgeofneutralthought.While“neutrosophic” (adjective),meanshavingthenatureof,orhavingthecharacteristicofNeutrosophy. Maji[7]combinedneutrosophicsetandsoftsetandestablishedsomeoperationson thesesets.Wangetal.[15]introducedintervalneutrosophicsets.Deli[5]introduced theconceptofinterval-valuedneutrosophicsoftsets.
Similaritymeasureisaveryeffectivemethodtomeasuredegreeofsimilaritybetweentwofuzzysetsandbetweentwosoftsets.Wecanmeasuredegreeofsimilarity betweentwoobjectsorpatternswhenevertheycanbemodeledasfuzzysetsorsoft setsorneutrosophicsetsetc.SaidBroumiandFlorentinSmarandache[4]introduced theconceptofseveralsimilaritymeasuresofneutrosophicsetsandJunYe[16] introducedtheconceptofsimilaritymeasuresbetweenintervalneutrosophicsets. RecentlyA.MukherjeeandS.Sarkar[9, 10]introducedseveralmethodsofsimilarity measureforneutrosophicsoftsetsandintervalvaluedneutrosophicsoftsets.
Patternrecognitionproblemisaveryimportantandhighlyattractivetopicof researchfornearlylastfivedecadesandgrowingdaybydayduetoitsemergingapplicationsinvariousfields.Basicallypatternrecognitionproblemconsistsoftwokinds ofproblems:supervisedpatternrecognitionandunsupervisedpatternrecognition. Insupervisedpatternrecognitioninputpatternorunknownpatternisidentifiedas amemberofapredefinedclass.Inunsupervisedpatternrecognitionproblemthe patternisassignedtoahithertounknownclass.Thuspatternrecognitionproblem isaproblemofclassificationorcategorizationofpatterns,wheretheclassesaredefinedbythesystemdesignerincaseofsupervisedpatternrecognitionproblemand incaseofunsupervisedpatternrecognitionpatternproblemspatternsarelearned basedonthesimilarityofpatterns.
Inthisarticleweintroducesimilaritymeasurebetweentwointervalvaluedneutrosophicsoftsetsandstudysomebasicpropertiesofsimilaritymeasure.Analgorithm isdevelopedinintervalvaluedneutrosophicsoftsetsettingusingsimilaritymeasure. Usingthisalgorithmamodelisconstructedforsupervisedpatternrecognitionproblemusingsimilaritymeasure.
2. Preliminaries
Inthissectionwebrieflyreviewsomebasicdefinitionswhichwillbeusedinthe restofthepaper.
Definition2.1 ([17]). LetXbeanonemptycollectionofobjectsdenotedby x Then afuzzyset α inXisasetoforderedpairshavingtheform α = {(x,µα(x)): x ∈ X}, wherethefunction µα : X→ [0, 1]iscalledthemembershipfunctionorgradeof membership(alsocalleddegreeofcompatibilityordegreeoftruth)of x in α.The interval M =[0, 1]iscalledmembershipspace. 2
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Definition2.2 ([18]). Let D[0, 1]bethesetofclosedsub-intervalsoftheinterval [0, 1].Aninterval-valuedfuzzysetinX, X =ΦandCard(X)=n,isanexpression Agivenby A = {(x,MA(x)):x ∈ X},where MA :X → D[0, 1].
Definition2.3 ([3]). LetXbeanonemptyset.Thenanintuitionisticfuzzy set(IFSforshort)Aisasethavingtheform A = {(x,µA(x),γA(x)): x ∈ X} wherethefunctions µA : X → [0, 1]and γA : X → [0, 1]representsthedegreeof membershipandthedegreeofnon-membershiprespectivelyofeachelement x ∈ X and0 ≤ µA(x) ≤ γA(x) ≤ 1foreach x ∈ X.
Definition2.4 ([6, 8]). LetUbeaninitialuniverseandEbeasetofparameters. Let P (U )denotesthepowersetofUand A ⊆ U .Thenthepair(F,A)iscalleda softsetoverU,whereFisamappinggivenby F : A → P (U ).
