Combining Fuzzy Partitions Using Fuzzy Majority Vote and KNN

Transcription

1 JOURNAL OF COPUTERS, VOL. 5, NO. 5, AY Cobnng Fuzzy Parons Usng Fuzzy aory Voe and KNN Chun sheng L Yaonan Wang Hadong Yang Deparen of aheacs and Copuaona Scence, Guang Dong Unersy of Busness Sudes, Guangzhou, Chna, 500, Ea: cs58084@sna.co Coege of Eecrca and Inforaon Engneerng, Hunan Unersy, Changsha, Chna, 4008 Coege of Auoaon Scence and Engneerng, Souh Chna Unersy of Technoogy, Guangzhou, Chna, 5065 Absrac hs paper frsy generazes aory oe o fuzzy aory oe, hen proposes a cuser achng agorh ha s abe o esabsh correspondence aong fuzzy cusers fro dfferen fuzzy parons oer a coon daa se. Fnay a new cobnaon ode of fuzzy parons s bud on he bass of he proposed cuser achng agorh and fuzzy aory oe. Coparae resus show ha he proposed cobnaon ode s abe o foser srenghs and crcuen weaknesses of coponen fuzzy parons and o cobne he coponen fuzzy parons no a beer fuzzy paron han any of coponen fuzzy parons and hose resued fro wo curren cobnaon odes of fuzzy parons. Inde Ters fuzzy oe, fuzzy aory oe, cobnaon of fuzzy parons, eauaon of fuzzy paron I. INTRODUCTION Fuzzy cuserng has been proed preferabe o crsp cuserng and a nuber of fuzzy cuserng agorhs [-4] hae been proposed. Howeer, dfferen fuzzy cuserng agorhs ay produce dfferen fuzzy parons oer he coon daa se, and none of he are unersa enough o perfor equay we n any cases. For eape, FC [] perfors we on noseess daase wh hyper-spherca shape, G-k [] agorh on noseess daase wh hyper-epsoda shape and boh AFC[] and PFC[4] are sar o FC ecep ha hey are robus o noses. For a fe daase ay be of dfferen shapes, herefore no snge fuzzy cuserng agorh can accuraey dscoer s srucure and akes soe errors. Howeer he errors ade by dfferen fuzzy cuserng agorhs woud no necessary oerap. Ths suggess ha dfferen cuserngs poenay offer copeenary nforaon abou he paerns o be paroned, whch coud be harnessed o proe he perforance of paern recognon syses. Therefore, A prosng drecon for accurae dscoery of he daa srucure ay be o cobne derse fuzzy parons no a consodae one, whch s epeced o erge adanages of upe canddae fuzzy cuserngs no one whoe. Sar probes assocaed wh crsp cuserngs hae been suded eensey and here s an eense body of work on cobnng upe crsp cuserngs [5-8]. Howeer, he opc of cobnng fuzzy cuserngs has no receed he sae aenon. Egena Dradou[9] proposes a cobnaon schee for fuzzy cuserngs ha as o fnd a consensus fuzzy paron whch opay represens he se of coponen fuzzy cuserngs oer he sae daa se. A.D. Gordon [0] aso presens a cobnaon ode ha as o denfy a consensus fuzzy paron whch cosey fs he se of coponen fuzzy parons oer he sae daa se. Howeer, no heory guaranees ha a consensus fuzzy paron represenng or fng a se of fuzzy parons can represen or f he rea srucure of he daa se. The curren paper aso addresses he probe of cobnng fuzzy parons wh he sae nuber of cusers oer he sae daa se. There are wo dffcu probes n cobnng upe fuzzy parons. One s o esabsh he correspondences aong cusers of he coponen fuzzy parons so ha he frs cuser of one paron eans he sae as ha of anoher one, so s he second cuser and so on, he oher probe s o desgn he rue of cobnng upe fuzzy parons. To soe he frs probe, Egena Dradou [9] frs buds up he confuson ar beween he consensus fuzzy paron ha s nazed by one of he coponen fuzzy parons and he coponen fuzzy paron, hen he frs wo cusers assocaed wh he frs au eeen of he confuson ar correspond o each oher, so do he second wo cusers assocaed wh he second au eeen of he confuson ar, and so on. Snce he na consensus fuzzy paron s