A New Statistical competitive Data Clustering Approaches: SRPCL and SRPCCL

Transcription

1 Ieaioal Joual of Applied Eieei Reseach ISSN Volume, Numbe (7) pp Reseach Idia Publicaios. hp:// A New Saisical compeiive Daa Clusei Appoaches: SRPCL ad SRPCCL Eddaouich Souad *, Zouad Rabab** ad Hammouch Ahmed *** * Reioal Educaioal Cee CRMEF, Compue Sciece Depame, Raba, Moocco. ** ERIT, LRGE Laboaoy, ENSET, Elecical Eieei Depame, Mohammed V Uivesiy, Raba, Moocco. ***Posaduae School of Techical Educaio, Ese, B.P., 67 Aveue des Foces Amées Royales, Raba, Moocco. ORCID: Absac Fidi useful paes i daases has aaced cosideable iees ecely, ad oe of he mos widely sudied echiues is daa clusei. Clusei is he usupevised classificaio of paes (obsevaios, daa iems, o feaue vecos) io cluses (oups) ad i is oe of he mos impoa eseach ad applicaio aeas of eual ewos. This pape summaizes some of he mos impoa developmes i compeiive leai aloihms fo daa clusei ad peses wo ew fas leai appoaches based o he compeiive cocep. The fis oe used he ival compeiive leai (RPCL) coceps ad he secod oe ivesiaes he ival pealized coolled compeiive leai (RPCCL) coceps, o dyamically cool he ival-pealizi foces. Keywod: -meas compeiive leai RPCL RPCCL; mode deecio pocedue. INTRODUCTION Cluse aalysis is a impoa saisical mehodoloy used i a wide vaiey of fields icludi machie leai, pae ecoiio, imae aalysis, ifomaio eieval, ad bioifomaics ec. The objecive of clusei is o paiio M daa pois x x, x,... x } M io K o-empy subses such ha alie daa ae ouped oehe ad daa i diffee subses o cluses ae o alie. The saisical appoach i cluse aalysis posulaes ha he ipu pae ae daw fom a udelyi pobabiliy desiy fucio (pdf) which descibes he disibuio of he daa pois houh he daa space. Reios of hih local desiy, which mih coespod o siifica classes i he populaio, ca be foud fom he peas o he modes of he desiy fucio esimaed fom he available paes. The, he ey poblem is o paiio he daa space wih a mulimodal pdf io subspaces ove which he pdf is uimodal []. Up o ow, umeous echiues have bee poposed o deal wih he clusei poblem, bu eual ewos ae cosideed as he mos ieesi aleaive o he coveioal mehods [], [], [3]. The advaae of eual ewos lies i hei abiliy o adjus hemselves o he daa wihou ay explici specificaio of fucioal o disibuioal fom fo he udelyi model. Secod, hey ae oliea models, which mae hem flexible i modeli eal wold complex elaioships. Fially, eual ewos have show hei effeciveess i a vaiey as of eal wold: medical diaosis [], ecoiio of hadwie siaues, chaacesad diis [], [6], [7], weahe foecasi [8], [9], [], fiacial pedicios [], [] ad classificaio applicaios[3],[],[]. I he lieaue, - meas [6] is a popula compeiive leai aloihm; i sas wih K adom cees, he pics up a veco adomly fom he ipu daa se, compues he disace wih each cee, chooses he miimum oe ad moves he cee owads o he coespodi ipu veco by a aio. The -meas aloihm is easy o impleme bu i suffes fom wo majo dawbacs: ) he poblem of dead uis i.e. if he iiial posiios of some cees ae iappopiaely chose (fa away fom he ipus compai o he ohe cees), hey may eve be aied ad, heefoe, immediaely become dead uis. ) We mus specify he exac umbe of, if his umbe