Computing OWA weights as relevance factors

Transcription

1 Compug OWA wghs as rlvac facors Agl Caţaro Dparm of Elcrocs ad Compurs, TRANSILVANIA Uvrsy of Braşov, Romaa -mal: Răzva Ado Compur Scc Dparm, Cral Washgo Uvrsy, Ellsburg, USA -mal: Absrac Ordrd Wghd Aggrgao (OWA) opraors rprs a dsc famly of aggrgao opraors ad wr roducd by Yagr [ Thy compu a wghd sum of a umbr of crra ha mus b sasfd Th cral lm of h OWA opraors s ha h crra ar rordrd bfor aggrgao ad hrfor a parcular wgh s assocad o a poso Rlvac Larg Vcor Quazao (RLVQ) [2 s a xso of h Larg Vcor Quazao (LVQ) algorhm [3 ad prforms a hursc drmao of h rlvac facors of h pu dmsos Ths mhod s basd o Hbba larg ad assocas a wgh facor o ach dmso of h pu vcors W prs a LVQ mhod for o-l compug of h OWA wghs as rlvac facors Th mhod uss a wghd mrc basd o OWA rsrcos Th prcpal bf of our algorhm s ha cocs wo dsc opcs: RLVQ algorhm ad h coss mahmacal modl of h OWA opraors Idx Trms ordrd wghd aggrgao opraors, larg vcor quazao, rlvac facors, mach larg, ural wors I INTRODUCTION OWA opraors rprs a class of aggrgao opraors ha provd a aggrgad valu basd o a rordrg of h crra ha mus b sasfd Th wghs of h OWA opraor ar assocad o a poso of h rordrd argums ad o o a spcfc valu LVQ s a mhod of classfcao basd o a umbr of pars Th vcor quazao dfs a mappg from a spac of -dmsoal vcors o a f s of -dmsoal vcors rfrrd o as codboo Th vcors from h codboo ar h prooyps ad ach of hm s assgd o a parcular class LVQ mplms a algorhm ha ravly adaps h codboo vcors by opmzg global crra basd o h Euclda dsac A umbr of modfcaos of sadard LVQ algorhm wr proposd ordr o sur a fasr covrgc (OLVQ) or for a br adapao alog h bordrs (LVQ2, LVQ3) [3 Th sadard LVQ algorhm dos o ma a dsco bw h mor or lss flu faurs of h pu vcors Th Dsco Ssv Larg Vcor Quazao (DSLVQ) algorhm roducd [8 mploys a wgh for ach faur ad uss a wghd dsac for classfcao I uss a hurscally rav algorhm o adap h wghs o problms rqurms: rducs h fluc of h faurs ha frquly lad o a wrog classfcao ad amplfy h fluc of h faurs ha hav a larg corbuo o a corrc classfcao RLVQ s a varao of LVQ, smlar o DSLVQ, ha roducs h rlvac facors for ach faur W prs our algorhm as a mhod o compu h OWA wghs as rlvac facors paralll wh dfg a classfcao basd o a modfd LVQ algorhm Ths mhod cocs wo dffr approachs: RLVQ ad OWA W hav obad good rcogo ras o svral sadard daass ca also b usd as a chqu for rag h pu vcor s faurs A prlmary vrso of h algorhm appard [0, howvr whou h OWA dals ad whou som of h xprms dscrbd blow O lav of absc from h Dparm of Elcrocs ad Compurs, Trasylvaa Uvrsy of Braşov