randoy seeced fro he se of coponen fuzzy parons and hen updaed by each of he oher coponen fuzzy parons sep by sep, he resuan consensus fuzzy paron suffers fro boh he na consensus fuzzy paron and he order of he coponen fuzzy paron o ake par n updang he consensus fuzzy paron. Unke Egena Dradou[9], A.D. Gordon[0] frs buds up he dssary ar beween he consensus fuzzy paron ha s nazed randoy and each of he coponen fuzzy parons, hen reas he probe of cuser correspondence as he probe of assgnen and soes by Hungaran ehod[]. The resuan consensus fuzzy paron suffers fro he nazaon of he consensus fuzzy paron. To oercoe he sensy of he aboe approaches o he na consensus fuzzy paron n achng cusers fro dfferen fuzzy parons, we ransfor he 00 ACADEY PUBLISHER do:0.404/cp

2 79 JOURNAL OF COPUTERS, VOL. 5, NO. 5, AY 00 probe of esabshng he correspondence aong he cusers of coponen fuzzy parons no he probe of paronng her cuser ceners so ha he cuser ceners n he sae cuser correspond o each oher. The second probe s soed by generazng he aory ong rue for ensebe of crsp parons o he fuzzy aory ong rue for ensebe of fuzzy parons. Based on hs, a new cobnaon ode of fuzzy parons s bud, and s perforance are suded nensey by suaon eperens. The res of hs paper s organzed as foows. In secon II he reaed work s reewed. The radona aory ong rue s generazed o he fuzzy aory ong rue n secon III. An agorh for achng cusers of dfferen fuzzy parons s proposed n secon IV. A new cobnaon ode of fuzzy parons s proposed n secon V.Nuerca eperens and concusons are gen n secon VI and VII, respecey. II. Reaed Work A The Vong Agorh [9] The an dea of eraure [9] s o fnd paron P of a gen daa se X={,,, N } wh g cusers whch opay represens a gen se of parons of X. Each of hese parons s represened by an N g ebershp ar U h (h=,,, ). The fna paron P s encoded as an N g ar. The h eeenu of U h s he degree of ebershp of o he -h cass of he h-h paron. We denoe he -h row h of U h as u, ha s u h s he ebershp ecor of he paern for he paron U h. The fna paron P s encoded as a N g ar wh eeens p and rows p. The ask of fndng an opa paron s gen by he nzaon probe: n U, U,, U, P P () N h n n u p h,, PN,, h N s any peruaon of he couns of U h. P Where h U h Ths nzaon probe s soed by he ong agorh[9], whch s descrbed n Tabe I. Tabe Vong Agorh [9] Sep se P () =U and ˆ d (d eans denca peruaon); Sep for = o ˆ of ' ˆ U U ar P (a) copue he souon ' U a r by he foowng approaon agorh: ()bud up he confuson ar beween P and U; ()fnd he au eeen n hs confuson ar; ()assocae he wo cusers correspondng o he au eeen; (4)reoe hese wo cusers; (5)wh he reduced confuson ar go o (); (b) copue he ong resu P () afer runs as P P ˆ U P () denoes he ong resu afer he frs seps. B A ode for Fng a Fuzzy Consensus Paron o a Se of ebershp Funcons [0] Ths ode denfes he coses consensus fuzzy paron P fng s ebershp funcon ar U o he ebershp funcon arces {U h (h=,,,)}, ha hae been perued o bes ach g casses of P h wh g casses of P. The coses consensus fuzzy paron P of {P h (h=,,,)} can be obaned by song he foowng probe n he neger arabes Y h =[y hp ] and nonnegae ebershp funcons U u : r [P] n FY, Y,, Yr, U w h h UhYh U () N g g h w h p h up u y hp Subec o he consrans g y g h p hp,,, ;,, () y p g h hp,,, ;,, (4) y hp 0, p,,,, g; h,, (5) 0,,, N;,,, g (6) g N,,, (7) The consraned probe [P] can be nzed by eans of he aernang eas-square agorh (ALS) descrbed n Tabe II, ha aernaes beween nzng F(Y, Y,,Y, U ) wh respec o {Y h (h=,,,)} gen he curren esae of he edan ebershp funcon ar U ; and nzng F(Y, Y,,Y, U ) wh respec o U gen he curren achng of