is o eual o he ue cluse umbe; he pefomace of -meas aloihm deeioaes apidly. Eveually, some of he seed pois ae o locaed a he cees of he coespodi cluses. Isead, hey ae eihe a some bouday pois bewee diffee cluses o a pois biased fom some cluse cees. Afe he K-meas comes he sadad compeiive leai [7] o impleme he K-meas i euoal sucue ha has he advaae of paallelism, obusess ad faul oleace. To cicumve he poblems of dead uis, he feuecy sesiive compeiive leai (FSCL) [8] has bee poposed o pefom a bee clusei. Ohe aloihms developed wee Rival Pealized Compeiive Leai (RPCL) [9] ad is vaias such as DPRCL [] ad RPCCL [], which y o emove he poblem of pe-seleci he umbe. We popose, i he pese pape, a ew appoach fo he 7

2 Ieaioal Joual of Applied Eieei Reseach ISSN Volume, Numbe (7) pp Reseach Idia Publicaios. hp:// deecio of he mode wihou usi eihe diffeeial opeaos o ay pocedues of filei. This appoach is based o eual ewo [7,8], ad moe paiculaly o he ival compeiive ewo. I beis by he esimaio of he pobabiliy desiy fucio (pdf), followed by a aii compeiive eual ewo. This sae allows deeci he local maxima of he pdf, cosideed as maues of he modes of his fucio ad also as he pooypes of pese cluses i he daa se. I his coex, wo aleaives of aii ae poposed hee. The fis oe used he RPCL coceps ad he secod oe ivesiaes he RPCCL coceps, o dyamically cool he ival-pealizi foces. The secio II of his pape ioduce biefly each compeiive daa clusei appoach deived fom he - meas aloihm. Secio III is cosecaed o he peseaio of he New Saisical compeiive Daa Clusei Appoaches: SRPCL ad SRPCCL. Secio IV illusaes by meas of examples usi daa eeaed aificially, he advaae of he ew appoach, compaes he pefomace of he wo aii pocedues SRPCL ad SRPCCL. OVERVIEW OF THE APPROACHES Feuecy sesiive leai aloihm The feuecy sesiive compeiive leai (FSCL) [8] ioduced i 99 is a exesio of he -meas aloihm o ovecome he poblem of dead uis. The basic idea is o ioduce a elaive wii feuecy o cosciece em [] io he similaiy measueme bewee a ipu ad he seed pois. The lae he wii feuecy of a seed poi, he moe i is pealized. The elaive wii feuecy coefficie is defied as: () j The wie is seleced accodi o his euaio: Whee he cee A mi D C, X () K i is he cumulaive umbe of he occueces of bewee he cee C i he pas ad C X C ad he wie. D, is he disace Alhouh he FSCL aloihm ca almos pefom a successfully clusei wihou he dead uis poblem, bu i suffes fom he same secod poblem as -meas: he eed o specify he exac umbe of cluses. This aloihm has bee used exesively fo veco uaizaio of speech wavefoms [3] ad fo imae compessio []. Rival pealized compeiive leai RPCL is a impoveme of he FSCL aloihm ha auomaically deemies he umbe of classes i he pesece of a daa sample while eepi he chaaceisic of solvi he poblem of dead uis. The basic idea of his leai ule is o covee he wie cee owad o he ipu, ad push he ival cee i.e. he secod wie away fom he daa se [9]. Theefoe, he updai of he wie ad he ival cee is doe accodi o hese euaios: C ( ) C ( ) ( ) X C ( ) C ( ) C ( ) ( ) X C ( ) C C ( ) ( ) if if if e Whee: ( ) ad ( ) ae especively, he leai coefficies fo he wie cee ad is ival. Ad ) is much eae ha ( ). I some ( lieaues, ( ). ad ( ). Howeve, he use of RPCL clusei offes a ieesi aleaive o he K-meas ad FSCL aloihms, bu is coveece is vey sesiive o he choice of paamees () ad ) (. Amo he applicaios of RPCL, we meio colo imae semeaio [], oliea chael eualizaio [6] ad imaes feaues exacio [7]. Rival pealized cooled compeiive leai The RPCCL aloihm [] is simila o he RPCL because i has he same advaae of auomaically deemii he umbe of classes. Ideed, if he umbe of he classes fixed by he use, i he iiializaio phase, is hihe ha he eal umbe of classes, exa euos ae dive away fom all obsevaios. I compaiso wih he RPCL aloihm, he RPCCL aloihm applies a ew mechaism o dyamically cool he pealizaio of hose euos (he ivals) by ioduci a ew em called he pealizaio seh []:, C ( ), DC ( ), C ( ) DC ( ), C ( ) mi D X p( X, C ( ), C ( )) () Whee C () ad () C (3) ae especively, he wie cee ad is ival. The updai of he weih vecos of he ival becomes: C ( ) C ( ) ( ) p( X, C ( ), C ( )) X C ( ) Fom (), i ca be see ha he ival will be fully pealized wih he ae () if he disace bewee he wii () 76

3 Ieaioal Joual of Applied Eieei Reseach ISSN Volume, Numbe (7) pp Reseach Idia Publicaios. hp:// cee ad is fis ival is smalle ha he disace bewee he wii cee ad he ipu. Ohewise, he ival will be pealized wih he seh ( ) p( X, C ( ), C ( )). Coseuely, he RPCCL eeally dives he udesied cees fa away fom he cluses much fase ha he RPCL, because is de-leai ae is eae. THE SRPCL AND SRPCCL PROCEDURES The pdf s mode deecio pocedue is a euoal pobabilisic appoach peseed i [8]. This appoach is caied ou i hee saes pocessi: he fis oe cosiss i esimai he udelyi pdf usi a o-paameic esimao. I he secod sae, a aificial euoal ewo wih compeiive aii is used fo exaci he local maxima of he pdf. Fially i he hid sae, he pdf modes ae deeced usi a pobabilisic appoach. Based o ha pocedue ad also o he he coceps of he RPCL ad RPCCL appoaches, we popose i he followi secio a wo aleaives leai appoaches amed especively saisical ival pealised compeiive leai (SRPCL) ad saisical ival pealised cooled compeiive leai (SRPCCL) o deec he local maxima of he pdf. Each sep of he pocedue will be explaied below. The esimaio of udelyi pobabiliy desiy fucio Le X X,...,, X Q, be he se of Q N-dimesioal obsevaios of a adom vaiable X wih a pobabiliy desiy fucio P (X ).To esimae his udelyi desiy fucio whe wha is available is oly a se X x,, x,,..., x,,..., x, N, =,,...,Q of Q obsevaios, he aalys may use echiues. o-paameic The Paze widow mehod [9] poves well adaped o he poposed pocedue i his pape. Howeve, his esimaio pocedue eeds pohibiive calculus whe he dimesio of he space is vey impoa. So, we have oped fo he fas esimaio aloihm which is poposed by Posaie ad Vasseu [3]. Fis, he ae of vaiaio of each compoe of hese obsevaios, is omalised o he ieval [,R], whee R is a iee such as R, by meas of he asfomaio defied as: y x, mi x, R (6) max x, mi x,, * Each axis of he so omalized daa space is he paiioed io R exclusive ievals of ui widh. This disciisaio N defies a se of hypecubes of ui side leh. Each hypecube oed H (X ), is a sie defied by is N coodiaes x, x,..., x, x N which ae he iee pas of he coodiaes of is cee X. To be moe specific, le y, y,..., y