Regression Learning Vector Quantization

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1 Rrsson Larnn Vcor Quanzaon Mhalo Grbovc Darn of Cour and Inforaon Scncs Tl Unvrs hladlha, USA -al: Slobodan Vucc Darn of Cour and Inforaon Scncs Tl Unvrs hladlha, USA -al: Absrac Larnn Vcor Quanzaon (LVQ s a oular class of nars roo classfrs for ulclass classfcaon. Larnn alorhs fro hs fal ar wdl usd bcaus of hr nuvl clar larnn rocss and as of lnaon. In hs ar w roos an xnson of h LVQ alorh o rrsson. Jus l h LVQ alorh, h roosd odfcaon uss a survsd larnn rocdur o larn h bs roo osons, bu unl LVQ alorh for classfcaon, also larns h bs roo ar valus. Ths rsuls n h ffcv aron of h faur sac, slar o h on h K-ans alorh would a. Exrnal rsuls on bnchar daass showd ha h roosd Rrsson LVQ alorh rfors br han h nars roo coors ha choos roos randol or hrouh K-ans clusrn, classfcaon LVQ on uanzd ar valus, and slarl o h or-basd arzn Wndow and ars hbor alorhs. Kwords-rrsson, larnn vcor uanzaon I. ITRODUCTIO Larnn Vcor Quanzaon (LVQ [8] s a sl, unvrsal, and ffcn classfcaon alorh. I blons o a class of roo-basd larnn alorhs such as nars nhbor, arzn wndow, rnl rcron, and suor vcor achn alorhs. LVQ s dfnd b a s of roos {(, c, = 1 }, whr s a K- dnsonal vcor n h faur sac, and c s s class labl. Gvn an unlabld daa on x u, s class labl u s drnd as h class c of s nars roo, u = c, = ar n d(x u,, whr d s a dsanc asur (.. Eucldan. Th rann of LVQ sars wh lacn h roos a so nal osons n h faur sac. LVQ alorh hn sunall scans h rann daa ons and udas h roos. Thr ar svral dffrn LVQ alorhs ha dal wh roo udas n dffrn was [1, 8]. LVQ [8] has bn shown o achv conssnl ood accurac and s coonl usd as a rrsnav of h LVQ alorhs. Gvn a rann daa on (x,, hr condons hav o b for LVQ o uda s roos: 1 Class of h roo closs o x has o b dffrn fro, Class of h nx closs roo has o b ual o, and 3 x us sasf h wndow rul b falln nar h hrlan a h don bwn h closs ( and h scond closs roo (. Ths wo roos ar hn odfd as (+1 = ( α( (x ( (+1 = ( + α( (x (, whr couns how an udas hav bn ad, and α( s a onooncall dcrasn funcon of. L d and d b h dsancs bwn x and and. Thn, h wndow rul s sasfd f n (d / d, d / d > s, whr s s a consan coonl chosn bwn 0.4 and 0.8. Euaon (1 corrsonds o a hard dcson basd soll on h nars-nhbor roo, as s don n 1-ars hbor alorh. Whl hard dcsons ar cal of LVQ, svral sof LVQ vrsons hav bn roosd [16], ha rdc basd on a whd vo fro h nar roos. Sof vrsons of LVQ ar rnscn of arzn wndow and Suor Vcor Machns alorhs. LVQ s radonall usd for classfcaon. In hs ar, w roos svral LVQ alorhs suabl for rrsson. W os Rrsson LVQ as an ozaon robl wh wo dffrn cos funcons. Th frs cos funcon lads o Rrsson LVQ wh hard rdcons and h scond o LVQ wh sof rdcons. To dsn an LVQ larnn alorh, w xlord wo ozaon aroachs, on bn radn-basd and anohr xcaon-axzaon- (EM- basd. Th radn-basd aroach allows boh bach and sochasc (onln udas of roos, whl h EMbasd s naurall sud for h bach od of oraon. Fnall, w show ha a slfcaon of on of h rsuln alorhs lads o a rrsson varan of h oular LVQ Thr ar svral an usfcaons for usn LVQ alorhs n rrsson. On s ha allows a clar chans for dsnn a roo-basd rrssor on a fxd roo bud. Unl classfcaon, whr hods such as condnsn [5] hav bn roosd for nars nhbor classfrs, hr s no clar alrnav for buldn bud-basd nars nhbor and arzn wndow alorhs for rrsson. Anohr usfcaon s ha rrsson LVQ has a rlavl low rann cos wh lnar and consan or scaln wh rann sz. Ths as vr aracv for h onln larnn alcaons. Fnall, so rrsson robls rur roos for rrsnaonal uross. W should also dscuss h dffrnc bwn sof and hard rdcons n roo-basd alorhs. Frs, hard rdcons ar suboal n h sns ha h ar a scal