casses beween Y h and U (h=,,, ). Tabe The Aernang Leas-square Agorh (ALS) Sep Gen he esaes of he edan ebershp funcon ar U, new eas-squares esaes of he eeens of Y h (h=,,, ) can be deerned by song ndependen achng probes: N g n U, U, Y g h [Pa] h h u u p p h,,, Subec o consrans (), (4) and (5). ([Pa] can be effceny soed usng he we-known Hungaran ehod [] n O(g ) e copey). Sep reang he eeens of Y h (h=,,, ) as consrans, s necessary o soe: [Pb]: n F(Y, Y,, Y, U ) Subec o consrans (6) and (7). The souon s gen by g h u w u y w,,, N; g h p h p hp h h,,, C aory Vong Rue (AJ) Ths rue does no requre he a poseror oupus for each cass, and each cassfcaon ges ony one crsp cass oupu as a oe for ha cass. Then, he ensebe oupu s assgned o he cass wh he au nuber of oes aong a casses. For any sape X, for a group of cassfcaon n a g-cass probe, we denoe he decson of abe oupus for fro cassfcaon f() s c(), c() g. Seera ernooges are defned n he foowng. Defnon For a sape X, he crsp oe d, () for cass gen by cassfcaon f() s defned as y p 00 ACADEY PUBLISHER

3 JOURNAL OF COPUTERS, VOL. 5, NO. 5, AY c d (8) 0 c d, = eans ha cassfcaon f() oes for cass, and d, =0 eans ha cassfcaon f() oes agans cass. Defnon For a sape X, he dscrnang g for cass ( g) s defned as funcon g d (9) The dscrnang funcon represens he oa nuber of oes gen o a cass by a cassfcaons. The hgher aue of he dscrnang funcon g ndcaes ore suppors for cass. Consequeny, he oupu of an ensebe of cassfcaons s he cass abe wh he au aue of he dscrnang funcon. k g (0) arg a g III Fuzzy aory Vong Rue The defnon of crsp oe d, () ndcaes ha a crsp paron f() eher suppors or denes uery ha paern beongs o cass. Snce a fuzzy paron consders ha any paerns beong o a cusers wh dfferen ebershp degrees, he crsp oe has o be resed before s apped o fuzzy parons. Snce a crsp cuserng s a speca case of fuzzy cuserng, Defnon ndcaes ha he crsp oe d, () s acuay he ebershp degree of paern beongng o cass. Fro hs pon of ew he naura way of generazng a crsp oe o a fuzzy oe s o defne he fuzzy oe d gen o paern by he fuzzy cuserng f as he fuzzy ebershp degree u of paern beongng o cass dered fro he fuzzy cuserng f. Ths yeds ha he consensus fuzzy parons s he ean of a he coponen fuzzy parons, whch s he opa represenaon of a he coponen fuzzy parons, us as Egena Dradou ec a saed n he eraure[9]. The reason can be found n rearks a he end of hs secon. Howeer eperens n secon VI show ha he ean of a he coponen fuzzy parons s no sure o represen he rea srucure of he daa se. Consderng hs, we do no spy defne he fuzzy oe d as he fuzzy ebershp degree u, bu rea he casses dffereny, ha s, we drecy defne d u for cass arg a u, bu for oher casses, we g defne d u u k k a, k,,, g and k. g Eperens n secon VI show ha hs defnon of fuzzy oe yeds reaey good consensus fuzzy paron. The foowng defnons are he fuzzy counerpars of defnons -. Defnon for a paern X, he fuzzy oe gen o cass by he fuzzy paron U u s Ng defned as u arg au g d () u u a oherwse g Where N s he nuber of paerns, g he nuber of cusers and u he ebershp degree of paern beongng o cuser. Conrary o he crsp oe, he fuzzy oe ndcaes ha a fuzzy paron neher suppors nor denes uery ha a paern beongs o a cuser, bu suppors beongs o a cusers o dfferen eens. I s obous ha he crsp oe s he speca case of he fuzzy oe. Defnon 4 for a paern X, he fuzzy dscrnang funcon for cass ( g) s defned as g d () Where s he nuber of coponen fuzzy parons. Lke he dscrnang funcon defned by forua (9), he fuzzy dscrnang funcon of a cass aso represens he aoun of suppors gen o by a fuzzy parons. Hgher aue of he fuzzy dscrnang