2 II OWA OPERATORS W am o dscrb frs h gral aggrgao problm Th w wll roduc h OWA opraors ad hr fudamal proprs L us cosdr ha C,,C ar crra whch df a mulcrral problm W do by X h doma of h valus of hs crra ad by I h rval [0, Th aggrgao problm mas o formula a global dcso fuco D I has h propry ha, for ay alrav x X, h valu Dx ( ) I rflcs h dgr ha x ms h rqurd codos rspc wh h crra, wh C ( x) I, =,, s h dgr ha x sasfs h crra C W ca hrfor wr h followg rlao: Dx ( ) = FC ( x),, C( x) ( ) F rprss h aggrgao opraor whch mus b [: a) Mooo, ha s h mor a dvdual crra sasfd, h bggr h global dcso fuco s valu: C x C y =,, ad x, y X D x D y ( ) ( ) ( ) ( ) b) Symmrc, ha s h ordr of h crra s o mpora for h global dcso fuco s valu Th aalyss of a aggrgao opraor mas o sudy h rlaos bw h crra ha dscrb h problm A xrm s wh x mus sasfy all h C,,C crra ad w hav a adg appld o h valus A obvous xampl of such a opraor s Dx ( ) = MC ( ( x),, C ( x) ) Aohr xrm s wh a las a crro mus b sasfd ad hs s a org of h valus A opraor who blogs o hs cagory s Dx = MaxC x,, C x Th usual aggrgao ( ( ) ( ) ( ) opraors ar bw hs wo xrm cass Spcal cagors of aggrgao opraors ar h Ordrd Wghd Aggrgao (OWA) opraors A OWA opraor s a -dmsoal fuco: whr h opraor, wh [ W = w A [ = a a FR : R Fa (,, a) = wa = w ) s h wghs vcor assocad o w [ 0,, w = ad = a ascdg rordrg of h [ argums of A = a such as a h s h largs a a Th cral lm usg h OWA opraors s h rordrg sp A parcular argum a wll o logr b assocad o a parcular w, bu o h valu from h poso rsuld afr rordrg I ca b show ha a OWA opraor s commuav, mooo ad dmpo [ I also has h boudg propry: M a,, a F a,, a Max a,, a ( ) ( ) ( ) Basd o hs propry, Yagr [ roducd a masur calld orss whch s clos o f h opraor has a or characr: O( W ) = ( ) w = A umbr of mhods wr proposd o choos h OWA opraor s wghs O Haga [4 roducd a chqu basd o a gv orss valu Torra [5, [6 usd a parcular aggrgao opraor amd Wghd OWA ad drmd s paramrs by usg a procdur ha compus a dal oupu for ach rag par Karayas [7 usd wo OWA wghs famls coag a s of qual wghs ad a s of lar dscdg valus Blaov [8 approxmad h OWA opraors by solvg a problm amd Las Squars wh Equaly ad Iqualy (LSEI) Flv ad Yagr [4, [9 compud h OWA opraor s wghs by usg a dscdg grad mhod W wll prs a mhod o drma h OWA wghs by cosdrg hm as rlvac facors III RELEVANCE LVQ I s ow ha somms o all faurs of h pu vcor hav h sam fluc h dcso of a classfcao or a rcogo sysm Th Rlvac Larg Vcor Quazao (RLVQ) algorhm compus a s of rlvacs assocad o ach faur of h pu vcor Ths s a rav mhod basd o Hbba larg I s a hurscally algorhm ad a umbr of mprovms wr proposd ordr o avod usabl bhavor som parcular suaos Th RLVQ algorhm rforcs h rlvac facors of h faurs ha hav h hghs fluc for h corrc classfcao Ths algorhm dcrass h wghs of h faurs ha hav a gav fluc ovr h rcogo procss Th clusrg s ralzd by a s of prooyps ha ar ud by h comg faur vcor ad a sadard LVQ algorhm

3 Assum ha a clusrg of daa o C classs s o b lard ad a s of rag daa s gv: X = {( x, y) R {,, C} =,, M} Th compos of a vcor x ar [ x,, x LVQ chooss prooyp vcors R for ach class, so calld codboo vcors Do h s of all codboo vcors by { w,, wk} Th compos of a vcor wj ar [wj,, wj Th rag algorhm adaps h codboo vcors for as las as possbl quazao rror o all faur vcors, as follows [3: For a gv pu x, fd h closs codboo vcor w j, h wr, whch provds h las valu of h dsac x w If x ad w j hav h sam class labl, h faur vcor s corrcly classfd 2 Upda h wr codboo: w j + x w j), f x was corrcly classfd w j = w j η( x w j), ohrws whr η > 0 s h larg ra RLVQ uss a modfd wghd mrc hs algorhm: whr [ 2 x w j = ( x wj) =,, j s h rlvac vcor ad = Followg a smlar rul, h wghg = facors ar ravly adapd [2: f x was corrcly max { α x wj, 0}, = classfd + α x w j, ohrws for =,, α > 0 s h larg ra for h wghg facors 2 Normalz h wgh vcors Rlvac drmao ca b usd afr LVQ larg or smulaously, hs scod vrso s yldg a o-l algorhm Rpord rsuls [2 provd a br rcogo accuracy of RLVQ compard o h sadard LVQ IV OWA RELEVANCE LVQ ALGORITHM W prs ow h algorhm for compug h OWA wghs as rlvac facors smulaously wh h upda of h codboo vcors W rdf frs h LVQ algorhm by rplacg h Euclda dsac wh a wghd dsac: 2 Dj = x w j = Th rlvacs vcor,, has h proprs of h OWA opraor s wghs: =,,, [ 0, = By h xprsso: x wj w dod h h largs dffrc bw corrspodg compos of h pu vcor x ad h codboo vcor w j Th dsac D j s a parcular cas of h ordrd wghd gralzd ma [7: p M = a = whr p s a ral, posv umbr By rplacg p = 2 w oba our modfd dsac whr o h dffrc x wj ad: p a xx wj x2 wj2 x wj w j = η I( x w j) w j = I( x w j) corrspods W ca rformula h LVQ algorhm by mmzg a objcv fuco basd o h modfd dsac Th codboo vcors ca b updad by compug f x was corrcly classfd accordg o h dsac ad I s h u dagoal marx I h cas wh h pu vcor x was o corrcly classfd, accordg o h sam dsac w us h followg upda formula: η I hs upda rlaos w dod wh ( x w j) D j a vcor obad by rordrg of s compos Th