2 cas of sof rdcons whr h whol wh of dcson s vn o h nars roo. I has lon bn nown ha sof rdcons hav a soohn ffc ha s usful for aroxan rrsson funcons. Howvr, hr ar svral suaons whr hard rdcons ar ncssar. Two xals ar dcnralzd saon [1] and alarnn n dsrbud daa nn [10, 11]. In boh alcaons, nsad of sndn raw daa or rdcons o h fuson cnr, h b or arora o snd ndx of h nars roo. Ths can b usful whn hr ar councaon channl and nr consrans n h fuson ss, or whn hr ar h daa rvac concrns. II. METHODOLOGY Th sarn on n h dsn of Rrsson LVQ (RLVQ alorhs s o nroduc robabl ( x of assnn obsrvaon x o roo ha s dndn on hr (Eucldan dsanc. L us assu ha h robabl dns (x of x can b dscrbd b a xur odl x 1 x, whr s h nubr of coonns, ( s h ror robabl ha a daa on s nrad b a arcular coonn and ( x s h condonal robabl ha coonn nras arcular daa on x. L us rrsn h condonal dns funcon ( x wh h noralzd xonnal for x K( x f ( x,, and consdr a Gaussan xur wh K ( = (π σ 1/ and f (x, = (x / σ. In hs cas, coonn s coll rrsnd b s an and sandard dvaon σ. Thrfor, w can dscrb coonn as roo and w wll follow hs convnon hrouh h rs of h ar. W should obsrv ha was assud ha all roos hav h sa sandard dvaon (wdh σ. Addonall, w wll assu ha ach roo has h sa ror, = 1/. Gvn hs, usn h Bas rul w can wr h assnn robabl as x x ( x. x ( x l 1 Sarn fro uaon (5, n scons II.A and II.B, w ar roosn wo cos funcons ha naurall lad o hard and sof vrsons of rrsson LVQ. A. Hard RLVQ To dvlo cos funcon for Hard RLVQ w roos h follown rdcon odl: daa on x s assnd o a snl roo * b chanc follown h assnn robabls ( x and s labl s rdcd as labl * of roo *. Th xcd Man Suard Error (MSE of such a odl can b xrssd as l D x ( 1 1, whr ( x s fro uaon (5. For h coacnss of noaon, w dfnd ( x and (. To s wh D 1 fro (6 s suabl as cos funcon for Hard RLVQ, consdr h cas whn σ 0. Thn, h assnn robabl ( x for h nars roo s on, whl h assnn robabl for ohr roos s zro. Ths s xacl h MSE of Hard LVQ. Th rason wh cass whn σ s nonzro ar nrsn has o do wh h rann rocdur, as wll b dscrbd shorl. Th obcv of LVQ larnn s o sa h unnown odl arars, nal roo osons and ar valus, = 1. Ths s don b nzn h cos funcon D 1 wh rsc o h arars. To do hs w roos o us h radn dscn hod. For hs, w hav o calcula drvavs D 1 / and D 1 / for = 1. Inrsnl, D 1 / = 0 can b xrssd n h closd for and, hrfor, roo labls could b udad xacl. Th rsuln larnn ruls for Hard RLVQ ar 1 1 ( 1 1 ( 1 ( x 1, whr s h raon nubr and ( s h larnn ra. A = 0, h rocdur sars b slcn a rando (or as h frs, n h on-ln cas rann ons as nal roos. Thn, roo labls ar calculad as n (7 and h rocdur s rad. Larnn s rnad whn h loss funcon D 1 sos rovn (.. b or han An oran ssu o b addrssd s choc of arar σ. Sn o a valu nar zro would hur h larnn rocss and ossbl lad o oor RLVQ odl, as obsrvd n h rlad ozaon robls [13, 16]. Thrfor, w nall s σ o a lar valu and slowl annal owards zro. Th uda rul can b drvd usn h radn dscn hod σ (+1 = σ ( α( D/ σ. Howvr, w xrnall drnd ha hs could lad o nsabls n h larnn rocss. A uch br aroach s o annal σ usn a scfc schdul. Follown h suson fro [16], h annaln was rford usn h schdul σ (+1 = σ (0 σ T /(σ T +, whr σ T s h arar ha conrols how fas s h annaln. For bach vrsons was s o σ T = 5 b dfaul, whl for sochasc vrsons w usd σ T = 5. Hard RLVQ alorh suarzd n (7 s bach-od rurs a ass hrouh whol rann s o rfor a snl raon. I can asl b convrd o h sochasc (onln od ha udas odl afr ach rann on, as 1 1 ( ( 1 ( (. x

