A COMPARISON OF MULTIVARIATE DISCRIMINATION OF BINARY DATA

Transcription

1 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK A COMPAISON OF MULTIVAIATE DISCIMINATION OF INAY DATA. I. Egbo;. S.I. Oyeagu & 3. D.D. Ekeze. Deatmet of Mathematcs Alva Ikoku Fedeal College of Educato Owe. Ngea. Deatmet of Statstcs Namd Azkwe Uvesty Awka Ngea 3. Deatmet of Statstcs Imo State Uvesty Owe Ngea ASTACT: The use of classfcato ules fo bay vaables ae dscussed ad evaluated. -softwae ocedues fo dscmat aalyss ae toduced ad aalyzed fo the use wth dscete data. Methods based o the full multomal otmal mamum lkelhood ule ad eaest eghbou ocedues ae teated. The esults obtaed aked the ocedues as follows: otmal mamum lkelhood full multomal ad eaest eghbou ule. Keywods: classfcato ules otmal mamum lkelhood full multomal eaest eghbou bay data. INTODUCTION Data ofte ase the eal wold volvg may obects wth a umbe of measuemets take fom them. These measuemets may be uattatve cotuous o dscete o ualtatve odeed o uodeed categoes. The latte may some cases be defed by oly two categoes ad ae the bay vaables. May motat outcomes ae bay such as ogam ecet labou maket status ad educatoal attamet. These outcomes ae feuetly msclassfed data sets fo easos such as mseotg suveys the eed to use a oy vaable o mefectly lked data. A bay vaable suffes fom msclassfcato f some zeos ae coectly ecoded as oes ad vce vesa whch ca ase fom vaous causes. ay classfcato s the task of classfyg the elemets of a gve set to two gous o the bass of a classfcato ule. Some tycal bay classfcato tasks ae; Medcal testg to deteme f a atet has ceta dsease o ot the classfcato oety s the esece of the dsease. Qualty cotol factoes.e.. decdg f a ew oduct s good eough to be sold o f t should be dscaded the classfcato oety s beg good eough. The oblem of dscmatg betwee two oulatos chaactezed by multomal dstbuto s ecevg etesve coveage the statstcal lteatue. Oe easo fo the ISSN Pt ISSN Ole 40

2 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK ebth of teest the aea s the feuet use of dscmat aalyss the socal ad behavoual sceces whee data ae ofte ot of teval o ato scale. I studes volvg uestoae data demogahc vaables moe ofte tha ot measued by a two thee o fou ot scale ae utlzed to dscmate betwee two o moe gous. I such cases t s moe atual to assume udelyg multomal stuctues ad oceed wth classfcato ocedues based o such chaactezatos tha to elect as most feuetly doe some vaat of Fshe s lea dscmat fucto. Seveal authos have studed multomal classfcato vayg degees of geealty ad wth vayg oetatos. Icluded ths lst ae Cocha ad Hoks 96 Hlls 967 Glbet 968 Glck 97 Mooe 973 Goldste ad abowtz 975 Kzaowsk 975 Ott ad Komal 976 Goldste ad Wolf 977 ad Oyeagu ad Osu 03. The eset study s le wth the wok of Glbet ad Mooe that t attemts to assess the efomace of vaous ocedues though Mote Calo samlg eemets ude dffeet oulato stuctues. I ths feetal settg the eseache ca commt oe of the followg eos. A obect fom may be msclassfed to. Also a obect fom may be msclassfed to. If msclassfcato occus a loss s cued. Let c / be the cost of msclassfyg a obect fom to. The obectve of the study s to fd the est classfcato ule. est hee meas the ule that mmzes the eected cost of msclassfcato ECM. Such a ule s efeed to as the otmal classfcato ule OC. I ths study we wat to fd the OC whee X s dscete ad to be moe ecse eoull.. The Otmal Classfcato ule Ideedet adom Vaables: Let ad be ay two multvaate eoull oulatos. Accodg to Oyeagu 003 Let / c be the cost of msclassfyg a tem wth measuemet fom to ad let be the o obablty o whee wth ad obablty mass Fucto f whee measuemet vecto to f t s some ego. Suose that we assg a tem wth ad to f s some ISSN Pt ISSN Ole 4