funcon g eans ore suppors for paern beongng o cass. Unke spe aory ong rue, nsead of assgnng a cass abe o paern, we cacuae he ebershp degree cu () of beongng o each cass, g, deerned by fuzzy parons ony as foows g cu g g g, () Forua () ndcaes ha he cobnaon of fuzzy parons s s a fuzzy paron. Ths s dfferen fro he consensus crsp paron. If forua () s repaced k g, s obous ha he wh arg a g aory ong rue s he speca case of he fuzzy aory ong rue. In he foowng we eepfy he fuzzy aory ong rue. Supposng ha here are hree coponen fuzzy parons oer he sae daa se, each of whch has hree cusers and s denoed by s fuzzy paron ar U () (=,, ). In he case ha he correspondence aong cusers fro he coponen fuzzy parons s esabshed,.e., he frs coun of he fuzzy coponen parons represens he sae cass, so are he second and hrd coun. Gen a paern, he ebershp degree of beongng o each cuser dered fro U () (=,,) s u,, u u 0.894,0.6894,0. u, u, u 0.487,0.76, 0.05 u, u, u 0.57,0.474,0.70 Defnon yeds, d, d ,0.6894,0.076 d. 00 ACADEY PUBLISHER

4 794 JOURNAL OF COPUTERS, VOL. 5, NO. 5, AY 00 d, d, d 0.487, 0.605, d, d, d 0.8,0.474,0.45 Foruar () ges g, g, g 0.60,.4, Forua () resus n (cu (), cu (), cu ())=(0.66, , 0.56). Rearks: f d u, hen g cu g g u g u u g u u IV A Cuser achng Agorh Based on KNN When no a pror cass nforaon for he paerns s aaabe, a drec appcaon of fuzzy aory ong rue o cobnng fuzzy parons s no possbe, for s no edaey cear whch cuser fro a specfc paron corresponds o wha n anoher. Therefore, s necessary o esabsh he correspondence aong cusers of a coponen fuzzy parons so ha he sae coun of U u,,,, defnes he N g sae cuser. Boh Egena Dradou[9] and A.D. GORDON[0] esabsh hs knd of correspondence by correspondng he cusers of each coponen fuzzy paron wh hose of he consensus fuzzy paron, whch s frsy nazed randoy or wh one of he coponen fuzzy parons, hen updaed adapey. They suffer fro he nazaon of he consensus fuzzy paron.the underyng dea of esabshng he correspondence beween cusers of one fuzzy paron and hose of anoher s ha he sar cusers correspond o each oher so ha he su of dssares beween wo fuzzy parons s nzed, as shown n he eraures [9, 0]. Inspred by hs dea, we ransfer he probe of parng cusers fro dfferen fuzzy parons no he probe of paronng he se of cuser ceners. Supposng ha here are fuzzy parons U, U,, U, each of whch has g cusers. Each cuser s represened by s cener. Ths yeds a se of cuser ceners, V,, g,,, g,,,, g.where s he -h cuser cener of he -h fuzzy paron. We defne he sary beween wo cusers as he Eucdan dsance beween her ceners. Consequeny, esabshng he correspondence aong he cusers of fuzzy parons s ransferred no paronng he se V of cener ecors no g cusers, each of whch conans cener ecors fro dfferen fuzzy parons. The cener ecors beongng o he sae cuser correspond o each oher,.e., f,,, beong o he sae cuser, hen he -h cuser ofu, he -h cuser of U,, he -h cuser of U defne he sae cuser, where s,,,, s,,, g. To assure he cener ecors n he sae cuser are fro dfferen fuzzy parons, we defne he dssary beween wo cuser ceners as s s s d,, (4), s, The K neares neghbors ehod (KNN) s epoyed o paron he daa se V no g cusers, each of whch conans cener ecors. Consequeny, an approach o esabshng he correspondence aong he cusers fro dfferen fuzzy parons s deeoped, whch s descrbed by he pseudo code n Tabe. Tabe The Cuser-achng Agorh Based on KNN Copue he cenre ecors of each fuzzy paron by k u X k u X k, k,, g;,, s Copue dssary d,,,,, g,, s,, usng (4); Fnd neares neghbours for each cenre ecor,,, g;,,. They for a -neares neghbourhood denoed by ; 4Copue he copacness