Modfed ose mlarty Measure betwee Itutostc Fuzzy ets hao-mg wag ad M-he Yag,* Deartmet of led Mathematcs, hese ulture Uversty, Tae, Tawa Deartmet of led Mathematcs, hug Yua hrsta Uversty, hug-l, Tawa msyag@math.cycu.edu.tw bstract. mlarty of tutostc fuzzy sets (s) s a mortat measure to dcate the smlarty degree betwee s. Recetly, Ye (0) roosed a smlarty measure betwee s based o the cose cocet. lthough ths cose smlarty measure has good cocet ad mert, the measure s ot satsfed the defto of a smlarty betwee s ad ot reseted well for aalyzg data. I ths aer, we modfy the cose smlarty measure betwee s. Ths modfed smlarty measure betwee s s ot oly to satsfy the defto of a smlarty betwee s, but also to mrove the effcecy of the Ye s measure. examle s used to demostrate ths heomeo. Keywords: Fuzzy set, Itutostc fuzzy set, mlarty measure, ose smlarty measure. Itroducto Fuzzy sets, frst troduced by Zadeh [], gve a aroach for treatg fuzzess. I fuzzy sets, the degree of obeloggess s just the comlemet to of the membersh degree. owever, humas who exresses the degree of membersh of a gve elemet a fuzzy set very ofte does ot exress a corresodg degree of omembersh as the comlemet to. Thus, taassov [] troduced the cocet of a tutostc fuzzy set () whch s a geeralzato of a fuzzy set. ce a ca reset the degrees of membersh ad omembersh wth a degree of hestacy, the kowledge ad sematc reresetato become more meagful ad alcable [3-4]. These s have bee wdely studed ad aled varous areas such as decso makg roblems [5], medcal dagoss [6], atter recogto [7]. mlarty measures are a mortat tool for determg the degree of smlarty betwee two objects. Dfferet smlarty measures betwee fuzzy sets have bee roosed ad smlarty measures betwee s are also wdely studed the lterature. Degfeg ad huta [8] roosed some smlarty measures betwee s used atter recogto roblem. Lag ad h [9] roosed smlarty measures betwee s whch they used umercal comarsos to show that Lag ad h s smlarty measures are more reasoable tha those of Degfeg ad huta. Mtchell [0] terreted s as esembles of ordered fuzzy sets from the * orresodg author. J. Le et al. (Eds.): II 0, LNI 7530,. 85 93, 0. rger-verlag erl edelberg 0
86.-M. wag ad M.-. Yag statstcal ot. ug ad Yag [] roosed several smlarty measures betwee s based o ausdorff dstace whch are well used wth lgustc varables. u ad he [] gave a comrehesve overvew of dstace ad smlarty measures betwee s. Recetly, Ye [3] roosed a cose smlarty measure betwee s. lthough ths cose smlarty measure has good cocet ad mert, the measure caot satsfy the defto of a smlarty betwee s. I ths aer, we modfy ths cose smlarty measure such that the modfed measure ca satsfy the defto of a smlarty betwee s. ome examles are used to demostrate the effcecy of the modfed cose smlarty measure. Itutostc Fuzzy et ad mlarty Measures Let { x, x,..., x} be the uverse of dscrete dscourses. osder two tutostc fuzzy sets (s) ad. We frst descrbe the asects of s dscussed by taassov [] as follows. Defto. (taassov []) tutostc fuzzy set () s defed as {( x, μ( x), ν ( x)) x )} where μ : [0,] ad ν : [0,] wth the codto 0 μ + ν ( x ), x. For each, the umbers μ ( x ) ad ( x) membersh ad o-membersh of x to, resectvely, ad the umber π ( x) μ ( x) ν ( x) deotes a hestacy degree of x to. I ths aer, we use s() to deote the class of all s of. ν deote the degree of Defto. If ad are two s of, the () f ad oly f x, u ( x) u ( x) ad v ( x) v ( x). () f ad oly f x, u ( x ) u ( x ) ad v ( x) v ( x). Measurg a smlarty betwee s s mortat s researches. ome methods had bee roosed to calculate the smlarty degree betwee s where L et al. [4] troduced the followg defto. Defto 3. (L et al. [4]) mag : s() s() [0,]. (, s sad to be the degree of smlarty betwee ad s() f (, satsfes the followg roertes: () 0 (, ; () (, ff ; (3) (, (, ) ; (4) (, ) (, ad (, ) (, ) f s(). (5) (, ) 0 ff s a crs set.