3 ego Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK whee ad 0. The eected cost of msclassfcato s gve by: ECM c / f / c/ f /. whee f / =classfyg to / =/. The otmal ule s the oe that attos such that ECM f / = classfyg to / =/ s a mmum. ECM c / f / c/ f /. c / c/ f / c/ f /.3 ECM s mmzed f the secod tem s mmzed. ECM s mmzed f s chose such that c / f / c/ f / 0.4 c / f / c/ f /.5 / f / f / c/ c /.6 Theefoe the otmal classfcato ule wth esect to mmzato of the eected cost of msclassfcato ECM s gve by classfy obect wth measuemet 0 to f f f c/.7 c/ Othewse classfy to. Wthout loss of geealty we assume that / ad c/ = c/. The the mmzato of the ECM becomes the mmzato of the obablty of msclassfcato ISSN Pt ISSN Ole 4

4 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK mc ude these assumtos the otmal ule educes to classfyg a tem wth measuemet 0 to f f 0 / ot :.8 f / 0 Othewse classfy the tem to. Sce s multvaate eoull wth P>0 = = the otmal ule s: classfy a tem wth esose atte to f.9 Othewse classfy the tem to. Ths ule smlfes to: Classfy a tem wth esose atte to f I I.0 Othewse classfy to.. The Otmal ule fo a case of two vaables Suose we have oly two deedet eoull vaables. The the ule becomes: classfy a tem wth esose atte to f: : I I I I.. Othewse classfy the tem to. Wtte aothe fom the ule smlfes to: classfy a tem wth esose atte to f: : w w c.. Othewse classfy the tem to whee ISSN Pt ISSN Ole 43

5 w w Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK I I I..3 I I..4 I c I..5 To fd the dstbuto of z we ote that { [ / ]..6 0 othewse Sce z w w w..7. Otmal ule fo a case of thee vaables. Suose we have thee deedet vaables accodg to Oyeagu 003 the ule s: classfy a tem wth esose atte to f: : I I I 3 3 I othewse classfy the tem to. Wtte aothe fom the ule smlfes to: classfy a tem wth esose atte to f: : w w w3 3 c.. othewse classfy the tem to. w I w I w 3 I c I Otmal ules fo a case of fou vaables Suose we have fou deedet eoull vaables the ule s classfy a tem wth esose atte to f ISSN Pt ISSN Ole 44

6 4 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK : I I I I I I I I Othewse classfy the tem to. Wtte aothe fom the ule smlfes to: classfy a tem wth esose atte to f: w w w3 3 w4 4 c c c3 4 othewse classfy the tem to 4 c Fo the case of fou vaables let w I w I w3 I 3 3 w 4 I c 3 4 I c I c3 I c4 I The z w w w 3 3 w 4 4 w Pobablty of msclassfcato I costuctg a ocedue of classfcato t s desed to mmze o the aveage the bad effects of msclassfcato Oyeagu 003 chad ad Dea 988 Oludae 0. Suose we have a tem wth esose atte fom ethe o. We thk of a tem as a ot a -dmesoal sace. We atto the sace to two egos ad whch ae mutually eclusve. If the tem falls we classfy t as comg fom ad f t falls we classfy t as comg fom. I followg a gve classfcato ocedue the eseache ca make two kds of eos classfcato. If the tem s actually fom the eseache ca classfy t as comg fom. Also the eseache ca classfy a tem fom as comg fom. We eed to kow the elatve udesablty of these two kds ISSN Pt ISSN Ole 45

