Z Similarity Measure Among Fuzzy Sets

Transcription

1 Z Smlarty Measure Amog Fuzzy Sets Zora Mtrovć Assocate Professor Faculty of Mechacal Egeerg Uversty of Belgrade Srđa Rusov Assocate Professor Faculty of Traffc ad Trasort Egeerg Uversty of Belgrade The exstg smlarty measures betwee fuzzy sets have bee aalyzed at the begg of the aer The exstg measures have bee defed as the smlarty measures of the two fuzzy sets The costrats these measures clude are that they are aled for the fuzzy sets whose membersh fuctos are dscotuous Lkewse, the roertes of the exstg measures have bee aalyzed It has bee oted that most freuet cases ot eve the basc roertes these measures should fulfll have bee fulflled O the bass of the exstg measures, the smlarty measure Z A, B betwee two fuzzy sets has bee aalyzed Ths measure fulflls the basc roertes I certa cases ths measure gves the defte result The soluto to ths roblem has bee reseted ths aer Wth ths soluto, the measure Z A, B has bee determed to the full each case, ad the costrats do ot affect the ualty of the soluto The ew measure wth whch we ca determe the smlarty measure of more fuzzy sets has bee defed usg the measure Z B Wth the troducto of the costrats for the ew smlarty measure amog more fuzzy sets the ew measure rovdes good results all cases Keywords: smlarty measures, fuzzy sets, Z B measure, Z measure INTRODUCTION I the cases whe we ca ot recsely eough descrbe a heomeo or a rocess, we troduce certa assumtos, e smlfcatos Ths s a classcal way of obtag a model of a object or a rocess The uesto the arses whether by troducg ths assumto we have formed the model correctly eough, e whether some of the basc roertes have bee eglected However, the roblem may arse the case we have modeled a object well eough We do ths most freuetly geeral umbers Whe we have to make secfc calculatos, we are ot sure whch values certa arameters should have I ths case fuzzy umbers may hel If we have more defte values oe model, the the eed to troduce more fuzzy umbers arses The uesto arses whether all these fuzzy umbers (sets) are mutually deedet or coected I order to determe the degree of ther correlato t s ecessary to troduce certa values whch ca measure ths correlato For ths urose smlarty measures betwee or amog fuzzy sets have bee troduced The frst aers ths feld aeared 980s [] Afterwards, dfferet authors roosed varous smlarty measures of fuzzy sets I 993 Pas ad Karacalds [] were amog Receved: May 006, Acceted: August 006 Corresodece to: Zora Mtrovć Faculty of Mechacal Egeerg, Kraljce Marje 6, 0 Belgrade 35, Serba ad Moteegro E-mal: zmtrovc@masbgacyu the frst oes to make a classfcato of the measures kow u to the They made a comarso of these measures aalyzg ther advatages ad dsadvatages Later, ew, mroved measures were defed [3] Lkewse, three more measures were troduced [4] The detaled aalyss was coducted for these measures The roblem of smlarty betwee fuzzy sets was also aalyzed from the mathematcal ot of vew, usg the correlato theory [5] I ths case, the correlato uotet was used as a smlarty measure of fuzzy sets I md 990s aother comarso of kow measures was made [6] Certa cases of solvg the dsadvatages of exstg measures were reseted [7] ad [8] as well BASIC DEFINITIONS Let us assume that A ( =,,, m) are fuzzy sets Wth X we ca deote the uverse of dscourse for each of the fuzzy sets A, e X = { x, x,, x} whe fuzzy sets are rereseted by ther membersh fuctos, the A = { µ A ( x), a x a }, () where µ A ( x) s a membersh fucto of fuzzy set A, for whch µ A ( x): x [0,], =,,, m Measures a ad a are such that: x < a x > a a a ; µ A ( x) = 0 Orgally, the dstace fucto has bee troduced to [] as Faculty of Mechacal Egeerg, Belgrade All rghts reserved FME Trasactos (006) 34, 5-9 5