Definition2.5 ([12, 13, 14]). AneutrosophicsetAontheuniverseofdiscourseX isdefinedas A = {(x,TA(x),IA(x),FA(x)),x ∈ X} where T,I,F : X →] 0, 1+[and 0 ≤ TA(x) ≤ IA(x) ≤ FA(x) ≤ 3+
Fromphilosophicalpointofview,theneutrosophicsettakesthevaluefromreal standardornon-standardsubsetsof] 0, 1+[.Butinreallifeapplicationinscientific andengineeringproblemsitisdifficulttouseneutrosophicsetwithvaluefromreal standardornon-standardsubsetof] 0, 1+[.Henceweconsidertheneutrosophic setwhichtakesthevaluefromthesubsetof[0,1]thatis
0 ≤ TA(x) ≤ IA(x) ≤ FA(x) ≤ 3.
Definition2.6 ([7]). LetUbetheuniversalsetandEbethesetofparameters. Alsolet A ⊆ E and P (U )bethesetofallneutrosophicsetsofU.Thenthecollection (F,A)iscalledneutrosophicsoftsetoverU,whereFisamappinggivenby F : A → P (U ).
Definition2.7 ([7]). Let E = {e1,e2,e3,.......,em} bethesetofparameters,then thesetdenotedby E anddefinedby E = { e1, e2, e3,......., em},where ei = notei,iscalledNOTsetofthesetofparametersE.Where and aredifferent operators.
Definition2.8 ([15]). LetUbeaspaceofpoints(objects),withagenericelement inU.Anintervalvalueneutrosophicset(IVN-set)AinUischaracterizedbytruth membershipfunction TA,aindeterminacy-membershipfunction IA andafalsitymembershipfunction FA.Foreachpoint u ∈ U ; TA,IA and FA ⊆ [0, 1].Thusa IVN-setAoverUisrepresentedas
A = {(TA(u),IA(u),FA(u)): u ∈ U }
Where0 ≤ sup(TA(u))+ sup(IA(u))+ sup(FA(u)) ≤ 3and(TA(u),IA(u),FA(u))is calledintervalvalueneutrosophicnumberforall u ∈ U
Definition2.9 ([5]). LetUbeaninitialuniverseset,Ebeasetofparametersand A ⊆ E.LetIVNS(U)denotesthesetofallintervalvaluedneutrosophicsubsetsof U.Thecollection(F,A)iscalledtheintervalvaluedneutrosophicsoftsetoverU, whereFisamappinggivenby F : A → IVNS(U ).
Theintervalvaluedneutrosophicsoftsetdefinedoverauniverseisdenotedby IVNSS.
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Definition2.10 ([5]). LetUbetheuniversalset,Ebethesetofparameters and(N1,E),(N2,E)betwointervalvaluedneutrosophicsoftsetsoverU(E).Then (N1,E)iscalledanintervalvaluedneutrosophicsoftsubsetof(N2,E)if infTN1 (xi)(ej ) ≤ infTN2 (xi)(ej ) supTN1 (xi)(ej ) ≤ supTN2 (xi)(ej ) infIN1 (xi)(ej ) ≥ infIN2 (xi)(ej ) supIN1 (xi)(ej ) ≥ supIN2 (xi)(ej ) infFN1 (xi)(ej ) ≥ infFN2 (xi)(ej ) supFN1 (xi)(ej ) ≥ supFN2 (xi)(ej )
3. Similaritymeasureforintervalvaluedneutrosophicsoft sets(IVNSS)
