On Distance and Similarity Measures of Intuitionistic Fuzzy Multi Set

Transcription

1 IOSR Joural of Mathematics (IOSR-JM) e-issn: Volume 5, Issue 4 (Ja. - Feb. 03), PP O Distace ad Similarity Measures of Ituitioistic Fuzzy Multi Set *P. Raaraeswari, **N. Uma * Departmet of Mathematics, Chikkaa rts College, Tirupur, Tamil Nadu. (INDI). ** Departmet of Mathematics, SNR Sos College, Coimbatore, Tamil Nadu. (INDI). bstract: I this paper, three distace measures ad their correspodig similarity measures of Ituitioistic Fuzzy Multi sets (IFMS) are itroduced ad compared. The measures are based o Hausdroff distace measure, Geometric distace measure ad the Normalized distace measure. Key Words - Multi set, Ituitioistic fuzzy set, Ituitioistic Fuzzy Multi sets, Distace Measure, Similarity measure I. Itroductio. The traditioal Fuzzy sets (FS) itroduced by Lofti. Zadeh [] i 965 was the geeralisatio of Crisp sets preseted by George Cator. The fuzzy set allows the obect to partially belog to a set with a membership degree () betwee 0 ad. Later, Krasssimir T. taassov [, 3] proposed the Ituitioistic Fuzzy sets (IFS) as the geeralisatio of the Fuzzy set. The IFS represet the ucertaity with respect to both membership ( [0,]) ad o membership (θ [0,]) such that + θ. The umber π = θ is called the hesitiatio degree or ituitioistic idex. Usig the distace ad the similarity measures, the IFSs defied o the same uiverse are compared. d the study of distace ad similarity measure of IFSs gives lots of measures, each represetig specific properties ad behaviour i real-life decisio makig ad patter recogitio works. Fuzzy Multi set (FMS) cocept was itroduced by R. R. Yager [4]. Multi set [5] allows the repeated occurreces of ay elemet ad hece the Fuzzy Multi set ca occur more tha oce with the possibly of the same or the differet membership values. The ew cocept Ituitioistic Fuzzy Multi sets (IFMS) was proposed by T.K Shio ad Suil Jacob Joh [6]. This paper is a extesio of the distace ad similarity measure of IFMS. The umerical results of the examples show that the developed distace ad similarity measures are well suited to use ay liguistic variables. The orgaizatio of this paper is as follows: I sectio, the Fuzzy Multi sets, Ituitioistic Fuzzy Multi sets ad the distace ad similarity measures of IFS are preseted. The methodologies of three differet distace ad similarity measures are proposed for the IFMS i sectio 3.The sectio 4, aalyses the umerical evaluatio of the proposed methods. II. Prelimiaries Defiitio:. Let X be a oempty set. fuzzy set i X is give by = x, x / x X -- (.) where : X [0, ] is the membership fuctio of the fuzzy set (i.e.) x 0, is the membership of x X i. The geeralizatios of fuzzy sets are the Ituitioistic fuzzy (IFS) set proposed by taassov [, ] is with idepedet memberships ad o memberships. Defiitio:. Ituitioistic fuzzy set (IFS), i X is give by = x, x, θ x / x X -- (.) where : X [0,] ad θ : X [0,] with the coditio 0 x + θ x, x X Here x ad θ x [0,] deote the membership ad the o membership fuctios of the fuzzy set ; For each Ituitioistic fuzzy set i X, π x = x x = 0 for all x X that is π x = x θ x is the hesitacy degree of x X i. lways 0 π x, x X. The complemetary set c of is defied as c = x, θ x, x / x X -- (.3) Defiitio:.3 Let X be a oempty set. Fuzzy Multi set (FMS) i X is characterized by the cout membership fuctio Mc such that Mc : X Q where Q is the set of all crisp multi sets i [0,]. Hece, for ay x X, Mc(x) is the crisp multi set from [0, ]. The membership sequece is defied as ( x, x, p x ) where x x p x. 9 Page