cop( ) of each -neares neghbourhood by cop d,, ; 5Seec g os copac and dson -neares neghbourhoods. The cenre ecors beongng o he sae -neares neghbourhood represen he sae cuser and he couns correspondng o he n U () (=,,,) are abeed as he sae cass abe. In he foowng we eepfy he proposed cuserachng agorh. Supposng ha here are hree coponen fuzzy parons, each of whch has hree cusers. Ther cuser ceners are sed n Tabe 4. The dssary beween any par of cuser ceners s dered fro forua (4). The -neares neghbourhood of each cener ecor and s copacness are sed n Tabe 4, where =. The se of nne cuser cener ecors s paroned no hree dson cusers:,,,,, and,,. The cuser,, eans ha he frs cuser of he frs fuzzy paron, he second cuser of he second paron and he second cuser of he hrd fuzzy paron defne he sae cuser, so do,, and,,. V A Cobnaon Schee for Fuzzy Parons Usng Fuzzy aory Vong Rue and KNN Supposng ha here are fuzzy parons oer he daa se X, each of whch s denoed by a fuzzy paron ar U (),=,,,. They are ached by he cuser achng agorh n Tabe, hen he we 00 ACADEY PUBLISHER

5 JOURNAL OF COPUTERS, VOL. 5, NO. 5, AY ached fuzzy parons are cobned no a consensus fuzzy paron usng fuzzy aory ong rue. The pseudo descrpon of he proposed cobnaon ode s gen n Tabe 5. Tabe The Cuser Ceners of Three Coponen Fuzzy Parons Tabe 4 he -neares neghbourhood of each cuser cener and s copacness -neares copacness neghbourhood {,, } 0.87*0-5 {,, } *0-5 {,, } 0.59*0-5 {,, } *0-5 {,, } 0.87*0-5 {,, } 0.59*0-5 {,, } *0-5 {,, } 0.87*0-5 {,, } 0.59*0-5 Tabe 5 A Cobnaon ode of Fuzzy Parons Based on Fuzzy aory Vong Rue and KNN Sep npu he fuzzy paron arces U () (=,,, ) and paern sapes X; Sep esabsh he correspondence aong cusers fro fuzzy parons usng he cuser-achng agorh n Tabe ; Sep cobne fuzzy parons usng fuzzy aory ong rue descrbed n secon III; V Eperens An poran consderaon n he cobnaon of parons s ha uch beer resus can be acheed f derse parons, raher han sar parons, are cobned. To creae derse fuzzy parons we epoy hree fuzzy cuserng agorhs FC[], PFC[] and AFC[], each of whch (ecep PFC[] ha s nazed by he oupu of FC) s nazed by hree cenre nazaon ehods CCIA[], kd-ree[] and ST[4], respecey. Therefore, here are oay nne fuzzy parons denoed by FC-CCIA, FC-ST, FC-kd-ree, PFC-CCIA, PFC-ST, PFC-kdree, AFC-CCIA, AFC-ST, AFC-kd-ree, respecey. They are cobned no a consensus fuzzy paron, denoed by FV, by he cobnaon ode n Tabe 5, n he way depced n Fg.. To es he perforance of he proposed cobnaon ode, we copare wh wo cobnaon ehods ong [9] and ALS[0] on four rea daa ses, whch are descrbed n Tabe 6. For a fuzzy cuserng agorhs we use he foowng Copuaona Proocos; conergence er ε=0.000, au nuber of eraons=00, he Fg. he fowchar of cobnng fuzzy parons (FV (=,, ) eans he cobnaon resus of he frs e and FV he fna resu of cobnaon.) Pa-Indansdabee [5] sgude[6] onosphere[6] Sa.age[6] Tabe 6 The bref descrpon of daa ses Nuber of nsances Nuber of arbues Nuber of cusers fuzzfer =. The paraeers of PFC are nazed as foows: =, η=.5, a=, b=. ALS suffers fro he nazaon of he consensus fuzzy paron. We naze wh each of nne coponen fuzzy parons respecey. The ong [9] agorh suffers fro he sequence of he coponen fuzzy parons o ake par n he cobnaon of fuzzy parons. We pace nne coponen fuzzy parons n he order of FC-CCIA, FC-ST, FC-kd-ree, AFC-CCIA, AFC-ST, AFC-kd-ree, PFC-CCIA, PFC-ST, PFC-kdree, hen naze he consensus fuzzy paron P () wh each of he aboe nne fuzzy parons, respecey and f ohers n her paces. We eauae he fuzzy paron usng paern recognon rae PR ha s a sandard eauaon nde, paron coeffcen PC[8] ha easures he fuzzy degree of fuzzy parons, fuzzy Rand ndee R and reaed ndees-fuzzy Jaccard coeffcen JC, fuzzy Fowkes-aows nde, fuzzy nkowsk F easure and fuzzy sasc [7], whch are obece crera for he eauaon of fuzzy parons, as R. J. G. B. Capeo saed [7]. The bg aues of he ndees R, JC, F and ndcae he good 6 00 ACADEY PUBLISHER

