Fuzzy clustering of intuitionistic fuzzy data

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1 It. J. Busess Itellgece ad Data Mg, Vol. 3, No., Fuzzy clusterg of tutostc fuzzy data Nos Peles* Deartmet of Iformatcs, Uversty of Praeus, Praeus, Greece E-mal: *Corresodg author Dmtrs K. Iaovds Deartmet of Methodology, Hstory ad Theory of Scece, Uversty of thes, thes, Greece E-mal: Evagelos E. Kotsfaos Deartmet of Iformatcs, Uversty of Praeus, Praeus, Greece E-mal: Ioas Koaas Techologcal Educatoal Isttute of Crete, Heralo Crete, Greece Webste: bstract: Challeged by real-world clusterg roblems ths aer rooses a ovel fuzzy clusterg scheme of datasets roduced the cotext of tutostc fuzzy set theory. More secfcally, we troduce a varat of the Fuzzy C-Meas (FCM) clusterg algorthm that coes wth ucertaty ad a smlarty measure betwee tutostc fuzzy sets, whch s arorately tegrated the clusterg algorthm. We descrbe a tutostc fuzzfcato of color dgtal mages uo whch we aled the roosed scheme. The exermetal evaluato of the roosed scheme shows that t ca be more effcet ad more effectve tha the well-establshed FCM algorthm, oeg ersectves for varous alcatos. Keywords: fuzzy clusterg; Fuzzy C-meas; FCM; tutostc fuzzy sets; tutostc smlarty metrcs. Referece to ths aer should be made as follows: Peles, N., Iaovds, D.K., Kotsfaos, E.E. ad Koaas, I. (2008) Fuzzy clusterg of tutostc fuzzy data, It. J. Busess Itellgece ad Data Mg, Vol. 3, No., Coyrght 2008 Iderscece Eterrses Ltd.

2 46 N. Peles et al. Bograhcal otes: Nos Peles receved hs BSc Degree from the Comuter Scece Deartmet of the Uversty of Crete (998). He has subsequetly joed the Deartmet of Comutato the Uversty of Machester Isttute of Scece ad Techology (UMIST) to ursue hs MSc Iformato Systems Egeerg (999) ad hs PhD Movg Object Databases (2002). Curretly, he s ost-doctoral researcher at the Database Grou of the Iformato Systems Laboratory of the Deartmet of Iformatcs, Uversty of Praeus (htt://sl.cs.u.gr/db/). Hs research terests clude temoral, satal ad satotemoral databases, maagemet of locato-based servces, data mg ad geograhcal formato systems. Dmtrs K. Iaovds s curretly a seor researcher at the Uversty of thes, Greece. He holds a BSc Degree Physcs (997), a MSc Degree Cyberetcs (200) ad a PhD Iformatcs (2004). He has authored ad co-authored over 60 research aers teratoal jourals ad coferece roceedgs, ad he has bee actvely volved more tha te Euroea ad Natoal R&D rojects. Hs research terests clude mage rocessg ad aalyss, data mg, atter recogto ad cultural, bomedcal ad busess tellgece alcatos. Evagelos E. Kotsfaos s a PhD caddate at the Deartmet of Iformatcs, Uversty of Praeus (UP). He receved hs Bachelor (200) ad Master (2003) Degree Iformato Systems from the Deartmet of Iformatcs of thes Uversty of Ecoomcs ad Busess. Hs research terests clude atter maagemet, data mg ad scetfc databases. He also has a rofessoal exerece software egeerg. Ioas Koaas s a ssstat Professor ad Head of the Det. of Commerce ad Maretg at the Techologcal Isttuto of Crete. He holds a Dloma Comuter Scece from the Uversty of Crete (998), Greece, a MSc formato Techology (999), ad a PhD Comutato (2003), both from UMIST, UK. He s the scetfc Drector of the e-busess Itellgece Lab at the Ceter for Techologcal Research Crete. Hs research terests clude data mg, vsual data mg, ad busess tellgece ad he has ublshed 23 aers jourals ad refereed cofereces. Itroducto The creasg amout of data beg geerated ad collected at varous oeratoal domas has made data mg a research area wth creasg mortace. mog the usuervsed data mg tass, clusterg ossesses a vot osto. Clusterg s the orgasato of objects to searate collectos, ad volves arttog of a feature set to clusters (grous), so that each cluster cotas roxmate feature vectors accordg to some dstace measure. lcatos of clusterg clude data aalyss, atter recogto ad boformatcs (Theodords ad Koutroumbas, 2006). Clusterg aroaches based o fuzzy logc (Zadeh, 965), such as FCM (Bezde et al., 984) ad ts varats (Yog et al., 2004; Thtmajshma, 2000; Chumsamrog et al., 2000) have roved to be comettve to covetoal clusterg algorthms, esecally for real-world alcatos. The comaratve advatage of these aroaches s that they do ot cosder shar boudares betwee the clusters, thus allowg each feature vector to belog to dfferet clusters by a certa degree (the so-called soft