3 W can obsrv ha h an dffrnc s n h wa roo ars ar udad hr, w could no us h closd-for uda of (7 bcaus would rur a ass hrouh whol rann daa s, hus dfan h uros of sochasc udan. B. Sof RLVQ To dvlo cos funcon for Sof RLVQ, w us h rdcon odl ha assns daa on x o h roos robablscall and rdcs basd on h whd avra of h roo labls. Assun hs xur odl, w can xrss h osror robabl ( x as x 1 x, ha was oband b obsrvn h condonal ndndnc bwn x and vn, ( x, = (. For ( x w us h Gaussan dsrbuon fro (5. For h robabl of nran ar b roo, (, w also assu Gaussan rror odl wh an ( and sandard dvaon σ. Th rsuln cos funcon D for Sof RLVQ can b wrn as h nav lo-llhood L D 1 ( x ln L 1 1 ln 1 1 ( (,, whr, for h coacnss of noaon, w dfnd ( x and (, σ. L us frs drv h radn dscn rul for nzaon of D, slar o h sochasc radn alorh (8, 1 1 ( ( (. Coarn o h larnn rul for n (8 w s ha (11 uss a whn r ha as no accoun how wll roo dos rlav o h ohr roos. Th bach vrson of uda ruls n (11 can b oband usn h bach radn dscn n a srahforward annr. In h roosd Sof LVQ vrson, w us h sa annaln schduls for σ and α. Th an dffrnc s ha w do no wsh σ o dro o a valu nar zro, bu nsad o s o an arora osv valu. To achv hs, w rsor o usn h valdaon daa s and connu o dcras σ as lon as D valu s bn rovd on h valdaon daa s. Sof RLVQ uss σ valu achvd a h rnaon of rann. x, Coard o Hard RLVQ, Sof RLVQ has σ as an addonal arar. In our alorh, arar σ s s as h an suard rror of h currn RLVQ odl. Gvn an unlabld daa on x u, s ar valu u s rdcd n a sof wa, usn h an funcon, as a whd avra of h roo ar valus u 1 As an alrnav o h radn dscn alorh for nzaon of D w xlord h Excaon- Maxzaon (EM aroach [3]. Th rsuln uda ss for h alorh arars (no shown du o lnh consran ar slar bu alcabl onl o bach od. C. Rrsson LVQ Sarn fro Hard RLVQ and slfn h assnn robabls such ha onl wo closs roos ar consuld ach h rdcon s ndd, w wll donsra ha h rsuln roo udas closl rsbl LVQ larnn rul fro (. Gvn a rann daa on x wh labl, l us dno h closs roo as and h scond closs roo as. W wll nlc h rann roos. Ths s accabl as an aroxaon of h Hard LVQ, bcaus assnn robabls of h wo closs roos ar h hhs. L us consdr wo xr scnaros. In h frs, and ar n slar rox o x. As a consunc, hr assnn robabls wll b aroxal h sa, = = 0.5. Th uda rul fro uaon (7 could hn b xrssd as 1 1 ( ( ( ( u. ( x ( x, whr has bn nlcd for slc ( could b ncororad n h larnn ra arar anwa. Th uda rul s vr slar o LVQ uda n (. Th onl dffrnc s h ( r whch as h aoun of uda dndn on h dffrnc n rrors. I has an nuv nrraon. If rrors of h roos ar slar, hr osons wll no b udad. If rror of -h roo s larr, s ovd awa fro daa on x and - h roo s ovd owards. Ths s xacl h nuon usd n LVQ. Th dffrnc fro LVQ s h cas whn rror of -h roo s sallr. Thr, RLVQ s ovn h -h roo owards and -h roo awa fro h daa on. Unl, LVQ would sa u. Th scond scnaro s whn h closs roo s uch closr han h scond closs, whch as 0. Follown uda rul (7, could b aroxad ha 1 1. Ths rsul s analoous o h vr succssful wndow rul of LVQ. Thrfor, n our alcaon of RLVQ, w