6 796 JOURNAL OF COPUTERS, VOL. 5, NO. 5, AY 00 coseness beween he reference paron and he fuzzy paron o be eauaed, whe he ow aue of reeas good coseness beween he. To es he perforance of he proposed cobnaon ode of fuzzy parons, we copare wh a coponen fuzzy parons and wo consensus fuzzy parons, ong [9] and ALS[0], oer a sa sze and a arge sze daa se, onosphere [5] and sa.age [6]. We aso copare wh ony wo consensus fuzzy parons, ong [9] and ALS [0], oer wo dde sze daa ses, dabees and sgude. The coparae resus sed n Tabe 7-9 ndcae ha, n ers of a eauaon ndees for fuzzy parons, he proposed cobnaon ode FV ouperfors wo consensus fuzzy parons, ong [9] and ALS [0]. Tabe 7 and 9 show ha, n ers of paern recognon rae, he proposed cobnaon ode FV s coparabe o he bes coponen fuzzy parons FC-CCIA, FC-ST and FC-kdree, whe n ers of oher ndees, FV s preferabe o he oer he daa ses onosphere and sa. age, respecey. Tabe 0 ndcaes ha, n ers of CPU e, ong [9] s he cheapes, and FV s a e ore epense copuaon han ALS [0]. In a word, he proposed cobnaon ehod FV s abe o foser srenghs and crcuen weaknesses of coponen fuzzy parons and ouperfors ong [9] and ALS [0] a he cos of a e era copuaon n our eperens. I s poran for he consensus fuzzy paron o be coparabe, bu unceran o be preferabe, o he bes coponen fuzzy paron n any cases, for no fuzzy cuserng agorh can generaes good parons n a cases and we do no know whch fuzzy cuserng agorh ay produce a good cuserng n adance. Furherore, we do no know how o accuraey assess a fuzzy paron, uch ess seec he bes nddua fuzzy paron when no nforaon abou he daa se s aaabe. In hs sense, he consensus fuzzy paron obaned fro he proposed cobnaon ode s ore sabe and reabe han any coponen fuzzy paron, for can cobne he adanages of a coponen fuzzy parons and poo he no a consensus fuzzy paron ha s unceran o be beer han any coponen fuzzy paron, bu sure o be superor o an oerwheng aory of he coponen fuzzy parons n any cases. To copare FV wh ong [9], ALS [0] and coponen fuzzy parons, we es he oer he daa se onosphere. Tabe 7 shows ha a eauaon ndees agree wh our consensus fuzzy paron FV ouperfors ALS and ong a e. Copared o nne nddua fuzzy parons, n ers of paern recognon rae, he consensus fuzzy paron generaed by he proposed cobnaon ode s as good as he bes coponen fuzzy paron FC-CCIA, FC-ST and FC-kdree, beer han oher s coponen fuzzy parons. Tabe 7 aso shows ha oher eauaon ndees agree wh ha FV ouperfors a he coponen fuzzy parons. In genera, he proposed consensus fuzzy paron FV s abe o cobne he adanages of a coponen fuzzy parons and a eas coparabe o he bes coponen fuzzy paron. Conrary o FV, n ers of paern recognon rae, ALS and ong are worse han he bes coponen fuzzy paron. Furherore, ALS and ong are worse han par of he coponen fuzzy parons n ers of oher eauaon ndees. In a word, ALS and ong fa o cobne he adanages of he coponen fuzzy parons and are nferor o he bes coponen fuzzy parons. I s aso reeaed by Tabe 7 ha when he paern recognon raes of FV, FC-ST and FCkdree are equa, oher ndees for fuzzy parons s can dsngush he hree fuzzy parons. Ths ndcaes ha s no suffcen o eauae fuzzy parons ony by paern recognon rae and oher eauaon ndees for fuzzy parons are aso powerfu oos for assessng fuzzy parons. To furher copare he perforances of