3 Fuzzy clusterg of tutostc fuzzy data 47 clusterg cotrast to hard clusterg roduced by covetoal methods). The degree of membersh of a feature vector to a cluster s usually cosdered as a fucto of ts dstace from the cluster cetrods or from other reresetatve vectors of the cluster. major challege osed by real-world clusterg alcatos s dealg wth ucertaty the localsato of the feature vectors. Cosderg that feature values may be subject to ucertaty owg to mrecse measuremets ad ose, the dstaces that determe the membersh of a feature vector to a cluster wll also be subject to ucertaty. Therefore, the ossblty of erroeous membersh assgmets the clusterg rocess s evdet. Curret fuzzy clusterg aroaches do ot utlse ay formato about ucertaty at the costtutoal feature level. Ths aer accets the challege to deal wth such d of formato, ad troduces a modfcato to the FCM. The ovel varat of the FCM algorthm assumes that the features are rereseted by tutostc fuzzy values,.e., elemets of a tutostc fuzzy set. Itutostc fuzzy sets (taassov, 986, 989, 994a, 994b, 999) are geeralsed fuzzy sets (Zadeh, 965) that ca be useful cog wth the hestacy orgatg from merfect or mrecse formato (Vlachos ad Sergads, 2007). The elemets of a tutostc fuzzy set are charactersed by two values reresetg ther beloggess ad o-beloggess to ths set, resectvely. To exlot ths formato for clusterg, we defe a ovel dstace metrc esecally desged to oerate o tutostc fuzzy vectors. The lethora ad mortace of the otetal alcatos of tutostc fuzzy sets have draw the atteto of may researchers who have roosed varous ds of smlarty measures betwee tutostc fuzzy sets. Examle alcatos clude detfcato of fuctoal deedecy relatoshs betwee cocets data mg systems, aroxmate reasog, atter recogto ad others. Smlarty measures betwee tutostc fuzzy sets have bee roosed by Che (995, 997) wth S C measure, by Hog ad Km (999) wth S H, by Fa ad Zhagya (200) wth S L, ad L et al. (2002) who roosed the S O measure. Degfeg ad Chuta (2002) roosed the S DC measure, Mtchell (2003) roosed a modfcato of the S DC measure, the S HB measure, Zhzhe ad Pegfe (2003) roosed three measures S, ad e Ss S h ad three more measures have bee roosed by Hug ad Yag (2004), the S, 2, ad 3 S S. L et al. (2007) rovde a detaled comarso of these measures, otg out the weaesses of each oe. Some measures, such as S C, S H, S L, S HB ad S, 2, ad 3 S S focus o the aggregato of the dffereces betwee membersh values ad dffereces betwee the o-membersh values whle others aly dstaces such as Mows, for S DC, or Hausdorff, for S, S 2, ad S 3, to calculate the degree of smlarty of the fuzzy h sets. S DC, Ss ad S focus also o the dfferece betwee membersh values ad o-membersh values. s regards the effectveess of these measures, some of them, such as S C ad do ot satsfy the roertes of a smlarty metrc defed betwee tutostc fuzzy sets, whereas all of the above-metoed measures fal secfc cases that L et al. (2007) meto wth couter-tutve examles. The rest of ths aer s structured as follows: Secto 2 rovdes a overvew of the tutostc fuzzy set theory. I Secto 3, we troduce a ovel tutostc reresetato of colour dgtal mages, as a aradgm of tutostc fuzzfcato of data. The roosed smlarty measure s defed Secto 4, where t s assessed

4 48 N. Peles et al. comarso wth other measures defed betwee tutostc fuzzy sets. I Secto 5, we aalyse the roosed tutostc fuzzy clusterg algorthm, ad Secto 6, the results of ts exermetal evaluato wth real-world data are aosed. Fally, the coclusos of ths study alog wth deas for future wor are summarsed Secto 7. 2 Itutostc fuzzy sets The theoretcal foudatos of fuzzy ad tutostc fuzzy sets are descrbed Zadeh (965) ad taassov (986). Ths secto brefly outles the related otos used ths aer. Defto (Zadeh, 965): Let a set E be fxed. fuzzy set o E s a object of the form = { x, µ ( x) x E} () where µ : E (0,) defes the degree of membersh of the elemet x E to the set E. For every elemet x E, 0 µ ( x ). Defto 2 (taassov, 986, 994): tutostc fuzzy set s a object of the form = { x, µ ( x), γ ( x) x E} (2) where µ : E (0,) ad γ : E (0,) defe the degree of membersh ad o-membersh, resectvely, of the elemet x E to the set E. For every elemet x E, t holds that 0 µ ( x). 0 γ ( x) ad 0 µ ( x) + γ ( x). (3) For every x E, f γ ( x) = µ ( x), reresets a fuzzy set. The fucto π ( x) = µ ( x) γ ( x) (4) reresets the degree of hestacy of the elemet x E to the set E. For every two tutostc fuzzy sets ad B, the followg oeratos ad relatos are vald (taassov, 986, 994) B ff x E, µ ( x) µ ( x) ad γ ( x) γ ( x) (5) B B = B ff B ad B (6) C = { x, γ ( x), µ ( x) x E} (7) B = { x, m( µ ( x), µ ( x)), max( γ ( x), γ ( x)) x E} (8) B B B = { x,max( µ ( x), µ ( x)),m( γ ( x), γ ( x)) x E} (9) B B = x, ( µ ( x) + µ B( x)), ( γ ( x), γb( x)) x E 2 2 (0)