4 us xacl h wndow rul usd n LVQ o dcd whhr h roo osons should b udad. In addon o uda of roo osons, n h Rrsson LVQ cas w hav o uda h roo ar valus as wll. W drv sochasc uda rul fro (8 and bach uda rul fro (7 as sohasc: bach 1 1 n ( ( ( ( :. n n Gvn an unlabld daa on x u, s ar valu u s drnd as h ar valu c of s nars roo c Y u = c, c = ar n d(x u,. Fnall, n F. 1 w suarz h rsuln RLVQ.1 alorh n h sochasc od. Valu of s s chosn bwn 0.4 and 0.8. Rcv a nw rann on (x, Fnd wo closs roos and f n (d / d, d / d > s // h wndow rul us (13 and (15 ls us (14 and (15 Fur 1. Sochasc RLVQ.1 alorh I s nrsn o no ha, n conras o LVQ.1, Hard RLVQ was drvd va ozaon of a scfc cos funcon. Inrsnl, as w showd, can b rducd o a sl hursc n a vr naural wa. D. T Colx T colx of ach run of Rrsson LVQ hrouh all rann xals s ( M, whr s h nubr of rann ons, M s h nubr of faurs, and s h nubr of roos. T colx of calculan h assnn robabls of ach on x, s ( M, and of calculan h rror ach roo as s ( M. T colx of ar valu rdcon for an unlabld on s ( M. Whl all roosd RLVQ alorhs hav sa colx, RLVQ alorh wors h fass bcaus dos no hav o dal wh sof assnns. I also has h fws hrarars, snc onl uss α fro h basc classfcaon LVQ and dos no us σ or σ ndd for Sof RLVQ and Hard RLVQ alorhs. III. EXERIMETAL RESULTS All roosd vrsons of Rrsson LVQ alorh wr valuad on varous rrsson ass, boh ral-lf and snhc. W wr nrsd n coarn h roosd n, Hard and Sof Rrsson LVQ alorhs and Rrsson LVQ wh svral bnchar alorhs. Onl rsuls of h sochasc Rrsson LVQs ar shown dus o lac of sac. Coard wh h bach vrsons, sochasc vrsons rsuld n slhl hhr accuracs, as was xcd []. A. Bnchar Rrsson Alorhs In h follown, w dscrb rrsson alorhs usd for bncharn of roosd Rrsson LVQ alorhs. K-ars hbor alorh s a radonal roo-basd alorh ha wors boh for classfcaon and rrsson. roos ar call all rann ons. Thr, K corrsonds o h nubr of nhbors consuld n an h rdcon. In rrsson scnaro, rdcon s vn as h avra labl of K nars nhbors. arzn Wndow Rrsson s an alorh ha bhavs as a sof vrson of h ars hbor alorh. I as h rdcon as h whd avra of roo labls, whr h wh s drnd basd on h rnl dsanc fro h ur daa on. Th soohn srnh of h whn s dcdd b h rnl wdh arar. K-Mans Rrsson s basd on slcon of roos usn sandard K-ans alorh [19] ha arons rann obsrvaons no clusrs. Th roos ar dfnd as h clusr cnrs. Gvn such a s of roos, boh nars nhbor and arzn wndow alorh can b usd o rovd rdcons. Ths aroach nsurs fxd roo bud, conssn wh h RLVQ alorhs. Thrfor, s arora for bncharn. Rando roos s a sl aroach whr roos ar slcd a rando fro h rann daa. Aan, boh nars nhbor and arzn wndow alorhs can b usd wh such roos. Quanzaon LVQ uss classfcaon o solv rrsson robls. I uanzs h ar varabls no M dscr valus and ras h as M classs. Thn rfors LVQ.1 on M classs usn roos. B. Daa Dscron W valuad Rrsson LVQ alorhs on svral bnchar rrsson daass fro SaLb, Dlv and UCI ML Rosors. Daa ss w usd can b dvdd n wo rous: srs daa ss (Tabl II and h rs (Tabl I. TABLE I. O-TIME SERIES DATA SETS Daa S Sourc Trann Sz Ts Sz Faurs Abalon UCI Ar M10 SaLb Ar O SaLb Bod Fa SaLb Boson UCI Concr UCI CU sall Dlv Fors UCI Houss SaLb MG [4] Moor SaLb MG UCI Sac SaLb Tcaor SaLb