2 r(, ) = = r d a b a b, () ad t reresets the dstace of r th order betwee ots a ad b, - dmesoal sace I secal cases, e for r = ad r =, ad d( a, b) = a b = d ( a, b) max a b r, (3) = (4) These relatos are ecessary sce they hel to defe smlarty measures betwee fuzzy sets whch, relatos (), (3) ad (4), are rereseted as ots There are several aroaches to defg the smlarty measures betwee fuzzy sets [6] The frst aroach s such that t volves geometrc dstaces -dmesoal sace These are measures based o the model of geometrc dstace These measures are aled oly to defe the smlarty measures of two fuzzy sets Let us assume that the fuzzy sets are A ad B These measures are LAB, = max a b, =,, (5) LAB, = d ( a, b), (6) a b = LAB, = (7) a + b = Aalyzg the dsadvatages of these measures, the smlarty betwee fuzzy sets A ad B has bee gve [6] the followg way WA, B = a b = (8) The secod aroach to defg smlarty measures betwee fuzzy sets s whe they are based o the so called set-theoretc aroach It s the assumed that fuzzy sets are defed through ther cotuous membersh fuctos ( µ A ( x) ) If the scalar cardalty (ower) of fuzzy subset A s defed as A = µ ( x)dx, (9) the, eg the smlarty measure of fuzzy sets A ad B s as follows A A B S B =, (0) A B or SAB, = su µ A B( x) () x X By aalyzg the revous measures, Pas suggested that relatos (0) ad () should be modfed the followg way, so that the ew measure should be A B MAB, = SAB, = A B, () whle the cotuous membersh fuctos ths measure s TA, B = S A, B = su µ A B ( x) (3) x X The thrd aroach to formg smlarty measure betwee fuzzy sets s based o the so called matchg fucto S [6] I ths case, vectors a ad b, whch are reresetatves of fuzzy sets A ad B, are observed The the smlarty measure betwee fuzzy sets A ad B ca be defed as follows a b S( a, b) = (4) max ( a a, b b) Ths measure has retaed ts orgal form, but s freuetly deoted as P B = S( a, b) O the bass of the stated measures, the formg of measure Z wll be show later It s therefore ecessary to exla the roertes of these measures, e ther advatages ad dsadvatages Bearg md the three dfferet aroaches to solvg ths roblem, the roertes to be cosdered refer to measuresw B, T B ad P B 3 PROPERTIES OF SIMILARITY MEASURES BETWEEN FUZZY SETS A AND B The smlarty measure of the sets A ad B, based o geometrc dstaces ( W B, [6]) has the followg roertes: (W) W A, B = WB, (W) A = B W B =, (W3) A B = 0 W B = 0, (W4) W A, A =, (W5) W, A = 0 A = I A = 0 (W6) ~ B W A W A C, A, B These are just the basc roertes of ths measure However, t ca be cocluded at frst sght that some of these roertes have ot bee fulflled We should bear md that each of the measures should fulfll these basc roertes I ths case (W3) ad (W6) are the roertes whch are ot always fulflled Eg there are cases where A B = 0 but thew A, B 0 Geerally, (W6) should also be vald, but t ca be show that t s ot Regardg the bascty of these roertes, whch are ot always fulflled, t may be cocluded that ths smlarty measure betwee fuzzy sets ca ot be geerally acceted I the cases whe we observe the smlarty measure betwee fuzzy sets A ad B based o the set-theoretc 6 VOL 34, No, 006 FME Trasactos

3 aroach, ths measure s followg roertes T, (T) T A, B = TB, (T) A = B T B =, (T3) A B = 0 T B = 0, (T4) A ~ B T A C, TA, B It should fulfll the O the bass of these fudametal roertes, t may be cocluded that ths measure s ot sutable for use most cases It may be very easly cocluded that the characterstc (T) has ot bee fulflled geerally Ths dsadvatage may be overcome f the membersh fuctos are stadardzed As ths s ot always the case, ths measure s ot accetable geerally sce the characterstc (T) must be fulflled The roerty (T4) s ot always fulflled ether, bearg md the ecessty for the measure (T4) to be fulflled, whch s ot always the case, the measure T