y y,,,,..., N,, =,, Q be he Q obsevaios i he omalized space. Each obsevaio Y is foud iside a o-empy hypecube wih he coodiaes x i( y,), =,,..,N, whee i( y,) desiaes he iee pas of y,. If seveal obsevaios fall i he same hypecube, his oe appeas may imes o he lis of o empy hypecubes. Fuhemoe, he umbe of imes he hypecube H(X ) appeas i ha lis idicaes he umbe of daa pois H(X ) which fall i his hypecube. Subseuely, he value of he local desiy esimaed is: H( X ) p( X ) (7) Q Sice he volume of H (X ) is eual o uiy. So, his fas pocedue allows oly he esimaio of he udelyi pobabiliy desiy fucio a he cees of he o-empy hypecubes whose umbe eve exceeds he umbe Q of available obsevaios. A he cees of he hypecube cells, which ae o o ha lis, he desiy esimaes ae ow o be ull. A he ed of his fas aloihm, all he available ifomaio fo clusei is i he discee se X of esimaed values of he udelyi pobabiliy desiy fucio p (X ). The exacio of local maxima by eual ewo I ode o deec he local maxima of he pdf (he pdf s modes), a compeiive leai aloihm wih a abiay umbe of hese local maxima has bee used [8]. The achiecue ad he acivaio fucios of he elaboaed ewo fo he exacio of hese maxima will be he same as hose of he eual ewos wih compeiive aii (RNAC) [8]. I he aii aloihm, we wo, ad oly, o he pdf by pesei seueially o he ewo he cees of he o-empy hypecubes of he 77

4 Ieaioal Joual of Applied Eieei Reseach ISSN Volume, Numbe (7) pp Reseach Idia Publicaios. hp:// se X isead of he Q obsevaios lie i he RNAC. This echiue allows us o evisae a ea educio of coveece ime of he elaboaed ewo because he umbe of he o-empy hypecubes peseed o he ewo is widely less ha he obsevaios peseed o RNAC (see he expeimes secio).. The eual ewo elaboaed is composed of wo layes: he ipu laye ad he oupu laye ( Fi. ). The fis oe is made of N uis I, =,,...,N, such ha ui I is solicied by he aibue X of he o-empy hypecube H(X ) whe his oe is peseed o he ewo. Howeve, each oupu euoe maeialises a hypecube which epeses he sie of oe local maximum of he pdf i he se X. The poposed mehod is usupevised, he umbe of he oupu uis, which is ha of local maxima, is fisly iiialised abiay. he disace D ( X ), H ( X ) of he pdf associaed o (X ) ad compai he values ad o H (X ). As we will see i he followi aloihm, he exposed ifomaio by he ew hypecube H (X ) is used o updae he mea veco ad he value of he pdf of he wie ad he ival euoes. Taii aloihm usi SRPCL: Iiialisaio phase: - Iiialise he mea vecos (X ) ; =,,...,K, of he K oupu eual, wih a abiay choice of K o-empy hypecubes fom he se X. - Iiialise he coefficies of he aii fucio ad. - Iiialise he ieaio paamee o zeo. - Iiialise he se io a empy se:. Pocessi phase:. = +;. Pese o he ewo, wih a abiay pulli, a o-empy hypecube H ( X ) C. X Fiue. Compeiive Neual Newo Le H (X ), be a o-empy hypecube whee he coodiaes x, x,..., x, x N of is cee ae peseed o he ewo a he ieaio. Each oupu euoe defies a o-empy hypecube fom he subse X, ad is maeialised wih he mea veco (X ) of his euoe i which is associaed he value of he pdf fucio f (X ) esimaed i his hypecube. We oe by D ( X ), H ( X ) he disace which sepaaes he mea veco (X ) fom he cee of he peseed hypecube H (X ) [8]. The aii of he ewo has bee doe by he compeiive coceps, so as o esimae is paamees amely: he local maxima of he pdf ad hei locaios i he discee space X [8]. So, o evey ieaio we pese o he ewo he coodiaes of he cee of a o-empy hypecube. The K oupu euoe wih especively he mea vecos (X ), =,,...,K es o compeiio by calculai fo each oe 3. H (X ).. Seach he oupu euoe which is is mea veco; D D (X ) is he closes o H (X ). Ispied by he FSCL[8], i is poposed o muliply he disace bewee he mea veco of a euoe ad he cee of he hypecube which is peseed o he ey of he ewo wih a cosciece em pope o each euoe. The wie euoe is defied by calculai he disace so as: K ( X ), H ( X ) mi D ( X ), H ( X ) (8) The secod wie o he ival is defied by calculai he disace: K ( X ), H ( X ) mi D ( X ), H ( X ) (9) 78

5 Ieaioal Joual of Applied Eieei Reseach ISSN Volume, Numbe (7) pp Reseach Idia Publicaios. hp:// Compae he fucios P (X ) ad H (X ) If P ( X ) PH ( X ) P. updae he wie ad he ival euoes as follow : ( X ) ( X ) ( ) H i ( X ) ( X ) ( X ) ( X ) ( ) H i ( X ) ( X ) p ( X ) ph ( X ) () ad () Whee () aalys i he iiializaio phase. Else o o sep ae deemied by he o-gaussia classes eeaed aificially. The saisical paamees of Gaussias ae: Table. Saisical paamees of aussia classes Classes Saisical Paamees Meas Vaiaces Numbe of pois Soppi cieia: Compae (X ) o ( X ) fo =,,.,K. If ( ( X ) ( X )),,... K, o o sep. Else, ed of he pocessi. Taii aloihm usi SRPCCL: The SRPCCL aii aloihm is close o he SRPCL aloihm, bu uilizes a ovel mechaism o cool he ival pealizaio. The idea of his mechaism is ha he ival should be fully pealized if he wie suffes fom sevee compeiio fom he ival; ohewise, he pealizaio seh should be popoioal o he deee of compeiio level. Thus he SRPCCL aloihm is exacly he same as RPCL, he oly diffeece is ha he ival euo will be pealized i sep as follows: ( X ) Whee: ( X ) P H mi P H ( X ); ( X ), ( X ) ( X ); ( X ), ( X ) ( ) H ( X ) ( X ) i () D ( X ), ( X ), D ( X ), H ( X ) D ( X ), ( X ) The oal of his leai sae is o deec he local maxima of he pdf. To adjus he umbe K of hese hypecubes, which is he oupu euos, we popose a ew mehod ha cosiss i placi all he ival euos i a poie dui he leai phase ad o use ha poie fo elimiai all he exa local maxima i he fial sae of he pocess. Whee aibues descibi he obsevaios of wo classes oliealy disibued i he sample of his example ae defied by: x x A cos B A si B Wih A = A =. he saisical paamees of wo o- Gaussia classes is show i he able, TABLE I. Saisical paamees of o-aussia classes Classes 3 m 7 s Saisical Paamees B B Numbe of pois 3 m s 3 3 Fiue. The eeaed daa se 3 EXPERIMENTS To show he pefomace of he ew clusi appoaches, we coduced he followi expeime usi Gaussia ad 79

6 Ieaioal Joual of Applied Eieei Reseach ISSN Volume, Numbe (7) pp Reseach Idia Publicaios. hp:// a b Fiue 3(a) The fial posiios of cee of classes obaied via FSCL aloihm wih epoch = ad six euos as fial umbe of cluses. (b) The fial posiios afe deemii he exac umbe of classes. Fiue. The fial posiio of six cee of classes obaied via RPCL wih epoch =,, whee oly oe euo is fa away fom he ipu se. Fiue. The fial posiio obaied via RPCCL wih epoch =, whee all ahe exa euos ae fa away fom he ipu se (a) (b) (c) 7