Inthissectionweintroduceanewmethodformeasuringsimilaritymeasureand weightedsimilaritymeasureforIVNSSandsomebasicpropertiesarealsostudied. Definition3.1. Let U = {x1,x2,x3,.......,xn} betheuniverseofdiscourseand E = {e1,e2,e3,.....,em} bethesetofparametersand(N1,E),(N2,E)betwointervalvaluedneutrosophicsoftsetsoverU(E).Thenthesimilaritymeasurebetween twoIVNSSs(N1,E)and(N2,E)isdenotedby S(N1,N2)andisdefinedasfollows: S(N1,N2)= 1 3mn n i=1 m j=1 3 −|T N1 (xi)(ej ) T N2 (xi)(ej )|−|I N1 (xi)(ej )
I N2 (xi)(ej )|−|F N1 (xi)(ej ) F N2 (xi)(ej )| ................(1)
where T N1 (xi)(ej )= infTN1 (xi)(ej )supTN1 (xi)(ej )
I N1 (xi)(ej )= infIN1 (xi)(ej )supIN1 (xi)(ej )
F N1 (xi)(ej )= infFN1 (xi)(ej )supFN1 (xi)(ej )
T N2 (xi)(ej )= infTN2 (xi)(ej )supTN2 (xi)(ej )
I N2 (xi)(ej )= infIN2 (xi)(ej )supIN2 (xi)(ej )
F N2 (xi)(ej )= infFN2 (xi)(ej )supFN2 (xi)(ej )
Theorem3.2. If S(N1,N2) bethesimilaritymeasurebetweentwoIVNSSs (N1,E) and (N2,E) then (i) 0 ≤ S(N1,N2) ≤ 1 (ii) S(N1,N2)= S(N2,N1) (iii) S(N1,N1)=1 (iv)if (N1,E) ⊆ (N2,E) ⊆ (N3,E) then S(N1,N3) ≤ S(N2,N3)
Proof. (i)Obviousfromdefinition 3.1 (ii)Obviousfromdefinition 3.1 (iii)Obviousfromdefinition 3.1. (iv)Let U = {x1,x2,x3,......,xn} betheuniverseofdiscourseand E = {e1,e2,e3,....., em} bethesetofparametersand(N1,E),(N2,E),(N3,E)bethreeintervalvalued neutrosophicsoftsetsoverU(E),suchthat(N1,E) ⊆ (N2,E) ⊆ (N3,E).Nowby definitionofintervalvaluedneutrosophicsoftsubsets[5]wehave, 4
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infTN1 (xi)(ej ) ≤ infTN2 (xi)(ej ) ≤ infTN3 (xi)(ej ) supTN1 (xi)(ej ) ≤ supTN2 (xi)(ej ) ≤ supTN3 (xi)(ej ) infIN1 (xi)(ej ) ≥ infIN2 (xi)(ej ) ≥ infIN3 (xi)(ej ) supIN1 (xi)(ej ) ≥ supIN2 (xi)(ej ) ≥ supIN3 (xi)(ej ) infFN1 (xi)(ej ) ≥ infFN2 (xi)(ej ) ≥ infFN3 (xi)(ej ) supFN1 (xi)(ej ) ≥ supFN2 (xi)(ej ) ≥ supFN3 (xi)(ej )
⇒|T N1 (xi)(ej ) T N3 (xi)(ej )|≥|T N2 (xi)(ej ) T N3 (xi)(ej )|
|I N1 (xi)(ej ) I N3 (xi)(ej )|≥|I N2 (xi)(ej ) I N3 (xi)(ej )|
|F N1 (xi)(ej ) F N3 (xi)(ej )|≥|F N2 (xi)(ej ) F N3 (xi)(ej )|
⇒|T N1 (xi)(ej ) T N3 (xi)(ej )| + |I N1 (xi)(ej ) I N3 (xi)(ej )|+
|F N1 (xi)(ej ) F N3 (xi)(ej )|≥|T N2 (xi)(ej ) T N3 (xi)(ej )|+
|I N2 (xi)(ej ) I N3 (xi)(ej )| + |F N2 (xi)(ej ) F N3 (xi)(ej )|
⇒ (3 −|T N1 (xi)(ej ) T N3 (xi)(ej )|−|I N1 (xi)(ej ) I N3 (xi)(ej )|−
|F N1 (xi)(ej ) F N3 (xi)(ej )|) ≤ (3 −|T N2 (xi)(ej ) T N3 (xi)(ej )|−
|I N2 (xi)(ej ) I N3 (xi)(ej )|−|F N2 (xi)(ej ) F N3 (xi)(ej )|)
⇒ 1 3mn n i=1 m j=1 3 −|T N1 (xi)(ej ) T N3 (xi)(ej )|−|I N1 (xi)(ej )
I N3 (xi)(ej )|−|F N1 (xi)(ej ) F N3 (xi)(ej )| ≤ 1 3mn n i=1 m j=1 3−|T N2 (xi)(ej ) T N3 (xi)(ej )|−|I N2 (xi)(ej )
I N3 (xi)(ej )|−|F N2 (xi)(ej ) F N3 (xi)(ej )| ⇒ S(N1,N3) ≤ S(N2,N3)[Byequation(1)]