2 O Distace ad Similarity Measures of Ituitioistic Fuzzy Multi Set Therefore, FMS is give by = x, ( x, x, p x ) / x X -- (.4) Defiitio:.4 Let X be a oempty set. Ituitioistic Fuzzy Multi set (IFMS) i X is characterized by two fuctios amely cout membership fuctio Mc ad cout o membership fuctio NMc such that Mc : X Q ad NMc : X Q where Q is the set of all crisp multi sets i [0,]. Hece, for ay x X, Mc(x) is the crisp multi set from [0, ] whose membership sequece is defied as ( x, x, p x ) where x x p x ad the correspodig o membership sequece NMc(x) is defied as ( θ x, θ x, θ p x ) where the o membership i i ca be either decreasig or icreasig fuctio. such that 0 x + θ x, x X ad i =,, p. Therefore, IFMS is give by = x, x, x, p x, ( θ x, θ x, θ p x ) / x X -- (.5) where x x p x The complemetary set c of is defied as c = x, ( θ x, θ x, θ p x ), x, x, p x, / x X (.6) where θ x θ x θ p x Defiitio:.5 The Cardiality of the membership fuctio Mc(x) ad the o membership fuctio NMc(x) is the legth of a elemet x i a IFMS deoted as η, defied as η = Mc(x) = NMc(x) If, B, C are the IFMS defied o X, the their cardiality η = Max η(), η(b), η(c) }..6: Distace Measures of Ituitioistic Fuzzy Sets I the IFS, the commoly defied distace measures [7, 8] for sets, B i X = x, x, x } are I Hammig metrics, it is h d, B = i= x i + θ x i -- (.6.) ad with all three degrees take uder cosideratio, it is h d, B = i= x i + θ x i + π x i -- (.6.) I Hammig metrics, the Hausdroff distace is d h, B = max i= x i } -- (.6.3) ad with all three degrees take uder cosideratio, it is d h, B = max i= x i, π x i } -- (.6.4) The Geometric distace is D g, B = ( x i ) + (θ x i ) -- (.6.5) ad with all degrees take uder cosideratio, is D g, B = ( x i x B x i ) + (θ x i ) + (π x i ) -- (.6.6) Hece the ormalized Geometric distace is D G, B = D g, B -- (.6.7) The Normalized Hammig distace is h D, B = i= x i + θ x i -- (.6.8) ad with all degrees take uder cosideratio is h D x, y = i= x i + θ x i + π x i -- (.6.9). 7: Similarity Measures of Ituitioistic Fuzzy Sets Defiitio:.7 Let S : X x X [0, ] be a map. The S(, B ) is said to be the similarity measure betwee ad B, where, B X ad X is ad ituitioistic fuzzy set, if S(, B) satisfies the followig properties. S(,B) [0,]. S(,B) = if ad oly if = B 3. S(,B) = S(B, ) 4. If B C X, the S(,C) S(,B) S(,C) S(B, C) 5. S(,B) = 0 if ad oly if = φ ad B = (or) = B ad B = φ The various similarity measures betwee Ituitioistic fuzzy sets have bee defied durig the past years. The most otable similarity measures which have bee used i patter recogitio are the followig. The Similarity measure proposed by Yahog et al.[9] was S O, B 0 Page

3 O Distace ad Similarity Measures of Ituitioistic Fuzzy Multi Set S O, B = - ( ( i=i x i ) + (θ x i ) ) -- (.7.) Later Hug ad Yag [0] preseted their similarity measures based o Hausdorff distace as 3 S HY, B, S HY, B, S HY, B S HY, B = d H (, B) -- (.7.) S HY, B = ( e d H (,B) e )/( e ) -- (.7.3) 3 S HY, B = d H, B /( + d H (, B) ) -- (.7.4) where d H, B was the Hausdorff distace from equatio (.6.3 ad.6.4) The similarity measure based o geometric distace was proposed by S. Sebastia, J. Philip [] was S G, B = D G, B -- (.7.5) Usig the cocept of ormalized Hammig distace, the similarity measure preseted by E. Szmidt, J. Kacprzyk, [] was Sim, B = h D,B h -- (.7.6) D,B c III. Proposed Distace d Similarity Measures For Ituitioistic Multi Fuzzy Sets I IFS, the distace ad similarity measures are cosidered for the membership ad o membership fuctios oly oce. But i IFMS, it should be cosidered more tha oce; because of their multi membership ad o membership fuctios. d, their cosideratios are combied together by meas of Summatio cocept based o their cardiality. 