B caot be acceted as a measure, whch geerally rovdes a farly accurate cture of the smlarty of two fuzzy sets A ad B Bearg md the smlarty measures based o matchg fucto S, the the measure P, s defed The roertes of ths measure are smlar to the revous roertes, e (P) P A, B = PB, (P) A = B P B =, (P3) A B = 0 P B = 0, (P4) ~ B P A P A C, A, B I the case of ths measure we may otce that the majorty of basc roertes have bee fulflled It meas that the basc dsadvatages of the revous measures have bee overcome wth ths measure However, ths case the roerty (P4) eed ot always be fulflled ether Bearg md all the roertes whch occur wth the reseted measures, t s essetal to defe a measure whch wll at least fulfll the basc roertes Lkewse, t s ecessary to geeralze ths measure for the case of smlarty of more sets as well The attemt to troduce ths measure ca be foud [8] 4 Z SIMILARITY MEASURES BETWEEN FUZZY SETS AND THEIR PROPERTIES 4 Z smlarty measure betwee two fuzzy sets We are aalyzg sets A ad B defed by ther cotuous membersh fuctos µ A ad µ B Ther uverse of dscourse are X A ad X B resectvely, ad they eed ot be eual, e X A X B The set of ots of cross secto of the membersh fuctos µ A ad µ B s defed by { k : ( k ) = µ ( k ), k < k, =, r} K = µ A B +,, (5) whch meas that there are r ots of cross secto of these two membersh fuctos Let us assume that ad are such umbers so that µ A( x) > 0 µ B ( x) > 0, x (, ) (6) Sce the uverse of dscourse of fuzzy sets A ad B s defed by X A ad X B, e X A X B { x, a x a} { x, b x b } =, (7) =, (8) t may be cocluded that = a ad = b O the bass of [8], the smlarty measure betwee two fuzzy sets (A ad B) ca be exressed as follows Z µ ( x) dx + µ ( x) dx C A, B = a b a A b µ ( x) dx B (9) The membersh fucto µ C (x) ca be defed as µ A( x), µ A( x) µ B ( x) =, x X X B (0) µ B ( x), µ A( x) > µ B ( x) Whe defg the value dx, we should bear md that the calculato terval s dvded to several arts ad that t deeds o r- e o the umber of cross secto ots of the membersh fucto It meas that dx = k k = µ C( x) dx+ µ C( x) dx+ + µ C( x) dx k kr! () The roertes the stated measure Z, fulflls are (Z) Z A, B = Z A, B, (Z) A = B ( A 0 B 0) Z B =, (Z3) A B = 0 Z B = 0, (Z4) A ~ B Z A C, Z A, B Note: If some uverse of dscourse s such that = or =, the t most freuetly occurs that eg o the terval (, k) or o the terval ( k r, ) the followg s fulflled µ A ( x) = µ B ( x) = () I ths case, whe calculatg the measure Z B, usg the exresso (4) we obta the defte exresso I order to overcome ths, costrats must be mosed for the alcato of the exresso (4) These costrats may be mosed o the tervals FME Trasactos VOL 34, No, 006 7

4 x (, k) x ( kr, ), (3) o whch µ A ( x) = µ B ( x) = (4) I ths case the set also has a ulmted umber of elemets, e r = The, o the tervals where (3) ad (4) are vald t must be as follows so that Z B s a fal umber = 0 (5) 4 Z Smlarty measure amog fuzzy sets Let us assume that A ( =,,, ) are fuzzy sets, defed wth ther cotuous membersh fuctos µ A (x) If we assume that the uverse of dscourse s X, for each set A, resectvely It meas that X = { x, a x a } (6) O the bass of the revous dscusso, values ad ca be defed as = m ( a ), (7) ad = max ( a ) (8) The set of cross secto ots of the membersh fuctos are K = { kl : µ A ( k ) ( ), l = µ A k j l, j =,,, k < k, l =,,, r (9) l l+ Now the Z smlarty measure amog fuzzy sets s defed wth Z A = a = a µ C ( x) dx µ A ( x) dx } (30) The membersh fucto µ C (x), ths case s defed as = m ( µ A ( x)), =,,, (3) where we should take to cosderato that ths fucto s determed o each terval ( k l, k l+ ), l = 0,,,, r The value the umerator of the exresso (30) s determed comletely the same way as for the two sets, usg (0) I order to obta the fal value of the smlarty measure amog fuzzy sets