5 TABLE II. TIME SERIES DATA SETS Daa S Sourc Trann Sz Ts Sz La Balloon SaLn Ocan Shar SaLb Rvr Flow SaLb Sana F [17] Sun Sos SaLb Yld SaLb C. Rrsson on a fxd bud Th roosd Rrsson LVQ alorhs usd odra roo buds of 100 on all bu h 3 salls daa ss (hr, bud was s o 50. Th sa roo buds wr usd for K-Mans Rrsson and Rando roos bnchar alorhs. For h K-ars hbor w xlord K = 1, and 3. Th rnl wdh for arzn wndow was ozd usn cross-valdaon. Each of h Rrsson LVQ vrsons usd h sa hrarar valu α 0 = 0.04 and h uda s α( = α 0 α T /(α T +, whr s h and α T = 5, whr s h sz of h rann s. Th hrarahr σ was nalzd as h valu of whn-daa varanc and udad usn h schdul dscrbd n Scon II dndn on h of h alorh (Sof or Hard. Hrarahr σ usd b h Sof RLVQ was alwas s o h sandard dvaon of h suard rror ad b h roos. In h cass whr h daa s was no xlcl dvdd no rann and s ss b h auhors, w randol slcd /3 of h daa s for rann and 1/3 for sn. Ths rocss was rad 10 s and h avra R valus on h s ss ar rord n Tabl III. In addon o shown h R accuracs on ach daa s, n Tabl III w ror how an s ach alorh was aon To and Boo rforrs, as wll as h avra ran of ach alorh. Basd on h To and avra ran scors w can conclud ha all hr vrsons of Rrsson LVQ on a fxd bud rfor slarl o on-bud arzn wndow and K alorhs. Bud vrsons of 1 and arzn wndow wr lss accura hn Bud Rrsson LVQ alorhs. Addonall, K-ans clusrn slcon of roos was or succssful han rando roo slcon n boh Bud 1 and Bud arzn wndow hods. Fnall, Quanzaon LVQ rford oorl and dd no show u as a succssful da. Coarn of Rrsson LVQ alorhs shows ha Sof RLVQ s slhl br han Hard RLVQ. Inrsnl, ds s slc, RLVQ was u cov. To llusra h bhavor of h roosd alorhs n h hhl rsourc consrand scnaros, anohr s of xrns was rford whr h bud alorhs wr rsrcd o a vr h bud (rsuls no shown. As xcd, h rsuln R accuracs wr lowr han whn larr buds ar usd. Coarn wh K-ans and rando roo slcon aroachs for bud annanc, w obsrvd an vn larr donanc of TABLE III. ERFORMACE RESULTS O A FIXED BUDGET. COMARISO WITH BUDGET AD O-BUDGET ALGORITHMS Daa S Hard RLVQ Sof RLVQ RLVQ Bud QLVQ M=3 K-ans Rrsson Rando roos on-bud K- Hard Sof Hard 1- arzn 1- arzn K=1 K= K=3 on-bud arzn Abalon ArM ArO Balloon Balloon rs Boson Bod Fa Concr CU Fors Houss MG Moor MG Ocan Rvr Sana F Sac Sun So Tcaor Yld To Boo Avra Ran

6 RLVQ alorhs. Ths s xcd, bcaus ror roo slcon bcos vr or oran as h bud dcrass. IV. RELATED WORK Th da of sof aron of h faur sac for LVQ was nroducd n [16] and rsuld n drvaon of Sof LVQ alorh. Th auhors nroducd wo varans of Larnn Vcor Quanzaon usn Gaussan xur assuon abou roos. Th larnn rul was drvd b nzn an obcv funcon basd on a llhood rao usn h radn dscn hod. Th for of h rsuln larnn alorh rsbls h radonal LVQ alorhs. I was shown ha sof aron can lad o br classfcaon as coard o h radonal LVQ alorhs. Howvr, Sof LVQ s ld o rforn classfcaon ass and canno drcl b xndd o rfor rrsson. Drnsc Annaln alorh s a sof roo aron alorh frs nroducd for clusrn [14] and hn xndd for vcor uanzaon [1, 15], classfcaon [9] and rrsson [13]. Th nral da bhnd Drnsc Annaln (DA s o fnd h oal roo osons (and ar valus n h rrsson cas b nzn h obcv cos funcon subc o a consran on h nro of h soluon whch s whd b a raur r ha annals durn h rann rocdur. Our cos funcon D 1 can b drvd fro hs