5 Fuzzy clusterg of tutostc fuzzy data x, ( x ), ( x ) x E = µ γ. = = = () Defto 3 (Degfeg ad Chuta, 2002): Let S be a mag IFSs( E ), IFSs( E) [0,] where IFSs(E) deotes the set of all tutostc fuzzy sets E. S(, B) s sad to be the degree of smlarty betwee IFSs(E) ad B IFSs(E), f S(, B) satsfes the followg codtos: P S(, B) (0, ) P2 S(, B) = = B P3 S (, B) = S( B, ) P4 S(, C) S(, B) ad S(, C) S( B, C) f B C, C IFSs( E). Reresetg the data of a real-world clusterg roblem by meas of tutostc fuzzy sets s a challegg ssue, rovdg the oortuty to vestgate the effectveess of the tutostc fuzzy theory ractce. 3 Itutostc fuzzy reresetato of data The roosed tutostc fuzzy clusterg requres that each data elemet x of a uverse E belogs to a tutostc fuzzy set E by a degree µ ( x) ad does ot belog to by a degree γ ( x ). The data elemets ca be of ay d. For the uroses of ths study, whch focuses to the clusterg of mage data, we exted the defto of the tutostc fuzzy reresetato of a greyscale dgtal mage (Vlachos ad Sergads, 2005) for the reresetato of a colour dgtal mage. Defto 4: colour dgtal mage P of a b xels sze, comosed of ξ chaels P, =, 2,, ξ, dgtsed q quatsato levels er chael, s rereseted as the tutostc fuzzy set { θ, ( ), ( ),,2,,,,2,,,,2,, } j µ Φ θj γ Φ θj θ j P a j b ξ Φ= = = = (2) where θ j s the value of P at the osto (, j), ad µ Φ ( θ j ) ad γ Φ ( θ j ) defe the membersh ad the o-membersh of θ j to P, resectvely. s a membersh fucto µ Φ ( θ ), we cosder the robablty of occurrece of θ [0, q ] a mage chael h( θ ) µ Φ ( θ) =, θ [ 0, q ], ab where { j } h( θ) = (, j) P θ = θ; =,..., a; j =,..., b, =,2,..., ξ, (4) (3)

6 50 N. Peles et al. s the crs hstogram of the xel values the chael, ad reresets the cardalty of the eclosed set. The robablty dstrbuto descrbed by equato (3) comrses a frst-order statstcal reresetato of the mage chael that s easy to comute, ad t s varat to the rotato ad traslato. Cosderg that real-world dgtal mages usually cota ose of varous orgs, ad mrecso the chael values, the degree of beloggess of a testy value θ a mage chael as exressed by µ Φ (θ ) s subject to ucertaty. To model ths stuato, a ealty factor (θ ) s troduced so that θ belogs less to the mage chael f h(θ ) dverges more from the fuzzy hstogram h ( θ ). The fuzzy hstogram, orgally roosed by Jawahar ad Ray (996), s defed as wth { j } h ( θ ) = (, j) P µ ( θ ); =,..., a; j =,..., b, =, 2,, ξ (5) θ x θ µ ( x) = max 0, θ (6) ψ where arameter ψ cotrols the sa of the fuzzy umber θ : R [0, ] reresetg a fuzzy testy level θ. Ths meas that a xel of a gve chael value wll cotrbute ot oly to ts secfc b, but also to the b cout of the eghbourg bs the hstogram. Thus, the fuzzy hstogram becomes smoother ad more sestve to ose tha the corresodg crs hstogram as ψ creases. ccordg to the roosed formulato, the o-membersh of θ to a mage chael ca be exressed as γ Φ ( θ ) = ( µ ( θ)) ( ( θ)). (7) The term ( θ ) s chose to be roortoal to the dstace betwee the crs h( θ ) ad the fuzzy hstogram h ( θ ), so that equato (3) s satsfed h( θ) h ( θ) ( θ) = λ max( h( θ) h ( θ) ) where [ 0,] θ λ s costat ad the deomator facltates ormalsato uroses. The hyscal meag of ths o-membersh fucto s that the o-beloggess of a testy value θ to a mage chael creases by a factor that s roortoal to the smoothess of the crs hstogram. So, as the ose levels the mage chael crease, the crs hstogram becomes coarser ad the hestacy the determato of the testy value θ creases. The membersh ad the o-membersh, defed by equatos (3) ad (7) over the values of the mage chaels, wll be cosdered to form feature vectors. To evaluate the smlarty betwee these vectors, a ovel smlarty measure s roosed. (8)