Modfed ose mlarty Measure betwee Itutostc Fuzzy ets 87 osder two s ad s(), Degfeg ad huta [8] roosed a smlarty measure betwee them as follows: (, ) m ( ) m ( ) D where m () ( ( ) ( ))/, u x + v x m ( ) ( ( ) ( ))/, u x + v x <. Whe, D (, ) (, ) where (, ) was he [5] as follow: (, ) ( u ( ) ( )) ( ( ) ( )) x v x u x v x og ad Kg [6] roosed ew smlarty measures (, ), (, ) L ad (, ) O (also see L et al. [4]) as follows: u x u x v x v x (, ) ( ) ( ) ( ) ( ) + L 4 4 + (, ) ( u ( x ) v ( x )) ( u ( x ) v ( x )) u ( x ) u ( x ) v ( x ) v ( x ) u x u x v x v x (, O ) ( ( ) ( )) ( ( ) ( )) + Lag ad h [9] roosed a smlarty measure betwee ad as follows: (, ) ( ( ) + ϕ ( )) where ϕ () u ( x ) u ( x ) / e ϕt f / t, ϕ () ( v ( x )) ( v ( x )) / f ad <. To get more formato o s, Lag ad h [9] gave aother smlarty measure as follows: (, ) s ( s( ) ( )) ϕ + ϕ where ϕ () ( ) ( ) / s m x m x s ad ϕ () ( ) ( ) / s m x m x wth m () ( u ( x ) m ( ))/ +, m ( ) ( ( ) ( ))/ m v x +, m () ( ( ) ())/ u x + m ad m ( ) ( m ( ) v ( x ))/ +. Mtchell [0] terreted s as esembles of ordered fuzzy sets from statstcal a vewot ad roosed a smlarty measure betwee ad as follows:
88.-M. wag ad M.-. Yag (, ) ( ρu (, ) + ρv (, )) where ρ (, ) ( ) ( ) v v x v x ρ (, ) u u ( x ) u ( x ),. ug ad Yag [] roosed several smlarty measures of s based o ausdorff dstace. For two s ad s(), they frst defed I ( x ) [ u ( x ), v ( x )] ad I ( x ) [ u ( x ), v ( x )],,...,. The ausdorff dstace ( I ( x ), I ( x )) betwee I ( x ) ad I ( x ) was the defed as follows: ( I ( x ), I ( x )) max{ u ( x ) u ( x ), v ( x ) ( v ( x )) They defed the dstace d (, betwee ad wth d (, ) ( I ( x ), I ( x )). I ug ad Yag [], they roosed three smlarty measures of ad as follows: d (, ) Y (, ) d (, ), e e Y (, 3 d (, ) ), (, ) Y. e + d (, ) 3 Modfed ose mlarty Measure betwee s y cosderg the formato carred by the membersh ad omembersh degrees tutostc fuzzy sets (s) as a vector reresetato, Ye [3] roosed a cose smlarty measure betwee s ad based o the cose cocet as follows: (, u ( x ) u ( x ) + v ( x ) v ( x ) ( u ( x )) + ( v ( x )) ( u ( x )) + ( v ( x )) I Ye [3], he also roved that (, satsfes the codtos: (P) 0 (, ; (P) (, (, ) ; (P3) (, f. owever, the above three codtos are oly for the correlato coeffcet. If we cosder the geeral defto of a smlarty betwee s as show Defto 3 of ecto, Ye s [3] cose smlarty measure (, betwee s does ot satsfy. For examle, f we gve { x,0.,0. } ad { x,0.4,0.4 }, the (,. That s, the codto () Defto 3 s ot satsfed. Furthermore, f { x,0.0,0. } ad { x,0.,0.0 } are gve, the (, 0. Ths s also ureasoable. We fd that Ye s [3] cose smlarty measure (, s oly to cosder oe sde of the formato betwee membersh ad omembersh degrees, but ot cosder the mddle-sde ad ooste-sde
Modfed ose mlarty Measure betwee Itutostc Fuzzy ets 89 formato betwee them. If we also cosder the mddle formato wth ( u ( x ) v ( x + )) ad ( u( x) + v ( x )) ad the ooste-sde formato wth ( u ( x )),( u ( x )),( v ( x )) ad ( v ( x )), we could get a good smlarty measure betwee s ad through the cose cocet. We ext utlze them to roose a ew smlarty measure betwee s. We frst defe the two tems (, ) ad (, ) as follows: + u ( ) ( ) ( ) ( ) ( ) ( ) ( ) x v x + u x v x + v x v x (, ) + u ( x ) v ( x ) + u ( x ) v ( x ) (, ) ( ) + ( v ( x)) ( ) + ( v ( x)) (( + u ( x ) v ( x )) ( + u ( x ) v ( x )) + 4 v ( x ) v ( x ) ( + u ( x) v ( x)) + ( v ( x)) ( + u ( x) v ( x)) + ( v ( x)) ( u ( x )) ( u ( x )) + ( v ( x )) ( v ( x )) ( u ( x)) + ( v ( x)) ( u ( x)) + ( v ( x)) We roose a ew smlarty measure betwee s ad as follows: (, ( (, + (, + (, ) () 3 s defed by equato (), we ca show that (, s a smlarty measure. Proosto. The measure (, s a smlarty measure betwee s ad. Proof: It s trval to clam (, )) (, )) that s the codto (3) of Defto 3. To clam the codtos () ad (3) of Defto 3, we frst cosder the followg ducto. For ay a 0, b 0, x 0, y 0, we have that ( ay bx) 0 ff ( ax by) ( a b )( x y ) + + + ff 0 ( ax + by) a + b x + y ff ax + by 0 a + b x + y. ce ( ), ( ), ( ) ad v ( x ) are all betwee 0 ad, t s easy to show that 0 (, ) (, ) ad 0 (, ) ax + by 0 a + b x + y for a 0, b 0, x 0, y 0. We the have 0 (, ))
90.-M. wag ad M.-. Yag mlarly, we cosder the followg ducto: (, ff ( (, ) + (, ) + (, )) 3 ff (, ), (, ) ad (, ) ff u ( x ) v ( x ) v ( x ) u ( x ), (( u ( x ) v ( x )) v ( x ) v ( x ) ( + u ( x ) v ( x )) +, ( u ( x )) ( v ( x )) ( v ( x )) ( u ( x )) ff u ( x ) u ( x ) ad v ( x ) v ( x ) ff. Thus, we clam (, ff that s the codto () Defto 3. We ext rove that (, satsfes the codto (4) Defto 3 as follows. If, the for each x, we have that u ( x ) u ( x ) u ( x ) v ( x ) v ( x ) v ( x ). We cosder a fucto f wth f x ax by x y ( ) ( + ) +. The, we have that d ( ) ( ) y ay f x bx 3/ dx ( x + y ) b v ( x ), ad. I ths case, f a u ( x), x u ( x ) ad y v ( x ), the d f ( x) < dx 0. That s, (x ) f s a decreasg fucto of x. Let us cosder aother fucto g wth g( y) ( ax+ by) x + y. The, we have d ( ) ( ) x bx g y ay. I ths case, f 3/ a u ( x ), b v ( x ), dy ( x + y ) x u ( x ) ad y v ( x ), the d g( y) > 0. That s, g (y ) s a creasg dy fucto of y. ece, we have u ( x) u ( x) v ( x) v ( x ) (, + ) (( u ( x )) + ( v ( x )) ) (( u ( x )) + ( v ( x )) ) u ( x ) u ( x ) + v ( x ) v ( x ) + + (( u ( )) ( ( )) ) (( ( )) ( ( )) ) x v x u x v x (, )
Modfed ose mlarty Measure betwee Itutostc Fuzzy ets 9 mlarly, we ca clam that (, ) (, ). We kow that ad ax + by have a smlar form as wth so that, a + b x + y for, we ca clam that (, ) (, ), (, ) (, ), (, ) (, ), (, ) (, ). ece, we have (, ) (, ad (, ) (, ) f. Ths s the codto (4) Defto 3. We ext rove that (, satsfes the codto (5) Defto 3 as follows. c (,( ) ) 0 ff (,( ) c c ) 0, (,( ) ) 0, (,( ) c ) 0 ff u ( x ) v ( x ) 0, (( + u ( x ) v( x )) ( + v( x ) u ( x )) + 4 v( x ) u ( x ) 0, ( u ( x )) ( v ( x )) 0 ff u ( x ), v ( x ) 0 or u ( x ) 0, v ( x ) ff s a crs set. Thus, the roof s comleted. If we follow the roof of Proosto, we ca fd that the cose smlarty measure betwee s roosed by Ye [3] ca satsfy (4), but t caot satsfy () ad (5). We ext use a examle to demostrate ths heomeo ad also make comarsos of our modfed smlarty measure wth some exstg measures. Examle. We cosder the data used Ye [3] where the data were frst used L et al. [4]. The sx data sets are as follows: { (x, 0.3, 0.3), (x, 0.4, 0.4)}, { (x, 0.3, 0.4), (x, 0.4, 0.3)}, 3 { (x,, 0), (x, 0, 0)}, 4 { (x, 0.5, 0.5), (x, 0, 0)}, 5 { (x, 0.4, 0.), (x, 0.5, 0.3)}, { (x, 0.4, 0.), (x, 0.5, 0.)}. 