FV, ALS and ong, we es he oer wo dde sze daa ses, dabees and sgude. Tabe 8 shows ha FV Tabe 7 he coparae resus aong consensus fuzzy parons and nddua fuzzy parons oer onosphere Fuzzy parons PR(%) PC R JC F FV ALS ean sd ong ean sd FC-CCIA FC-ST FC-kdree AFC-CCIA AFC-ST AFC-kdree PFC-CCIA PFC-ST PFC-kdree The bod nuber eans he opa aue of eauaon nde ouperfors ALS and ong by huge argns n ers of paern recognon. The paron coeffcens PC sed n Tabe 8 show ha ALS and ong aerage confcng ebershps owards /c (c=) oer he daa se dabees. The fuzzy rand nde and reaed ndees sed n Tabe 8 agree wh ha FV s a e beer han ALS and ong. Tabe 8 aso shows ha ALS s as bad as ong n ers of a eauaon ndees. Tabe 8 he coparae resus aong consensus fuzzy parons oer wo sa sze daa ses wh wo cusers 00 ACADEY PUBLISHER

7 JOURNAL OF COPUTERS, VOL. 5, NO. 5, AY Daa se dabees sgude Consensus fuzzy parons PR(%) PC R JC F FV ALS ean sd ong ean sd FV ALS ean sd ong ean sd To furher nesgae he perforance of FV, we copare wh wo consensus fuzzy parons, ong [9] and ALS [0], and a he coponen fuzzy parons oer a arge sze daa se wh s cusers. Tabe 9 shows ha hree consensus fuzzy parons, FV, ong [9] and ALS [0], and hree nddua fuzzy parons, FC-CCIA, FC-ST and FC-kdree, are ery cose o each oher and FV s a e beer han he bes nddua fuzzy paron n ers of paern recognon rae. In ers of oher eauaon ndees, ong [9] and ALS [0] are nferor o FC-CCIA, FC-ST, FCkdree, PFC-CCIA and PFC-ST, whe FV s superor o a coponen nddua fuzzy parons, as shown by Tabe 9. Ths eperen once ore confrs ha FV s abe o cobne he adanages of coponen nddua fuzzy parons and a eas coparabe o he bes coponen fuzzy paron n he case of arge sze daa ses wh upe cusers, whe ong [9] and ALS [0] can no. Tabe 9 he coparae resus aong consensus fuzzy parons and nddua fuzzy parons oer Sa. age Fuzzy parons PR(%) PC R JC F FV ALS ean sd Vong ean sd FC-CCIA FC-ST FC-kdree AFC-CCIA AFC-ST AFC-kdree PFC-CCIA PFC-ST PFC-kdree The perforance of an agorh s one poran aspec and he copuaona copey s anoher poran aspec of he agorh. So he copuaona copees of he proposed cobnaon ode FV, Vong[9] and ALS[0] are aso copared n ers of CPU es. The coparae resus sed n Tabe 0 show ha Vong[9] s of he cheapes copuaon and FV s a e ore epense copuaon han ALS[0]. Tabe 0 Copuaon Copey of Three Cobnaon ehods Daa se CPU e (second) FV ALS ong onosphere sgude dabees Sa. age The copuer syse s of Genune Ine CPU 40,.60GHz and.60ghz, GB eory. CPU e s ony coposed of he runnng e of FV, ALS and ong, bu no ha of any nddua fuzzy cuserng agorh. VII CONCLUSIONS Ths paper generazes he radona aory ong rue o he fuzzy aory ong rue and proposes a cuser achng agorh, based on whch a cobnaon ode of fuzzy parons s deeoped. We copare our cobnaon ehod wh oher wo cobnaon ehods - ong [9] and ALS [0] and nddua fuzzy parons. Coparae resus show ha our cobnaon ehod ouperfors ong [9] and ALS [0] n ers of a eauaon ndees used n hs paper. The reason ay be ha boh ong [9] and ALS [0] a o fnd he consensus fuzzy paron ha opay represens and cosey fs he se of coponen fuzzy parons, and he opa represenaon and fng of a se of fuzzy parons do no equa he opa represenaon of he rea srucure of he daa se, ha s, f he consensus fuzzy paron opay represens or fs a coecon of fuzzy parons, does no guaranee o represen he rea srucure of he daa se. We aso fnd ha ong [9] and ALS [0] are a e worse han soe of he coponen nddua fuzzy parons, whe FV s a eas coparabe o he bes one of he coponen nddua fuzzy parons n a cases. Ths confrs ha FV s abe o foser srenghs and crcuen weaknesses of coponen fuzzy parons, whe ong [9] and ALS [0] can no. In a word, FV s no ony superor o ong [9] and ALS [0], bu aso ore sabe and reabe han any nddua fuzzy paron n soe cases. 00 ACADEY PUBLISHER