7 Fuzzy clusterg of tutostc fuzzy data 5 4 Proosed smlarty measure I ths secto, we roose a ovel smlarty measure betwee tutostc fuzzy sets, based o the membersh ad o-membersh values of ther elemets. Gve a tutostc fuzzy set, we defe two fuzzy sets, amely M, Γ F ( E) where F(E) s the set of all fuzzy subsets of a elemet x E. The membersh ad o-membersh of these sets s defed as M = { µ ( x)}, Γ = { γ ( x)} x E. I ths coecto, ca be descrbed by the ar ( M, Γ ). Defto 5: Cosderg two tutostc fuzzy sets = (M, Γ ), B=(M B, Γ B ), where M, MB, Γ, ΓB F( E), ad cosderg E as a fte uverse E = { x, x2,, x }, we defe the smlarty measure z betwee the tutostc fuzzy sets ad B by the followg equato: where Z (, B) z ( M, M ) + z ( Γ, Γ ) 2 B B = (9) m( ( ), ( )), B φ (, m( ( ), )) (20), B φ x B x = z B )= x B ( x = wth, B F ( E). To accet z as a smlarty metrc, we eed to rove that z satsfes the roertes defed Defto 3. It s straghtforward to rove that roertes P, P2 ad P3 are satsfed by z. We suly the roof for the 4th roerty. Lemma: For all, B, C F ( E), where F(E) s the set of all fuzzy subsets of a elemet x E ad cosderg E as a fte uverse E = { x, x2,, x }, f B C the z (, C ) z (, B ) ad z (, C ) ( B, C ). Proof: By B C, t mles that ( x ) B ( x ) C ( x ) x E ad x C x x = = x C x C x = = m( ( x ), ( )) ( ) B x x = = max( ( x ), ( )) ( ) B x B x = = m( ( ), ( )) ( ) z (, C ) = = max( ( ), ( )) ( ) z (, B ) = = m( B ( x ), ( )) ( ) C x B x = = (, ) = =. max( B ( x ), ( )) ( ) C x C x = = z B C

8 52 N. Peles et al. Thus hece ( x ) ( x ) ( x ) B ( x ) = = = =, C ( x ) B ( x ) C ( x ) C ( x ) = = = = z (, C ) z (, B ) ad z (, C ) z ( B, C ). Sce, BC, IFSsE ( ) ad B C, we have µ ( x) µ ( x) µ ( x) ad γ ( x) γ ( x) γ ( x) x E, =,2,, B C B C therefore, z( M, MB) ad z( Γ, Γ B) satsfy all roertes P P4 ad so z also satsfes these roertes. Thus, z s a smlarty metrc. To demostrate the roosed measure, a smle umerc examle s gve here. Examle: ssumg three sets, BC, IFSsE ( ) wth = {x, 0.4, 0.2}, B = {x, 0.5, 0.3}, C = {x, 0.5, 0.2}, we wat to fd whether B or C s more smlar to. Usg the equatos (9) ad (20), we comute the smlarty of B ad C to set Z (, B) = = 0.733, Z(, C) = = So, we coclude that C s more smlar to tha B. The roosed tutostc smlarty measure uses the aggregato of the mmum ad maxmum membersh values combato wth those of the o-membersh values. lthough t s very smle to calculate, t s sestve to small value chages ad t deals well wth all the couter-tutve cases whch other measures fal. Most of the smlarty measures revewed Secto 4 fal to evaluate to a vald tutostc value for secfc cases. Some of them evaluate to 0 or suggestg that the comared sets are ether totally rrelevat or detcal, whle t s obvous that ths s ot true, ad some other measures result a hgh smlarty value for obvously dfferet sets. More secfcally, Table, we reset all the couter-tutve cases that L et al. (2007) have defed ad the other measures fal, alog wth the calculato of the roosed measure for those cases. I case (I) of Table, measure values S C (, B) ad S DC (, B) mly that ad B are totally smlar. I cases (II) ad (IV), other measures result a rather bg smlarty value, our measure s ot that otmstc. Moreover, case (IV), t s obvous that set s more smlar to C tha to B ( ad C have the same o-membersh value), somethg that other measures do ot tae to accout. I (III), whle B ad C are totally dfferet, measures SH, SHB, S e gve a smlarty value of 0.5. O the cotrary, (V), measures 2 3 S, S, S gve a smlarty value of 0 eve f the o-membersh value of both ad B s the same, suggestg a level of smlarty betwee the two sets. I (VI) ad (VII), 2 3 measures S, S, S result a rather hgh smlarty value ad (VII) they do ot recogse that s more smlar to C tha to B, owg to the same o-membersh value of ad C.