DA cos funcon b on h nro consran. Th nuon bhnd annaln h raur s ha can avod shallow local na and roducs a hard soluon a h l of zro raur. Is drawbac s ha, snc h cos funcon s dfnd and nzd a ach raur, h soluon canno b found n an on-ln annr and can a u lon o roduc du o h slow annaln rocss. Mxur of local xrs [6, 9 and 7] s a survsd larnn rocdur ha dcooss a colx as no a nubr of slr larnn robls. Th odl consss of a nubr of scalzd rdcors and a an funcon ha dcds how o cobn h o a a fnal rdcon. B usn h radn dscn [6] or EM [7, 9] hod ach xr larns o handl a subs of h col daa s and h an funcon larns how o cobn h xr s rdcons. Ths rocdur can b vwd as an assocav vrson of cov larnn. RLVQ rsbls h xur of xrs aroach f w ra roos as xrs and us hr dsancs fro a ur on o calcula h an whs. In fac, h cos funcon D roosd n Sof RLVQ closl rsbls h cos funcon usd n h xur of xrs [6]. V. COCLUSIO In hs ar w rsnd a rrsson xnson of h oular LVQ alorh for classfcaon. Afr a horouh sud of sof aron rrsson and xr rlad alorhs, w ralzd ha h xr nwors can b rlacd b uch or asl lnd and nuvl clarr LVQ roos. As a rsul, w roosd hr dffrn vrsons of LVQ alorh for rrsson, on of h usn a sof aron of faur sac and h ohr wo usn h hard aron. Th xrnal rsuls showd ha all vrsons of our alorh ar vr ffcacous and ha h achv consdrabl rovn n rdcon accurac whn coard o ohr roo basd hods and rfor slar o or unconsrand hods. ACKOWLEDGMET Ths wor was suord b h U.S. aonal Scnc Foundaon undr Gran IIS REFERECES [1] Bhl M., Ghosh A., Har B., Dnacs and nralzaon abl of LVQ alorhs, Th Journal of Machn Larnn Rsarch, vol. 8, , 007. [] Boou L., Sochasc larnn, Advancd Lcurs on Machn Larnn , 003. [3] Dsr A.., Lard. M. and Rubn D. B., Maxu llhood fro ncol daa va h EM alorh, J. Roal Sascal Soc B, vol. 39,. 1-38, [4] Fla G. W. and Lawrnc S., Effcn SVM rrsson rann wh SMO. Machn Larnn, vol 46, , 00. [5] Har. E., Th condnsd nars nhbor rul, IEEE Trans. Infor. Thor, vol. IT-14, , [6] Jacobs R. A., Jordan M. I., owlan S.J and Hnon G.E., Adav xurs of local xrs, ural Couaon, vol. 3, , [7] Jordan M. I., Jacobs R. A., Hrarchcal xurs of xrs and h EM alorh, ural Couaon, vol. 6, ,1994. [8] Kohonn T., Th Slf-oranzn Ma, rocdns of h IEEE, vol 78, , [9] Mllr D., Rao A. V., Ros K., and Grsho A., A lobal ozaon chnu for sascal classfr dsn, IEEE Trans. Snal rocssn, vol. 44, , [10] rodrods A., Chan., Solfo S., Advancs n Dsrbud and aralll Knowld Dscovr, MIT rss Cabrd, MA, 000. [11] rodrods A., Chan., and Solfo S., Ma-larnn n dsrbud daa nn sss: Issus and aroachs. In H. Karua and. Chan, dors, Advancs n Dsrbud and aralll Knowld Dscovr. AAAI/MIT rss, Cabrd, MA, 000. [1] Rao A., Mllr D., Ros K. and Grsho A., A nralzd VQ hod for cobnd corsson and saon, IEEE Inrn. Conf. on Acouscs Sch and S. roc., vol. 4, , [13] Ros K., Drnsc Annaln for Clusrn, Corsson, classfcaon, rrsson and rlad ozaon robls, roc. IEEE, vol 86, , [14] Ros K., Gurwz E., Fox G., A drnsc annaln aroach o clusrn, arn Rconon Lrs, vol 11, , [15] Ros K., Gurwz E., Fox G., Vcor uanzaon b drnsc annaln, IEEE Transacons on Inforaon Thor, vol 38, , 199. [16] So S. and Obrar K.. Sof larnn vcor uanzaon, ural Couaon, vol. 15, , 003. [17] Wnd A. S., Manas M. and Srvasava A.., onlnar ad xrs for srs: dscovrn rs and avodn ovrfn, Inrnaonal Journal of ural Sss, vol. 6, , 1995.