8 798 JOURNAL OF COPUTERS, VOL. 5, NO. 5, AY 00 I s s poran for he consensus fuzzy paron o be coparabe o, bu no sure o ouperfor, he bes coponen fuzzy paron n any cases, for no fuzzy cuserng agorh can generaes good parons n a cases and we do no know whch fuzzy cuserng agorh ay produce a good cuserng oer a gen daa se n adance. Furherore, when no nforaon abou he daa se s aaabe, s hard o us accuraey eauae he fuzzy paron, uch ess pck ou he bes nddua fuzzy paron. In hs sense, he consensus fuzzy paron s ore sabe and reabe han any coponen nddua fuzzy paron, for s abe o cobne upe fuzzy parons no a consodae one ha s a eas coparabe o he bes coponen fuzzy parons n any cases. Acknowedgeens The proec s suppored by he Naona Naura Scence Foundaon under Gran No and he Scenfc Research Foundaon for Docor Gran No RERENCES [] C. Bezdek, Paern Recognon wh Fuzzy Obece Funcon Agorhs, New York: Penu Press, 98. [] D. E. Gusafson, W. C. Kesse, Fuzzy cuserng wh a fuzzy coarance ar, n Proc. IEEE Conf. Decson Conr., San Dego, CA, pp , 979. [] Kuo-Lung Wu, n-shen Yang, Aernae C-eans cuserng agorh, Paern Recognon, o.5,, pp.67-78, 00. [4] Nkh R. Pa, Kuhu Pa, Jaes. Keer, Jaes C. Bezdek, a possbsc fuzzy c-eans cuserng agorh, IEEE Trans. On Fuzzy syses, o.(4), pp57-50, 005. [5] Ana L.N. Fred, An K. Jan, Cobnng upe Cuserngs Usng Edence Accuuaon, IEEE Transacons on paern anayss and achne negence, Vo. 7(6), pp85 850, 005. [6] Aeander Topchy, An K. Jan and Wa Punch, Cuserng ensebes: odes of consensus and weak parons, IEEE Transacons on paern anayss and achne negence, VOL. 7(), pp866 88, 005. [7] Tanng Hu, Yng Yu, Jnzh Xong and Sa Yuan Sung, au kehood cobnaon of upe cuserngs, Paern Recognon Leers 7, pp , 006. [8] Aeander Topchy, An K. Jan, and Wa Punch, Cobnng upe Weak Cuserngs, Proceedngs of he Thrd IEEE Inernaona Conference on Daa nng (ICD 0). [9] Egena Dradou, Andreas Wengesse, and Kur Hornk, A cobnaon schee for fuzzy cuserng, Inernaona Journa of Paern Recognon and Arfca Inegence, Vo.6(7), pp.90-9, 00. [0] A.D. GORDON,. VICHI, Fuzzy paron odes for fng a se of parons, Psychoerka, VOL. 66(), pp.9 48, 00. []G. Carpeno, S. aeo, E. Toh, Agorhs and codes for he assgnen probe, In B. Seone, E Toh, G. Gao, E affo, & S. Paono (Eds.), Foran codes for nework opzaon. Annas of Operaons Research,, pp.9-4, 988. []S. Shehroz Khan, Ar Ahad, cuser cener nazaon agorh for K-eans cuserng, Paern Recognon Leers 5, pp.9-0, 004. []S.J.,Redond, C.,Heneghan, A ehod for nazng he K-eans cuserng agorh usng kd-rees, Paern Recognon Leers, o.8, pp , 007. [4] Yao-nan Wang, Chun-sheng L, Y Zuo, A Seecon ode for Opa Fuzzy Cuserng Agorh and Nuber of Cusers Based on Copee Coprehense Fuzzy Eauaon, IEEE Transacon On Fuzzy Syses, o 7(), pp , June 009. [5] The UCI achne Learnng Reposory, 99, hp:// [6] Chh-Chung Chang and Chh-Jen Ln, LIBSV : a brary for suppor ecor achnes, 00. hp:// [7] R.J.G.B Capeo, A fuzzy eenson of he Rand nde and oher reaed ndees for cuserng and cassfcaon assessen, Paern Recognon Leers 8, pp.8-84, 007. [8] E.Trauwaer,, On he eanng of Dunn s paron coeffcen for fuzzy cusers, Fuzzy Ses Syses 5, pp.7 4, 988. L Chun-sheng receed he B.S and he aser degree n aheac fro Jang X Nora Unersy n 99 and he Cenra Souh Unersy n 999 respecey, Chna. He now s a Ph.D. a he Coege of Eecrca and Inforaon Engneerng, Hunan Unersy. Hs research neress are copuaona negence, negen conro and negen nforaon processng. 00 ACADEY PUBLISHER