9 Fuzzy clusterg of tutostc fuzzy data 53 Table Proosed ad other smlarty measures wth couter-tutve cases Couter-tutve No. Measure cases I S C, S DC = {( x,0,0)}, B = {( x,0.5,0.5)} II SH, SHB, S = {( x,0.3,0.3)}, e B = {( x,0.4,0.4)}, C = {( x,0.3,0.4)}, D = {( x,0.4,0.3)} III SH, SHB, S = {( x,,0)}, e B = {( x,0,0)}, C = {( x,0.5,0.5)} IV S ad L S = {( x,0.4,0.2)}, S B = {( x,0.5,0.3)}, C = {( x,0.5,0.2)} V 2 3,, S S S = {( x,,0)}, B = {( x,0,0)} VI 2 3,, S S S = {( x,0.3,0.3)}, B = {( x,0.4,0.4)}, C = {( x,0.3,0.4)}, D = {( x,0.4,0.3)} VII 2 3,, S S S = {( x,0.4,0.2)}, B = {( x,0.5,0.3)}, C = {( x,0.5,0.2)} Proosed Measure values measure value S C (, B) = S DC (, B) = Z = 0 S H (, B) = S HB (, B) = (, B) = 0.9 S H (C, D) = S HB (C, D) = (C, D) = 0.9 S H (, B) = S HB (, B) = S H (B, C) = S HB (B, C) = S e S e SL(, B)= SS (, B) = 0.95 S (, C)= S ( C, D) = 0.95 L S 2 3 S (, B) = 0.5 e S (B, C) = 0.5 e Z (, B) = Z (C, D) = 0.75 Z (, B) = 0.5, Z (B, C) = 0 Z (, B) = 0.73 Z (, C) = 0.9 S (, B) = S (, B) = S (, B) = 0 Z (, B) = 0.5 S (, B) = S ( C, D) = 0.9 S (, B) = S ( C, D) = 0.85 S (, B) = S ( C, D) = S (, B) = S (, C) = 0.9 S (, B) = S (, C) = 0.85 S (, B) = S (, C) = Z (, B) = Z (C, D) = 0.75 Z (, B) = 0.73 Z (, C) = 0.9 The earler dcate the tutveess of the roosed measure, whch satsfes all the roertes of a smlarty metrc ad does ot fal cases that other measures fal. Furthermore, the roosed measure s easy to calculate ad does ot use exoets or other fuctos that sgfcatly slow dow the calculatos. 5 Clusterg tutostc fuzzy data Most clusterg methods assume that each data vector belogs oly to oe cluster. Ths s ratoal f the feature vectors resde comact ad well-searated clusters. However, real-world alcatos, clusters overla, meag that a data vector may belog artally to more tha oe cluster. I such a case ad terms of fuzzy set theory (Zadeh, 965), the degree of membersh of a vector x to the th cluster u s a value the terval [0,]. Rus (969) troduced ths dea, whch was later aled by Du (973), to roose a clusterg methodology based o the mmsato of a objectve fucto. I Bezde et al. (984), Bezde troduced the FCM algorthm, whch uses a weghted exoet o the fuzzy membershs.

10 54 N. Peles et al. FCM s a teratve algorthm ad ts am s to fd cluster cetrods that mmse a crtero fucto, whch measures the qualty of a fuzzy artto. fuzzy artto s deoted by a (c N)-dmesoal matrx U of real u [0,], c ad N, where c ad N s the umber of clusters ad the cardalty of the feature vectors, corresodgly. The followg costrat s mosed uo u : c N u =, 0 < u < N. (20) = = Gve ths, the FCM objectve fucto has the form: c N m 2 m(, ) ( ) = = J UV = u d (2) where V s a ( C)-dmesoal matrx storg the c cetrods, s the dmesoalty of the data, d s a -orm measurg the dstace betwee data vector x ad cluster cetrod v, ad m [, ) s a weghg exoet. The arameter m cotrols the fuzzess of the clusters. Whe m aroxmates, FCM erforms a hard arttog as the -meas algorthm does, whle as m coverges to fty, the arttog s as fuzzy as ossble. There s o aalytcal methodology for the otmal choce of m. Bezde et al. (984) roved that f m ad c are fxed arameters ad I, I are sets defed as: I = { c; d = 0}, N, I (22) = {, 2,, c}\ I, the, J ( U, V ) may be mmsed oly f: m ad 2 m ( d ), I, 2 = c m u ( ) j j = d = 0, I, I, u =, I I c N (23) c N m ( u ) x = N m ( u ) v = =. (24) By teratvely udatg the cluster cetrods ad the membersh degrees for each feature vector, FCM teratvely moves the cluster cetrods to the rght locato wth the data set. I detal, the algorthm that results the otmal artto s the Pcard algorthm, whch s descrbed Fgure. The arameter ε maes the algorthm to coverge whe the mrovemet of the fuzzy artto over the revous terato s below a threshold, whle F deotes the Frobeous orm.