Definition3.3. Let U = {x1,x2,x3,.......,xn} betheuniverseofdiscourseand E = {e1,e2,e3,.....,em} bethesetofparametersand(N1,E),(N2,E)betwointervalvaluedneutrosophicsoftsetsoverU(E).Nowifweconsiderweights wi of xi(i =1, 2, 3,.....,n)thentheweightedsimilaritymeasurebetweenIVNSSs(N1,E) and(N2,E)denotedby WS(N1,N2)isdefinedasfollows: WS(N1,N2)= 1 3m n i=1 m j=1 wi 3−|T N1 (xi)(ej ) T N2 (xi)(ej )|−|I N1 (xi)(ej ) I N2 (xi)(ej )|−|F N1 (xi)(ej ) F N2 (xi)(ej )| ................(2)
Where wi ∈ [0, 1],i=1,2,3,.......,nand n i=1 wi =1.Inparticularifwetake wi = 1 n i=1,2,3,.......,nthen WS(N1,N2)= S(N1,N2)
Theorem3.4. If WS(N1,N2) betheweightedsimilaritymeasurebetweentwo IVNSSs (N1,E) and (N2,E) then (i) 0 ≤ WS(N1,N2) ≤ 1 (ii) WS(N1,N2)= WS(N2,N1) (iii) WS(N1,N1)=1 (iv)if (N1,E) ⊆ (N2,E) ⊆ (N3,E) then WS(N1,N3) ≤ WS(N2,N3) 5
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Proof. (i)Obviousfromdefinition 3.3. (ii)Obviousfromdefinition 3.3. (iii)Obviousfromdefinition 3.3. (iv)Similartoproofof(iv)oftheorem 3.2
Definition3.5. Let(N1,E)and(N2,E)betwoIVNSSsovertheuniverseU.Then (N1,E)and(N2,E)aresaidbe α-similar,denotedby(N1,E) α ∼ = (N2,E)ifand onlyif S(N1,N2) >α for α ∈ (0, 1).WecallthetwoIVNSSssignificantlysimilarif S(N1,N2) > 1 2 .
Definition3.6. Let(N1,E)and(N2,E)betwoIVNSSsovertheuniverseU.Then (N1,E)and(N2,E)aresaidbesubstantially-similarif S(N1,N2) > 0 95andis denotedby(N1,E) ≡ (N2,E).
4. ApplicationofsimilaritymeasureofIVNSSsinpatternrecognition problem
Analgorithmforsupervisedpatternrecognitionproblemsusingsimilaritymeasurebetweentwointervalvaluedneutrosophicsoftsetsisdevelopedinintervalvalued neutrosophicsoftsetsetting.
Stepsoftheproposedalgorithmareasfollows:
Step1:constructIVNSS(s)ˆ µi(i=1,2,3........,n)asidealpattern(s).
Step2:constructIVNSS(s)ˆ νj (j=1,2,3........,m)forsamplepattern(s)whichis/are toberecognized.
Step3:calculatesimilaritymeasurebetweenIVNSS(s)foridealpattern(s)andsamplepattern(s).
Step4:recognizesamplepattern(s)undercertainpredefinedconditions.
Example4.1. Todemonstratetheproposedalgorithmforsupervisedpatternrecognitionweconsideraproblemofmedicaldiagnosis.Afictitiousnumericalexampleis giventodeterminewhetherapatientwithsomevisiblesymptomsissufferingfrom cancerornot.Forthisweassumethatifthesimilaritymeasurebetweentheideal patternforthediseaseandsamplepatternforthepatientliesintheinterval [0.7, 0.9](whichcanbedecidedwithhelpofmedicalexpertperson) thenpatient ispossiblysufferingfromthedisease,ifthesimilaritymeasureisgreaterthan 0.90 thenthepatientissurelysufferingfromthediseaseandifsimilaritymeasureisless than 0.7 thenthepatientmaynotsufferingfromthedisease.