7 Ieaioal Joual of Applied Eieei Reseach ISSN Volume, Numbe (7) pp Reseach Idia Publicaios. hp:// (d) (e) Fiue 6. The SRPCLad SRPCCL pocedues. (a) Udelyi pdf. (b) The modes of he pdf usi SRPCL. (b) The modes of he pdf usi SRPCCL (d) The classified daa usi SRPCL (eo=.66) (e) The classified daa usi SRPCCL(eo=.377). The expeimes have show he ousadi pefomace ad apidiy of he saisical aii appoaches i compaiso wih RPCL, RPCCL ad FSCL. The so deeced modes dui he sep of leai ae used i he las sep of his appoach fo he classificaio pocess fi 6-d ad fi 6-e. Compaed o he K-meas clusei o o he clusei appoaches based o he diffee compeiive leai schemes, he poposed appoach has pove, ha does o pass by ay hesholdi ad does o euie ay pio ifomaio o he umbe of classes o o he sucue of hei disibuios i he daase. coveece ime i secods (s) FSCL RPCL RPCCL SRPCCL SRPCL umbe of classes Fiue 7. Compaiso of he coveece ime of diffee aloihms To bee achieve he eliabiliy of he mode deecio pocedue, we compaed is pefomace wih hose of RPCL, FSCL ad RPCCL. Fi. 7 shows he coveece ime of he fou pocedues, applied o he above daa se, depedi o he umbe of he oupu euos of he ewo. Accodi o he esuls i Fi. 7, we see ha he poposed aloihm is moe efficie ha he FSCL, RPCL ad RPCCL. These aloihms pocess obsevaio by obsevaio, while he poposed ewo hadles he oempy hypecubes. Give ha he umbe of hese hypecubes is siificaly less ha he oal umbe of obsevaios of he sample, he poposed aloihm covees fase. CONCLUSION We have fuhe ivesiaed some impoa aloihms i he compeiive leai. A usupevised clusei mehod based o mode deecio of udelyi pdf has bee ioduced; his appoach has pove, addiioally o is speed, he abiliy o auomaically deemie he umbe of classes wihou ay huma ieveio. REFERENCES ( PT) [] E.W. Foy, Cluse Aalysis of Mulivaiae Daa: Efficiecy Vesus Iepeabiliy of Classificaios, Poc. Biomeic Soc. Meeis, 96. [] H.Q. Wa ad D.S. Hua, A Novel Clusei Aalysis Based o PCA ad SOMs fo Gee Expessio Paes, Lecue Noes i Compue Sciece, vol. 37, pp. 76-8, Spie-Vela,. [3] L. Xu, A. Kzyz ja, ad E. Oja, Rival Pealized Compeiive Leai fo Clusei Aalysis, RBF Ne, ad Cuve Deecio, IEEE Tas. Neual Newos, vol., pp , 993. [] Amao, Filippo, e al. "Aificial eual ewos i medical diaosis." Joual of applied biomedicie. (3): 7-8. [] Miah, Mohammad Badul Alam, e al. 7

8 Ieaioal Joual of Applied Eieei Reseach ISSN Volume, Numbe (7) pp Reseach Idia Publicaios. hp:// "Hadwie Couesy Amou ad Siaue Recoiio o Ba Cheue usi Neual Newo." Ieaioal Joual of Compue Applicaios 8. (). [6] Ya, Jie, e al. "Robus Muli-Laye Hieachical Model fo Dii Chaace Recoiio." JOURNAL OF ELECTRICAL ENGINEERING & TECHNOLOGY. (): [7] Shama, Ai, ad Dipi R. Chaudhay. "Chaace ecoiio usi eual ewo." Ieaioal Joual of Eieei Teds ad Techoloy (IJETT)-Volume (3): [8] Rajuma, R., A. James Albe, ad D. Chadaala. "Weahe Foecasi usi Fuzzy Neual Newo (FNN) ad Hieachy Paicle Swam Opimizaio Aloihm (HPSO)." Idia Joual of Sciece ad Techoloy8. (). [9] Shivasava, Gyaesh, e al. "Applicaio of aificial eual ewos i weahe foecasi: a compehesive lieaue eview." Ieaioal Joual of Compue Applicaios.8 (). [] Maou, Mohammed, e al. "The climae chae implicaio o Joda: A case sudy usi GIS ad Aificial Neual Newos fo weahe foecasi."joual of Taibah Uivesiy fo Sciece 7. (3): -. [] Aydi, Alev Dile, ad Seyma Calisa Cavda. "Pedicio of Fiacial Cisis wih Aificial Neual Newo: A Empiical Aalysis o Tuey." Ieaioal Joual of Fiacial Reseach 6. (): 36. [] Ge, Ruibi, Idail Bose, ad Xi Che. "Pedicio of fiacial disess: A empiical sudy of lised Chiese compaies usi daa mii." Euopea Joual of Opeaioal Reseach. (): [3] Haiuma, R. "Pefomace aalysis of eual ewos fo classificaio of medical imaes wih waveles as a feaue exaco." Ieaioal Joual of Imai Sysems ad Techoloy. (): 33-. [] Levi, Gil, ad Tal Hasse. "Ae ad ede classificaio usi covoluioal eual ewos." Poceedis of he IEEE Cofeece o Compue Visio ad Pae Recoiio Woshops.. [] Capizzi, Giacomo, e al. "Auomaic classificaio of fui defecs based o co-occuece maix ad eual ewos." Compue Sciece ad Ifomaio Sysems (FedCSIS), Fedeaed Cofeece o. IEEE,. [6] J.B. MacQuee, Some Mehods fo Classificaio ad Aalysis of Mulivaiae Obsevaios, Poc. Fifh Beeley Symp. Mah. Saisics ad Pobabiliy, vol., Beeley, Calif.: Uiv. of Califoia Pess, pp. 8-97,967. [7] Ruhmelha, D.E. e Zipse. D. Feaue discovey by compeiive leai. coiive sciece vol. 9, pp 7-, 98. [8] S.C. Ahal, A.K. Kishamuhy, P. Che, ad D.E. Melo, Compeiive Leai Aloihms fo Veco Quaizaio, Neual Newos, vol. 3, pp. 77-9, 99. [9] L. Xu. A. Kyza, ad E. Oja, "Rival Pealized Compeiive Leami fo Clusei Aalysis, REF Ne, ad Cuve Deecio", IEEE Tasacios o Neual Newos. Val., No., pp. 6369, 993. [] Coia Booca, Geoea Budua, Complex daa clusei usi a ew compeiive leai aloihm, ELEC. ENERG. vol. 9, o., pp. 6-69, 6. [] Yiu-mi Cheu: O Rival Pealizaio Coolled Compeiive Leai fo clusei wih auomaic Cluse Numbe selecio [J], IEEE Tas. Kowlede ad Daa Eieei, VOL.7, NO., 83-88, NOV.. [] S.C.AHA & A.K.KRISHNAMURTHY & CHEN. Paoo Compeiive Leai Aloihms fo Veco uaizaio.. Neual Newos, vol-3, pp. 77-9, 99. [3] Ahal, S. C., A. K. Kishamuhy, P. Che ad D.E. Melo. Compeiive leami aloihms fo veco uaizaio. Neual Newos, Vol. 3, pp 77-9, 99. [] Bouzedoum, A. ad T. J. Alo. Imae compessio usi feuecy sesiive compeiive leami. Poc. of IEEE Woshop o Visual Sial Pocessi & Commuicaios, pp Sepembe 993, Melboue, Ausalia. [] B. Fize, Gowi Cell Sucues A Self- Oaizi Newo fo Usupevised ad Supevised Leai, Neual Newos, vol. 7, o. 9, pp. -6, 99. [6] S.C. Ahal, A.K. Kishamuhy, P. Che, ad D.E. Melo, Compeiive Leai Aloihms fo Veco Quaizaio, Neual Newos, vol. 3, pp. 77-9, 99. 7

9 Ieaioal Joual of Applied Eieei Reseach ISSN Volume, Numbe (7) pp Reseach Idia Publicaios. hp:// [7] P. Guo, C.L. Philip Che, ad M.R. Lye, Cluse Numbe Selecio fo a Small Se of Samples Usi he Bayesia Yi-Ya Model, IEEE Tas. Neual Newos, vol. 3, o. 3, pp ,. [8] Souad Eddaouich ad Abdeahmae Sbihi, Neual Newo fo Modes Deecio i Pae Classificaio, ICTIS 7. Ifomaio ad Commuicaio Techoloies Ieaioal Symposium, Moocco, Fez, 3- Apil 7,pp [9] E. Paze. ''O Esimaio of a Pobabiliy Desiy Fucio ad Mode''. A. Mah. Sa., vol. 33, pp [3] J.-G. Posaie e C. P. A. Vasseu "A fas Aloihm fo o Paameic Pobabiliy Desiy Esimaio". IEEE, Tas. o Pae Aal. ad Machie Iel. PAMI-, 6, pp ,