3.: HUSDROFF MESURE I Hammig metrics, the Hausdroff distace is defied as d h, B = η η = i= max x i } -- (3..) ad with all three degrees, it is d h, B = η η = max i= x i, π x i } -- (3..) Hece the Similarity measure based o the Hausdroff distace becomes S H, B = d h (, B) -- (3..3) S H, B = ( e d h (,B) e )/( e ) -- (3..4) S H 3, B = d h, B /( + d h (, B) ) -- (3..5) 3.: GEOMETRIC MESURE The Geometric distace of the Ituitioistic Multi Fuzzy set is defied as D g, B = η η = ad whe all degrees are take uder cosideratio, it D g, B = η η = ( x i ) i= + (θ x i ) } -- (3..) ( x i ) + (θ x i ) i= + π } -- (3..) Where the Normalized Geometric distace is D G, B = D g, B -- (3..3) Therefore the Similarity measure based o geometric distace is S G, B = D G, B -- (3..4) 3.3: NORMLIZIED HMMING MESURE I the IFMS, the Normalized Hammig distace is N D, B = η η = i= x i + θ x i } -- (3.3.) ad with all three degrees take uder cosideratio it becomes N D, B = η η = i= x i + θ x i + π x i } } -- (3.3.) Usig the cocept of Normalized Hammig distace, the Similarity measure i IFMS is x i Page

4 O Distace ad Similarity Measures of Ituitioistic Fuzzy Multi Set Sim, B = N D, B N D, B c Where B c is the complemet set of B such that B c = x, ( θ x, θ x, θ p x ), x, x, p x / x X, where θ x θ x θ p x. -- (3.3.3) IV. Numerical Evaluatio EXMPLE: 4. Let X =,, 3, 4, 5... } with =,, 3, 4, 5 } ad B = 6, 7, 8, 9, 0 } such that the IFMS ad B are defied i terms of membership ad o-members = 0.6,0.4, 0.5, 0.5, 0.5,0.3, 0.4, 0.5, 3, 0.5, 0., 0.4, 0.4, 4 0.3,0., 0.3, 0., 5 0.,0., 0., 0. } B = 6 0.8,0., 0.4, 0.6, 7 0.7,0.3, 0.4, 0., 8, 0.4, 0.5, 0.3, ,0.7, 0., 0.8, 0 0.,0.6, 0, 0.6 } Here, the cardiality η = 5 as Mc() = NMc( ) = 5 ad Mc(B) = NMc(B) = 5 The IFMS Hausdroff distace measure d h, B = 5 5 = max i= x i } = The similarity measure S H, B = 0.67, S H, B = 0.555, S H, B = The IFMS Geometric distace measure is D g, B = 5 5 = ( x i ) i= + (θ x i ) } = D G, B = Therefore the Similarity measure S G, B = I the IFMS, the Normalized Hammig distace is N D, B = 5 5 = i= x i + θ x i } Hece usig Normalized Hammig distace, the Similarity measure Sim, B = 0.45 = EXMPLE: 4. Let X =,, 3, 4... } with =,, }ad B = 3, 4 } are the IFMS ad B defied as = 0.4,0.,0., 0.3, 0., 0., 0., 0., 0., 0., 0.4, 0.3, 0.6,0.3,0, 0.4, 0.5, 0., 0.4, 0.3, 0., 0., 0.6, 0. } B = ,0.,0.3, 0.4, 0., 0.3, 0.4, 0., 0., 0., 0., ,0.6,0., 0.4, 0.5, 0, 0.3, 0.4, 0., 0., 0.4, 0. } The cardiality η = as Mc() = NMc( ) = ad Mc(B) = NMc(B) = The IFMS Hausdroff distace measure = 4 i= max x i, π x i } = d h, B = 4 3 The similarity measure S H, B = 0.85, S H, B = 0.795, S H, B = The IFMS Geometric distace measure is D g, B = 4 = 4 i= ( x i ) + (θ x i ) + π } = D G, B = = Therefore the Similarity measure based o Geometric distace S G, B = I the IFMS, the Normalized Hammig distace is N D, B = 4 = 4 i= x i + θ x i + π x i } } Hece usig Normalized Hammig distace, the Similarity measure is Sim, B = = EXMPLE: 4.3 Let X =,, 3, 4... } with =,, 3 } ad B = 6 } such that the IFMS ad B are = 0.6,0.,0., 0.4, 0.3, 0.3, 0., 0.7, 0., 0.5, 0.4, 0., 0., 0.6, 0., 0.7,0.,0., 0.3, 0.6, 0., 0., 0.7, 0., 0.6, 0.3, 0., 0.3, 0.4, ,0.4,0., 0.4, 0.4, 0., 0, 0.8, 0., 0.7, 0., 0., 0.4, 0.4, 0. } B = 6 0.8,0.,0., 0., 0.7, 0., 0.3, 0.5, 0., 0.5, 0.3, 0., 0.5, 0.4, 0. } Here L(, B ) = η = 3 as Mc() = NMc( ) = 3 ad Mc(B) = NMc(B) = Hece, their cardiality η = Max η(), η(b) } = max 3,} = 3. The IFMS Hausdroff distace measure d h, B = 3 3 = max 5 i= x i, π x i } = The similarity measure S H, B = , S H, B = , S H, B = x i Page

5 O Distace ad Similarity Measures of Ituitioistic Fuzzy Multi Set The IFMS Geometric distace measure becomes D g, B = = 5 i= ( x i ) + (θ x i ) + π } = D G, B = Therefore the Similarity measure based o geometric distace S G, B = I the IFMS, the Normalized Hammig distace is N D, B = = 5 i= x i + θ x i + π x i } } Hece usig Normalized Hammig distace, the Similarity measure is Sim, B = = From the examples 4. ad 4. it is clear that the ew measures perform well i the case of two represetatives of IFMS (membership, o- membership fuctio) ad three represetatives of IFMS (membership, omembership ad hesitatio fuctio). The example 4.3 depicts the effective measure to check the distace ad similarity betwee the IFMS ad IFS. V. Coclusio Three methods of distace ad similarity measure of IFMS are preseted ad aalyzed. The proposed methods are mathematically valid ad ca be applied to ay decisio makig problems or patter recogitios. From the umerical evaluatio is clear that the proposed similarity measure also satisfy the coditio ad properties of similarity measure (Defiitio:.7). The uique feature of this proposed method is that it cosiders multi membership ad o membership for the same elemet. The preset study costitutes a first study of distace measures based o IFMS ad future research will establish the proposed methodology as a cocrete patter classificatio framework. Refereces [] Zadeh L.., Fuzzy sets, Iformatio ad Cotrol 8 (965) [] taassov K., Ituitioistic fuzzy sets, Fuzzy Sets ad System 0 (986) [3] taassov K., More o Ituitioistic fuzzy sets, Fuzzy Sets ad Systems 33 (989) [4] Yager R. R., O the theory of bags,(multi sets), It. Jou. Of Geeral System, 3 (986) [5] Blizard W. D., Multi set Theory, Notre Dame Joural of Formal Logic, Vol. 30, No , (989). [6] Shio T.K., Suil Jacob Joh, Ituitioistic Fuzzy Multi sets ad its pplicatio i Medical Diagosis, World cademy of Sciece, Egieerig ad Techology, Vol. 6 (0). [7] Szmidt E., Kacprzyk J., O measurig distaces betwee Ituitioistic fuzzy sets, Notes o IFS, Vol. 3 (997) - 3. [8] Szmidt E., Kacprzyk J., Distaces betwee Ituitioistic fuzzy sets. Fuzzy Sets System, 4 (000) [9] Yahog L., Olso D., Qi Z., Similarity Measures betwee Ituitioistic Fuzzy sets : Comparative alysis, Patter Recogitio Letters, 8() (007) [0] Hug W.L., Yag M. S., Similarity Measures of Ituitioistic Fuzzy Sets Based o Hausdorff Distace, Patter Recogitio Letters, 5 (004) [] Souriar Sebastia ad James Philip, Similarity Measure for Ituitioistic Fuzzy sets : Ituitive pproach, Joural of Comp. & Math. Sci. Vol. () (0) [] Szmidt E., Kacprzyk J., similarity measure for Ituitioistic fuzzy sets ad its applicatio i supportig medical diagostic reasoig. ICISC 004, Vol. LNI 3070 (004) [3] Mitchell H., O the Defgfeg Chutia Similarity Measure ad its pplicatio to Patter Recogitio, Patter Recogitio Letters, 4 (003) x i 3 Page