Z e, so that t would ot be, smlar costrats should be mosed as the case of the measure Z B It meas that, o the tervals k, ) o whch ( l k l+ µ ( x) = =,, (3) A we should take to accout that µ ( x) = 0 (33) I ths case the measure 5 CONCLUSION C Z A s the fal umber I the troductory art we dscussed the eed to troduce smlarty measures betwee fuzzy sets The chroologcal aalyss of the scetfc vews ths feld has also bee reseted Havg md the freuet aearace of geeralzato of the results ths feld, the eed for further geeralzato of the exstg results arses, as well as the mrovemet of the exstg solutos It s artcularly mortat to rovde ew solutos whe the exstg oes are ot good eough The exstg smlarty measures are freuetly coected wth the fuzzy umbers whch are defed wth dscotuous membersh fuctos Havg md the lmted techcal alcato of these fuzzy umbers t was ecessary to defe the smlarty measures betwee fuzzy sets determed wth ther cotuous membersh fucto 3 The roertes of the exstg smlarty measures have bee aalyzed It may be cocluded that the basc roertes, whch should be fulflled by these measures, are most freuetly ot fulflled 4 The Z B smlarty measure betwee two fuzzy sets has bee searately aalyzed, whe these sets are determed by ts cotuous membersh fucto 5 Although the measure Z B does ot fulfll all basc roertes whch should be fulflled by a certa measure, there are cases where ths measure has ot bee determed to the full, e t has the value The soluto to ths roblem has bee gve 6 The smlarty measure amog more fuzzy sets, e Z, =,,, has bee defed usg the A troduced measure Z B Lkewse, the soluto whe ths measure has a defte value, has bee gve REFERENCES [] Zwck, R, Carlste, E, Budescu, D: Measures of smlarty amog fuzzy cocets: a comaratve aalyss, lterat J Aroxmate Reasog, -4, 987 [] Pas, C, Karacalds, N: A comaratve assessmet of measures of smlarty of fuzzy values, Fuzzy Sets ad Systems 56, 7-74, VOL 34, No, 006 FME Trasactos

5 [3] Hyug, L, Sog, Y, Lee, K: Smlarty measure betwee fuzzy sets ad betwee elemets, Fuzzy Sets ad Systems 6, 9-93, 994 [4] Wag, W: New smlarty measures o fuzzy sets ad o elemets, Fuzzy Sets ad Systems 85, Vol 3, , 997 [5] Gerstekor, T, Ma'ko, J: Correlato of tutostc fuzzy sets, Fuzzy Sets ad Systems 44, 39-43, 99 [6] Che, S, Yeh, M, Hsao, P: A comarso of smlarty measures of fuzzy values, Fuzzy Sets ad Systems 7, Vol, 79-89, 995 [7] Mtrovć, Z: Measures of coecto of fuzzy sets ad ther elemets, Trasactos, Vol XXVII, ssue, 3-5, 998 [8] Mtrovć, Z, Rusov, S, Mladeovć, N: About smlarty measures betwee fuzzy sets, Bullets for Aled & Comuter Mathematcs, No CVII/005, 79-84, 005 Z МЕРЕ ПОВЕЗАНОСТИ ИЗМЕЂУ FUZZY СКУПОВА Зоран Митровић, Срђан Русов У раду се најпре анализирају постојеће мере повезаности између fuzzy скупова Постојеће мере дефинисане су као мере повезаности два fuzzy скупа Ограничења која ове мере имају су да се оне примењују за fuzzy скупове чије функције припадности су прекидне Такође, анализиране су особине постојећих мера Уочено је да најчешће ни основне особине, које би ове мере требало да задовољавају, нису испуњене На основу постојећих мера, анализирана је и Z B - мера повезаности између два fuzzy скупа Ова мера задовољава основне особине У одређеним случајевима ова мера даје неодређени резултат облика У раду је дато решење овог проблема Овим решењем мера Z B, у потпуности је одређена у сваком случају, а ограничења која су наметнута у поступку њеног израчунавања, не утичу на квалитет решења Користећи меру Z B дефинисана је нова мера којом се може утврдити повезаност између више fuzzy скупова Са уведеним ограничењима за меру Z B, која је прилагођена за нову меру повезаности између више fuzzy скупова, нова мера даје квалитативно добре резултате у свим случајевима FME Trasactos VOL 34, No, 006 9