4 rlvacs ar usd afr h rordrg sp As OWA opraor s proprs suggs, h compos of h vcors w j wll corrspod o h rsulg posoal wgh compos W sablshd h mhod o adap h prooyp vcors cosdrg h modfd dsac D j W wll s ow how w ca upda h rlvac facors Th modfd dsac D j s a crro o dcd h corrc classfcao of h pu vcors W cosdr ha x s corrcly classfd f our modfd dsac o h codboo w j s mm ad h wo vcors blog o h sam class: D < D, l j Do whr j l [ d = d,, d d = x w, =,, j I h cas wh x s corrcly classfd accordg o h modfd dsac D j, a small valu of d should lad us o a larg valu of O h ohr had, a larg valu of h dsac d should hav a small fluc for h rlvac valus ad h magud s mmal Thrfor, ha s If w cosdr: d h w ca wr: ' < d > ' ' d > d > = αd = α x wj Wh h classfcao of h pu vcor x s o corrc accordg o h modfd dsac D j, h upda formula ca b dvlopd wh a smlar mhod A small valu of d ducs a small valu of ad a larg valu of d corrspods o a larg valu of Ths mas ha ad or d ' > d > = αd ' ' = α x wj Bcaus h rlvac facors ar wghs of a OWA opraor, w fally us h followg rasform: whch sur us ha =, =,, = = = ad [ 0, W ar rady ow o wr h procdur ha smulaously adaps h codboo vcors ad h rlvac facors Ialz h larg ras η ad α Assgg h al valus o h rlvac vcor: =, =,, 2 Ialz h codboo vcors 3 Upda h codboo vcors usg h modfd LVQ algorhm whch uss h dsac D j : ( ) ( ) w I x w x j + η j, f was corrcly classfd w j = w j ηi x w j, ohrws 4 Upda h rlvac facors: f x was corrcly α x wj, = classfd + α x w j, ohrws for all =,, 5 Normalz rlvacs: =, =,, = 6 Compu h wgh of ach faur as a avrag of s bfor ordrg poso dx h pu vcor, for all prvous sps 7 Rpa sps 3-6 for ach rag par Ths algorhm compus h rlvac facors ha ar aachd o a spcfc poso h ordrd vcor of dsacs o h codboo vcors I also compus h ra of ach faur ad hss valus hav a dffr mag I s aachd o a spcfc faur ad hs xplas h sp 6

5 V EXPERIMENTS W usd sadard bchmars [3 o s h OWA- RLVQ algorhm: Irs daabas, Vowl Rcogo daabas (Drdg daa) ad Ioosphr daas W sudd h rcogo ras ad h rsuld rlvac vcors ha ca also b rprd as OWA opraor s wghs Th Irs daabas coas 3 classs of pars wh 50 vcors ach Two classs ar o larly sparabl Th problm s o dc h classs cosdrg h 4 faurs of ach vcor W usd our rag procdur 6 codboo vcors ad w fally obad a rcogo ra of 9660% ad h followg rlvac vcor: [ For h larg cosas w usd h valus η = 03 ad α = 2 Th faur rag, dpcd Tabl, rflcd h sam rsuls as obad [ for RLVQ, wh h las faur cosdrd as h mos mpora Tabl Faur rag for h Irs daabas Ra RLVQ Faur Faur Faur Wgh Th Vowl Rcogo daabas coas vcors xracd from 5 dvdual spars prooucg vowls coxs, 6 ms ach Th problm s o us h proucaos of h frs 8 spars for rag ad h proucaos of h las 7 spars for rcogo ss W usd 59 codboo vcors ad h accuracy ra ha w obad hs xprm was 4675% comparg o 4632% obad wh RLVQ ad 448% wh LVQ Th valus of h larg ras wr η = 7 ad α = 9 W obad h followg rlvac vcor: [ Th faur umbr 2 was rad as h scod mos mpora by ad as mos mpora by RLVQ, as dscrbd Tabl 2 Tabl 2 Faur rag for h Vowl Rcogo daabas Ra RLVQ Faur Faur Faur Wgh RLVQ Faur Faur Faur Wgh Th Ioosphr daas cosss of 35 sacs of radar collcd daa, wh 34 couous faurs ach Th vcors, balacd bw posv ad gav xampls, ar labld wh bad or good, hs yldg a bary classfcao as Th frs 200 pars wr usd for rag ad h rmag 5 w usd for h rcogo ss By rag 8 codboo vcors, w obad a rcogo ra of 9337% wh, of 927% wh RLVQ ad of 9006% wh LVQ I Tabl 3 w prs h rag of h mos mpora 5 faurs as rsuld from our xprms W usd h valus η = 33 ad α = 35 for h larg paramrs Tabl 3 Faur rag for h Ioosphr daabas Oly h fv mos mpora faurs wr rprsd Ra RLVQ Faur Faur Faur Wgh A comparso of h rcogo ras bw LVQ, RLVQ ad s provdd Tabl 4, rflcg ha our algorhm prformd br all xprms rpord hr Tabl 4 Comparav rcogo ras obad wh LVQ, RLVQ ad Daabas LVQ RLVQ Irs 933% 9533% 9660% Vowl 4480% 4632% 4675% Ioosphr 9006% 927% 9337% VI CONCLUSIONS W hav prsd a mhod o compu h OWA opraor s wghs as rlvac facors of h pu faurs Th algorhm uss a modfd wghd mrc Th rlvac vcor s updad o-