7 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK of eos classfcato. Let the o obablty that a obsevato comes fom be ad fom be. Let the obablty mass fucto of be f ad that of be f. Let the egos of classfyg to be ad to be. The the obablty of coectly classfyg a obsevato that s actually fom to s / f ad the obablty of msclassfyg such a obsevato to s / f.4. Smlaly the obablty of coectly classfyg a obsevato fom to s / f ad the obablty of msclassfyg a tem fom to s / f.4. The total obablty of msclassfcato usg the ule s TPMC f f.4.3 I ode to deteme the efomace of a classfcato ule the classfcato of futue tems we comute the total obablty of msclassfcato kow as the eo ate. Lachebuch 975 defed the followg tyes of eo ates.. Eo ate fo the otmum classfcato ule ot. Whe the aametes of the dstbutos ae kow the eo ate s TPMC f f whch s otmum fo ths dstbuto. Actual eo ate: The eo ate fo the classfcato ule as t wll efom futue samles. Eected actual eo ate: The eected eo ates fo classfcato ules based o samles of sze fom ad fom ISSN Pt ISSN Ole 46

8 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK v The lug- estmate of eo ate obtaed by usg the estmated aametes fo ad. v The aaet eo ate: Ths s defed as the facto of tems the tal samle whch s msclassfed by the classfcato ule. The table above s called the cofuso mat ad the aaet eo ate s gve by P mc.4.4 Hlls 967 called the secod eo ate the actual eo ate ad the thd the eected actual eo ate. Hlls showed that the actual eo ate s geate tha the otmum eo ate ad t tu s geate tha the eectato of the lug- estmate of the eo ate. Mat ad adley 97 oved a smla eualty. A algebac eesso fo the eact bas of the aaet eo ate of the samle multomal dscmat ule was obtaed by Goldste ad Wolf 977 who tabulated t ude vaous combatos of the samle szes ad the umbe of multomal cells ad the cell obabltes. The esults demostated that the boud descbed above s geeally loose..5 Evaluatg the obablty of msclassfcato fo the otmal ule ot The otmal classfcato ule ot fo... whch s dstbuted multvaate eoull s: classfy a tem wth esose atte to f ot : I I.5. ISSN Pt ISSN Ole 47

9 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK Othewse classfy to We ca obta the obablty of msclassfcato fo two cases Case I Kow aametes a Geeal case whee b Secal case whee... wth the assumto.5.3 c Secal case b wth addtoal assumto that Fo case a the otmal classfcato ule fo... whch s dstbuted ot multvaate eoull s: Classfy a tem wth esose atte f ot : I I.5.5 Othewse classfy to Case b: Secal case whee... wth the assumto that the otmal classfcato ule fo the -vaate eoull models becomes: classfy a tem ot wth esose atte to f othewse classfy to. The obablty of msclassfcato usg the secal case of ot : I I s ot.5.6 / I I I.5.7 I y y y.5.8 y0 ISSN Pt ISSN Ole 48

10 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK 49 ISSN Pt ISSN Ole / I I I I.5.9 I I I I mc.5.0 Case c: Secal case b wth addtoal assumto that ad ad. The otmal classfcato ule ot fo... dstbuted multvaate eoull s: classfy the tem wth esose atte to f : P ot I I.5. ad to othewse. The obablty of msclassfcato usg the secal case of ot whe s / I I.5. / I I I I I I mc.5.3 Fo the fed values of ad dffeet values of ad Case : Ukow aametes a Geeal case... k

11 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK I ode to estmate ad we take tag samles of sze ad fom ad esectvely. I we have the samle k k k The mamum lkelhood estmate of s k k.5.5 Smlaly the mamum lkelhood of estmate of s.5.6 k k We lug ths estmate to the ule fo the geeal case a to have the followg classfcato ule: classfy a tem wth esose atte to f : I I.5.7 othewse classfy to b Secal case of b whee... I ths secal case wth the assumto that k ISSN Pt ISSN Ole 50

12 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK 5 ISSN Pt ISSN Ole s dstbuted k k s dstbuted The mamum lkelhood estmate of s k k.5.9 Lkewse the mamum lkelhood estmate of s k k.5.0 We lug these two estmates to the euato fo the secal case b to have the followg classfcato ule: classfy the tem wth esose atte to f I I.5. Othewse classfy to The obablty of msclassfcato s gve by I I I I mc.5.