11 Fuzzy clusterg of tutostc fuzzy data 55 Fgure Fuzzy c-meas algorthm The FCM algorthm mmses tra-cluster varace, but shares the same roblems wth -meas (MacQuee, 967). It does ot esure that t coverges to a otmal soluto, whle the detfed mmum s local ad the results deed o the tal choce of the cetrods. FCM tres to artto the data set by just loog at the feature vectors ad as such t gores the fact that these vectors may be accomaed by qualtatve formato, whch may be gve er feature. For examle, followg the dea of tutostc fuzzy set theory, a data ot x s ot just a -dmesoal vector ( x,, x ) of quattatve formato, but stead t s a -dmesoal vector of trlets [( x, µ, ),..., (,, )], γ x µ γ, where for each x measuremet there exsts t qualtatve formato, whch s rovded va the tutostc membersh µ ad t o-membersh γ of the curret data ot to the feature l. It s evdet that the FCM t algorthm does ot utlse trscally such qualtatve formato. I the alcato scearo of clusterg mages, a feature l may corresod to colour formato. Obvously, t would be of advatage f the clusterg methodology could tae to accout the degree of membersh ad the degree of o-membersh, regardg (for stace) how much red the mage s, ad how sure we are about our belef. The ma reaso that FCM s uable to effectvely utlse such tutostc vectors s that ts dstace fucto oerates oly o the feature vectors ad ot o the qualtatve formato, whch may be gve er feature. I ths aer, we roose a dfferet ersectve by substtutg the dstace fucto wth the tutostc fuzzy set dstace metrc troduced Secto. Usg the roosed dstace fucto, the FCM objectve fucto taes the form: c N IFS m m = IFS = = J ( U, V) ( u ) x v. (25) The mmsato of equato (26) ca be acheved term by term: where N IFS J ( UV, ) = ϕ ( U) (26) m N = c m U u x v IFS = ϕ ( ) = ( ). (27) The Lagraga of equato (28) wth costrats from equato (2) s: c c m Φ ( U, λ) = ( u) x v λ u IFS = = N (28)

12 56 N. Peles et al. where λ s the Lagrage multler. Settg the artal dervatves of Φ ( U, λ) to zero, we obta: ad c Φ ( U, λ) = u = 0 (29) λ N = Φ ( U, λ) = mu x v = u z c N z m ( z ) λ 0. IFS (30) Solvg equato (3) for u z, we get: m λ m uz = ( x vz IFS ). m From equatos (30) ad (32), we obta: λ m m = c j= m j IFS ( x v ). (3) (32) The combato of equatos (32) ad (33) yelds: m ( x vz IFS) uz = z c c N m ( x ) j v = j IFS. (33) IFS Smlarly wth Jm ( U, V ), J ( U, V ) may be mmsed oly f: m ( x v IFS) c m u ( x ) j vj IFS = = c N 0, I I m u =, I, I =,, I, whle the cetrods are comuted by equato (25). It should be clarfed that u corresods to the membersh of the th tutostc fuzzy vector to the th cluster ad has othg to do wth the teral tutostc fuzzy membershs of the vector. Furthermore, as our dstace fucto betwee two vectors s comuted solely uo the tutostc fuzzy membershs ad o-membershs of the vectors, after the comutato of the cetrods by equato (25) ad before the ext terato, where the u membershs to the ew clusters are udated, there s a eed for a addtoal ste, whch estmates the tutostc fuzzy membershs ad o-membershs of the ew (vrtual) cetrods. I other words, t s ecessary to deduce the membersh µ ad o-membersh γ l values of each feature l that corresods to l the lth dmeso of the th cetrod. t each terato ad for every cetrod, (34)

13 Fuzzy clusterg of tutostc fuzzy data 57 we extract the membersh degree µ of cetrod v l as the average of the membersh degrees of all the tutostc fuzzy vectors that belog to cluster. Smlarly, we extract the o-membersh degrees γ l. More formally, f P s a set defed as: P = { N; d < d, r N r } (35) c r the the tutostc fuzzy set c IFS v for cetrod v s defed as: IFS IFS. (36) v P From taassov (994), we obta: µ γ P l P l µ =, ν. l = l P P l (37) Gve the above dscusso, the modfed FCM algorthm that clusters tutostc fuzzy data s subsequetly descrbed: I comarso wth the lteral FCM algorthm, the clusterg scheme reseted Fgure 2 troduces a dfferet talsato tactc of the V matrx as our case cetrod vectors are tutostc fuzzy vectors (Ste ) a ew way of the calculato of the membersh degrees of a vector to a cluster, tag to accout both membersh ad o-membersh values of the tutostc fuzzy vectors (Ste 2) a method to udate the V matrx at each terato based solely o the theory of the tutostc fuzzy sets (Ste 3). Fgure 2 Modfed fuzzy c-meas algorthm for clusterg tutostc fuzzy data 6 Results Comrehesve exermets have bee coducted for the evaluato of the erformace of the roosed clusterg algorthm, comarso wth the well-establshed FCM. The alcato scearo for the exermetal evaluato volves clusterg of a 400-mage collecto sag four equally dstrbuted classes of dfferet colour themes cludg amhorae, acet moumets, cos, ad statues (Fgure 3). The mages have bee rovded by the Foudato of Hellec World, whch matas a ublcly avalable reostory of texts, mages ad multmeda data collectos of Gree hstorcal tems ad art (FHW). They are of dfferet szes ad have bee cosstetly acqured from