6 The degrees of some exstg smlarty measures betwee the two s ad are show Table. We fd that, for the st data set wth, Ye s [3] smlarty measure (, s gve wth (, ). Note that smlar cases from (, are also occurred for the 3 rd ad 4 th 3 data sets. Obvously, 4 (, ) does ot satsfy the codto () of Defto 3 for a smlarty measure. owever, most other smlarty measures gve the result wth (, ). O the other had, by comarg the 5 th data set ad the 6 th data set 5, the 6 smlarty ( 6) for 6 should be larger tha the smlarty ( 5) for 5. We fd that the smlarty measures,,, ad our O e reset the correct
9.-M. wag ad M.-. Yag case wth ( 6) > ( 5), but others do ot. Our modfed cose smlarty measure actually corrects the drawbacks of the Ye s measure. Data sets Table. mlarty measures betwee s ad 3 4 5 6 0.9 0.5 0.95 0.9 0.9 0.5 0.5 0.9 0.95 0.95 0.9 0.5 0.75 0.95 0.95 L 0.9 0.9 0.3 0.5 0.9 0.93 O 0.9 0.5 0.95 D 0.9 0.9 0.5 0.5 0.9 0.95 e 0.9 0.9 0.5 0.5 0.9 0.95 s 0.95 0.9 0.5 0.75 0.95 0.95 Y 0.9 0.9 0 0.5 0.9 0.9 Y 0.85 0.85 0 0.38 0.85 0.85 3 Y 0.8 0.8 0 0.33 0.8 0.8 0.96 0 0 0.997 0.9965 0.997 0.859 0.90 0.90 0.995 0.997 4 oclusos lthough Ye s [3] cose smlarty measure has good cocet ad mert, the measure s ot satsfed the defto of a smlarty betwee s. I ths aer we aalyzed the drawback of Ye s [3] smlarty measure ad the modfy t. We showed that the modfed measure satsfes the defto of a smlarty betwee s. The examle reseted better results of our modfed smlarty measure tha Ye s [3] measure ad some other exstg measures. Refereces. Zadeh, L..: Fuzzy ets. Iformato ad otrol 8, 338 356 (965). taassov, K.: Itutostc Fuzzy ets. Fuzzy ets ad ystems 0, 87 96 (986) 3. taassov, K.: Itutostc Fuzzy ets: Theory ad lcatos. Physca-Verlag, edelberg (999)
Modfed ose mlarty Measure betwee Itutostc Fuzzy ets 93 4. taassov, K., Georgev, G.: Itutostc Fuzzy Prolog. Fuzzy ets ad ystems 53, 8 (993) 5. Pakowska,., Wygralak, M.: Geeral IF-ets wth Tragular Norms ad Ther lcatos to Grou Decso Makg. Iformato ceces 76, 73 754 (006) 6. De,.K., swas, R., Roy,.R.: lcato of Itutostc Fuzzy ets Medcal Dagoss. Fuzzy ets ad ystems 7, 09 3 (00) 7. ug, W.L., Yag, M..: O the J-Dvergece of Itutostc Fuzzy ets wth Its lcato to Patter Recogto. Iformato ceces 78, 64 650 (008) 8. Degfeg, L., huta,.: New mlarty Measures of Itutostc Fuzzy ets ad lcato to Patter Recogto. Patter Recogto Letters 3, 5 (00) 9. Lag, Z., h, P.: mlarty Measures o Itutostc Fuzzy ets. Patter Recogto Letters 4, 687 693 (003) 0. Mtchell,..: O the Degfeg-huta mlarty Measure ad Its lcato to Patter Recogto. Patter Recogto Letters 4, 30 304 (003). ug, W.L., Yag, M..: mlarty Measures of Itutostc Fuzzy ets ased o ausdorff Dstace. Patter Recogto Letters 5, 603 6 (004). u, Z.., he, J.: Overvew of Dstace ad mlarty Measures of Itutostc Fuzzy ets. Iteratoal Joural of Ucertaty, Fuzzess ad Kowlege-ased ystems 6, 59 555 (008) 3. Ye, J.: ose mlarty Measures for Itutostc Fuzzy ets ad Ther lcatos. Mathematcal ad omuter Modellg 53, 9 97 (0) 4. L, Y., Olso, D.L., Q, Z.: mlarty Measures betwee Itutostc Fuzzy (Vague) ets: omaratve alyss. Patter Recogto Letters 8, 687 693 (007) 5. he,.m.: Measures of mlarty betwee Vague ets. Fuzzy ets ad ystems 74, 7 3 (995) 6. og, D.., Km,.: Note o mlarty Measures betwee Vague ets ad betwee Elemets. Iformato cece 5, 83 96 (999)