Let U = {u1,u2,u3} betheuniverseofdiscourse,where u1 =intense, u2 =durationand u3 =metamorphosisand E = {e1,e2,e2} bethesetofparameters(certain visiblesymptoms),where e1 =weakness, e2 =fatigueand e3 =headache.
Step1:constructanIVNSS(idealpattern)forthediseasecancer,whichcanbe constructedwiththehelpofamedicalexpertperson: ˆ (µ,E) e1 e2 u1 (0.4,0.6),(0.2,0.3),(0.3,0.4) (0.7,0.8),(0.2,0.4),(0.3,0.4) u2 (0.6,0.7),(0.1,0.2),(0.2,0.3) (0.8,0.9),(0.1,0.2),(0.4,0.5) u3 (0.5,0.6),(0.05,0.1),(0.3,0.4) (0.4,0.6),(0.2,0.3),(0.2,0.3) 6
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e3
u1 (0.2,0.3),(0.1,0.3),(0.4,0.5) u2 (0.6,0.7),(0.1,0.2),(0.4,0.5) u3 (0.9,1.0),(0.1,0.2),(0.1,0.2)
Step2:constructanIVNSS(samplepattern)forthepatient P: ˆ (ν1,E) e1 e2 u1 (0.4,0.55),(0.2,0.3),(0.25,0.35) (0.7,0.9),(0.2,0.3),(0.25,0.35) u2 (0.5,0.7),(0.1,0.15),(0.2,0.35) (0.75,0.9),(0.1,0.15),(0.4,0.5) u3 (0.4,0.6),(0.1,0.15),(0.2,0.4) (0.5,0.6),(0.15,0.25),(0.2,0.4) e3
u1 (0.1,0.3),(0.1,0.2),(0.4,0.5) u2 (0.5,0.7),(0.05,0.2),(0.35,0.45) u3 (0.8,0.9),(0.1,0.2),(0.2,0.3)
ConstructanIVNSS(samplepattern)forthepatient Q: ˆ (ν2,E) e1 e2 u1 (0.1,0.2),(0.6,0.7),(0.7,0.8) (0.2,0.3),(0.05,0.1),(0.8,0.9) u2 (0.2,0.3),(0.4,0.5),(0.8,0.9) (0.1,0.2),(0.4,0.5),(0.9,1.0) u3 (0.2,0.3),(0.4,0.5),(0.7,0.8) (0.2,0.3),(0.6,0.7),(0.7,0.8) e3 u1 (0.7,0.8),(0.6,0.7),(0.1,0.2) u2 (0.1,0.2),(0.4,0.5),(0.7,0.8) u3 (0.2,0.3),(0.5,0.6),(0.8,0.9)
Step3:calculatesimilaritymeasurebetweenˆ µ andˆ ν1 andbetweenˆ µ andˆ ν2 :
Nowbydefinition 3.1 similaritymeasurebetweenˆ µ andˆ ν1 isgivenby S(ˆµ, ˆ ν1)=0.96 andsimilaritymeasurebetweenˆ µ andˆ ν2 isgivenby S(ˆµ, ν2)=0.56.
Step4: sincesimilaritymeasure S(ˆµ, ˆ ν1)=0.96 > 0.9,thereforepatient P issurely sufferingfromthediseasecancer.Butsimilaritymeasure S(ˆµ, ˆ ν2)=0.56 < 0.7, thereforepatient Q possiblynotsufferingfromthediseasecancer.
5. Conclusions
Inthispaperweproposedsimilaritymeasureandweightedsimilaritymeasure forintervalvaluedneutrosophicsoftsets.Wealsostudysomedefinitionsandbasicpropertiesofsimilaritymeasure.Analgorithmisdevelopedinintervalvalued neutrosophicsoftsetsettingforsupervisedpatternrecognitionproblemusingsimilaritymeasure.Afictitiousnumericalexampleisgiventodemonstratethepossible applicationofproposedmodelinamedicaldiagnosisproblem.
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AnjanMukherjee (anjan2002 m@yahoo.co.in)–DepartmentofMathematics, TripuraUniversitySuryamaninagar,Agartala-799022,Tripura,INDIA SadhanSarkar (Sadhan7 s@rediffmail.com)–DepartmentofMathematics,Tripura UniversitySuryamaninagar,Agartala-799022,Tripura,INDIA
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