6 l, gvg h possbly of dyamcal adapao o h comg daa W hav obad good rcogo ras wh applyg our algorhm o sadard bchmars Th rlvac facors ca b usd as OWA wghs Thy ca also b usd for o-l faur rag ad for faur slco REFERENCES [ RR Yagr, O Ordrd Wghd Avragg Aggrgao Opraors Mulcrra Dcsomag, IEEE Tras Sysms, Ma, ad Cybrcs, 8, 88-90, 988 [2 T Bojr, B Hammr, D Schu ad KT vo Toschaowz, Rlvac Drmao Larg Vcor Quazao I MVrlys (Ed) Europa Symposum o Arfcal Nural Nwors'200 (D-sd publcaos) , 200 [3 T Koho, Slf-Orgazg Maps, Sprgr- Vrlag, 997 [4 RR Yagr ad D Flv, Iducd Ordrd Wghd Avragg Opraors IEEE Tras o Sysm, Ma, ad Cybrcs, 29, 4-50, 999 [5 V Torra, Larg Wghs for Wghd OWA Opraors, Proc of h IEEE Il Cofrc o Idusral Elcrocs, Corol ad Isrumao (IECON 2000), Nagoya, Japa, 2000 [6 V Torra, Wghd OWA Opraors for Syhss of Iformao, Proc of h 5 h IEEE Il Cofrc o Fuzzy Sysms, Nw Orlas, USA, , 2000 [7 NB Karayas, Sof Larg Vcor Quazao ad Clusrg Algorhms Basd o Ordrd Wghd Aggrgao Opraors, IEEE Tras o Nural Nwors,, , 2000 [8 G Blaov, How o Buld Aggrgao Opraors From Daa Iraoal Joural of Illg Sysms, 8, , 2003 [9 D Flv ad RR Yagr, O h Issu of Obag OWA Opraor Wghs, Fuzzy Ss ad Sysms, 94, 57-69, 998 [0 A Caţaro ad R Ado, RLVQ Drmao Usg OWA Opraors I M Hamza (Ed) Procdgs of h 3rd IASTED Iraoal Cofrc o Arfcal Illgc ad Applcaos (ACTA Prss) , 2003 [ B Hammr ad T Vllma, Bach-RLVQ I M Vrlys (Ed) Europa Symposyum o Arfcal Nural Nwors 2002 (D-sd publcaos), , 2002 [2 B Hammr ad T Vllma, Gralzd Rlvac Larg Vcor Quazao Nural Nwors, 5, , 2002 [3 CL Bla ad CJ Mrz, UCI Rposory of Mach Larg Daabas, Uvrsy of Calfora, Irv, Dp of Iformao ad Compur Sccs, hp://wwwcsucdu/ ~mlar/mlrposoryhml, 998 [4 D Flv ad RR Yagr, Larg OWA Opraor Wghs from Daa, Procdgs of h Thrd IEEE Iraoal Cofrc o Fuzzy Sysms, Orlado, IEEE Prss, , 994 [5 NB Karayas, A Axomac Approach o Sof Larg Vcor Quazao ad Clusrg, IEEE Tras o Nural Nwors, 0, 53-65, 999 [6 NB Karayas ad MM Radolph-Gps, Sof Larg Vcor Quazao ad Clusrg Algorhms Basd o No-Euclda Norms: Mulorm Algorhms, IEEE Tras o Nural Nwors, 4, 89-02, 2003 [7 RK D, N R Pal ad SK Pal, Faur Aalyss: Nural Nwor ad Fuzzy S Thorc Approachs, Par Rcogo 0(30), , 997 [8 M Prgzr, D Flozgr ad G Pfurschllr, Dsco Ssv Larg Vcor Quazao A Nos-Issv Classfcao Mhod, Procdgs of h IEEE Iraoal Cofrc o Nural Nwors (IJCNN 994), Orlado, Florda, , 994