14 58 N. Peles et al. dfferet sources, ad they have bee dgtsed 256 quatsato levels er RGB chael ad have bee dowscaled to ft to a boudg box. Fgure 3 Examle mages from the four classes used the exermets: (a) amhorae; (b) acet moumets; (c) cos ad (d) statues (a) (b) (c) (d) Based o the observato that colour s a dscrmatve feature for most of the avalable mage classes, each mage was rereseted by a tutostc fuzzy set accordg to equato (2), usg oly chromatc formato so as to be aroxmately deedet from testy varatos. To decorrelate the testy from the chromatc mage comoets, the mages have bee trasformed to the I I 2 I 3 colour sace accordg to the followg equato (Ohta et al., 980) I R I2 = G. (38) I B I ths colour sace, the I comoet exlas the hghest roorto of the total varace ad reresets testy, whereas I 2 ad I 3 corresod to the secod ad the thrd hghest roorto, resectvely, ad carry chromatc formato. very useful roerty of ths sace s that mage regos of dfferet colours ca be easly dscrmated by smle thresholdg oeratos. I other words, the hstograms roduced by the values of ts colour comoets exhbt eas corresodg to regos of dfferet colours the mage. mog the chromatc comoets of I I 2 I 3, we selected I 2 as the most dscrmatg for the colour regos comrsg the avalable mages. Ths s agreemet wth Ohta et al. (980), whch suggests that the dscrmato ower of I 2 could oly margally crease wth the cotrbuto of I 3. Moreover, we observed that the mage chael corresodg to the I 3 comoet exhbts a low dyamc rage of values, havg a sgle-ea hstogram that vares slghtly betwee mages belogg to dfferet classes. Examles of membersh ad o-membersh fuctos used for the tutostc fuzzy reresetato of colour mages are llustrated Fgure 4. The values of the arameters used equatos (6) (8) for the estmato of the membersh ad of the o-membersh fuctos are λ = ad ψ = 5. I Fgures 5(a) (d), the hgher of the two eas corresod to the whte bacgroud regos of the mages, whereas the lower eas corresod to the dected objects. Smlarly, Fgure 5(b), the hghest

15 Fuzzy clusterg of tutostc fuzzy data 59 ea corresods to the marble of the acet moumet ad the lower eas corresod to the sy rego. s regards the o-membersh fuctos, a tutve terretato could be gve by cosderg ther correlato wth the corresodg membersh fuctos. The correlato s usually less aroud the eas that corresod to less homogeeous mage regos. For examle, Fgure 5(b), the absolute correlato betwee the membersh ad the o-membersh fucto estmated for the rego of the acet moumet s 66%, whereas for the rego of sy s 78%. Smlarly, the absolute correlato betwee the membersh ad the o-membersh fuctos Fgure 5(a) (c), for the homogeeous whte bacgroud regos reaches 96.5%. Clusterg exermets were coducted wth all ossble class combatos, usg the roosed clusterg algorthm wth the tutostc fuzzy data FCM wth crs I 2 -hstogram data FCM wth fuzzy I 2 -hstogram data. I all the exermets, the same arameters (ε = , m = 2.0) ad talsato codtos were used. The clusterg erformace was evaluated terms of classfcato accuracy, algorthm teratos ad absolute executo tme. Classfcato accuracy was comuted as the orgal FCM algorthm, by assgg a mage to the cluster wth the hgher degree of membersh value. The exermets were executed o a PC wth Itel Petum M at.86 GHz, 52 MB RM ad 60 GB hard ds. The results are summarsed Fgure 5. Fgure 4 Membersh ad o-membersh fuctos corresodg to the mages of Fgure 3. The horzotal axes rereset the values of I 2 ormalsed wth the rage [0, 255], whereas the vertcal axes have bee rescaled to mrove the vsblty of the grahs. The grahs focus o the regos of the membersh ad o-membersh fuctos for whch the varace s hgher. The les that tersect the frame of the grahs extedg beyod the vsble area jo to ea membersh ad o-membersh values (a)

16 60 N. Peles et al. Fgure 4 Membersh ad o-membersh fuctos corresodg to the mages of Fgure 3. The horzotal axes rereset the values of I 2 ormalsed wth the rage [0, 255], whereas the vertcal axes have bee rescaled to mrove the vsblty of the grahs. The grahs focus o the regos of the membersh ad o-membersh fuctos for whch the varace s hgher. The les that tersect the frame of the grahs extedg beyod the vsble area jo to ea membersh ad o-membersh values (cotued) (b) (c)

17 Fuzzy clusterg of tutostc fuzzy data 6 Fgure 4 Membersh ad o-membersh fuctos corresodg to the mages of Fgure 3. The horzotal axes rereset the values of I 2 ormalsed wth the rage [0, 255], whereas the vertcal axes have bee rescaled to mrove the vsblty of the grahs. The grahs focus o the regos of the membersh ad o-membersh fuctos for whch the varace s hgher. The les that tersect the frame of the grahs extedg beyod the vsble area jo to ea membersh ad o-membersh values (cotued) (d) Fgure 5 Comaratve results of usg the roosed clusterg algorthm wth the tutostc fuzzy data, ad of usg the FCM wth the crs ad wth the fuzzy data as ut: (a) classfcato accuracy; (b) umber of teratos requred for the clusterg algorthms to coverge ad (c) executo tme requred secods (a)

18 62 N. Peles et al. Fgure 5 Comaratve results of usg the roosed clusterg algorthm wth the tutostc fuzzy data, ad of usg the FCM wth the crs ad wth the fuzzy data as ut: (a) classfcato accuracy; (b) umber of teratos requred for the clusterg algorthms to coverge ad (c) executo tme requred secods (cotued) (b) (c) Fgure 5(a) shows that all the exermets, the accuracy acheved by the roosed algorthm was hgher tha the accuracy obtaed by FCM for four or three classes. The maxmum accuraces acheved wth the roosed algorthm are 74.4% ad 93.3% for four ad for three classes, resectvely. These ercetages reduce to 64.4% ad 79.2%, the case of FCM clusterg wth fuzzy data. The results of the clusterg exermets erformed wth data from two classes show that the accuracy of the roosed algorthm ca be cosdered comarable wth or hgher tha, the accuracy obtaed by FCM. However, ths could be attrbuted to a smaller cotrbuto of the o-membersh values to the reresetato of the mages of the artcular classes. The maxmum accuracy obtaed by both algorthms reached 00% two cases (BC ad BD). Comarg the two algorthms terms of effcecy, Fgure 5(b) ad (c) show that the roosed algorthm has a cosderable advatage over FCM, as t requres less algorthm teratos ad most cases less tme to reach covergece. The average mrovemet absolute executo tme s 63 ± 27%.

19 Fuzzy clusterg of tutostc fuzzy data 63 7 Coclusos ad ersectves We reseted a ovel aroach to fuzzy clusterg of data sets roduced the cotext of tutostc fuzzy set theory. More secfcally, We roosed a ovel varat of the FCM clusterg algorthm that coes wth ucertaty the localsato of feature vectors owg to mrecse measuremets ad ose. We troduced a tutostc fuzzy reresetato of colour dgtal mages as a aradgm of tutostc fuzzfcato of data. We defed a ovel smlarty measure betwee tutostc fuzzy sets ad roved ts suerorty over other metrcs. Ths measure was cororated the roosed FCM varat. We have coducted a comrehesve set of exermets ad based o the results, t ca be cocluded that the roosed clusterg algorthm ca be more effcet ad more effectve tha the well-establshed FCM algorthm, esecally as the umber of clusters creases. Future ersectves of ths wor clude: Systematc evaluato of the roosed scheme comarso wth other clusterg schemes for the clusterg of varous ds of data sets after arorately reresetg them terms of tutostc fuzzy sets theory; t s worth otg that curretly, o referece tutostc fuzzy data set s avalable to bechmarg clusterg algorthms. Ehacemet of the roosed clusterg scheme so as to tae to accout ot oly the membersh, but also the o-membersh of each data vector to a cluster. cowledgemets Ths research wor s suorted by the roject MetaO, fuded by the Oeratoal Programme Iformato Socety of the Gree Mstry of Develomet, Geeral Secretarat for Research ad Techology, co-fuded by the Euroea Uo. The collecto of mages used ths study s courtesy of the Foudato of the Hellec World (FHW), htt:// Refereces taassov, K.T. (986) Itutostc fuzzy sets, Fuzzy Sets ad Systems, Vol. 20, taassov, K.T. (989) More o tutostc fuzzy sets, Fuzzy Sets Systems, Vol. 33, taassov, K.T. (994a) New oeratos defed over the tutostc fuzzy sets, Fuzzy Sets ad Systems, Vol. 6, taassov, K.T. (994b) Oerators over terval valued tutostc fuzzy sets, Fuzzy Sets Systems, Vol. 64,

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21 Fuzzy clusterg of tutostc fuzzy data 65 Yog, Y., Chogxu, Z. ad Pa, L. (2004) ovel fuzzy c-meas clusterg algorthm for mage thresholdg, Measuremet Scece Revew, Vol. 4, Sec.,. 9. Zadeh, L.. (965) Fuzzy sets, Iformato Cotrol, Vol. 8, Zhzhe, L. ad Pegfe, S. (2003) Smlarty measures o tutostc fuzzy sets, Patter Recogto Lett., Vol. 24, Webste FHW, Foudato of the Hellec World, htt://