13 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK 5 ISSN Pt ISSN Ole mc Whee I I y k y k y k y k c Secal case of b wth 0 we take tag samles of sze fom ad estmate by k k.5.4 Fo a fed value of The classfcato ule s: classfy the tem wth esose atte to f : I I.5.5 othewse classfy to. The obablty of msclassfcato s gve by I I I I mc.5.6

14 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK mc I.5.7 I 3. Mamum Lkelhood ule ML-ule The mamum lkelhood dscmat ule fo allocatg a obsevato to oe of the oulatos.. s to allocate to the oulato whch gves the lagest lkelhood to. Classfy f w / w / o to f w / w / 3.. whee w / s the osteo obablty whch ca be foud by the ayes ule. ut ths s the same as: classfy to f / w w / w w 3. whee / w s the class codtoal obablty desty fucto ad w s the o obablty. y deotg the classes as the mamum lkelhood classfe s based o the assumed multvaate omal obablty desty fucto fo each class gve by T f / e 3.3 whee s the estmated mea vecto fo class ad s the estmated vaace covaace mat fo class ad s the umbe of chaactestcs measued e the legth of each vecto to oe of the classes ecall that the desty fucto f / s evaluated fo each of the k classes ad the s assged to f assumg eual costs of msclassfcato ad eual a o obabltes oe has f / f / fo all 3.4 ISSN Pt ISSN Ole 53

15 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK We assumed that the data ca be modeled adeuately by a mult-omal dstbuto. If the class-codtoal obablty desty fucto / w s estmated by usg the feuecy of occuece of the measuemet vectos the tag data the esultg classfe s oaametc. A motat advatage of the o-aametc classfe s that ay atte howeve egula t may be ca be chaactezed eactly. Ths advatage s geeally outweghed by two dffcultes wth the o-aametc aoach. It s dffcult to obta a lage eough tag samle to adeuately chaacteze the obablty dstbuto of a mult-bad data set. Secfcato of a meagful -dmesoal obablty desty fucto eues a massve amout of memoy o vey cleve ogammg. 4. The Full Multomal ule Suose we have dscete adom vaables each assumg values 0 o. The ot obablty mass fucto mf accodg to Had 983 s gve by:! !!...! fo =0 fo each but wth the subect to the estcto The age sace of cossts of vectos... whee... ad fo ou uoses s assumed that t s geeated by a -adom vecto whose agumet s 0 o. Suose that two gous ad ae lage oulatos havg o obabltes ad whee ude the full multomal the obablty mass fucto deoted by f wth mmum vaace ubased estmatos s f 4. whee s the umbe of the dvduals a samle of sze fom the th oulato ISSN Pt ISSN Ole 54

16 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK havg esose atte. The classfcato ule s: classfy a tem wth esose atte to f : ad to f ad wth obablty f : 4.5 A tutve estmate based o D of the otmal eo s the aaet eo smly defed as the ooto of eos made by the ule. The aaet eo of the samle based full multomal classfcato ule assumes the fom: m D 4.6 whee ae the umbe of samle values fom oulato ad esectvely. The advatages of usg the full multomal ule accodg to Oyeagu 003 ae as follows: It s etemely smle to aly. Secodly the comutato of aaet eo does ot eue goous comutatoal fomula. The dsadvatages ae as follows: Thee ae howeve ceta obsevatos that ae aaet ad ot to otetal dffcultes alyg the so-called full-multomal ule. Pehas the most omet s the oblem of state olfeato made esecally toublesome actce by the avalablty of elatvely small samle szes vaables each assumg oly k dstct values geeate k states. Obvously a lage umbe of obsevatos elated to the umbe of vaables s eued f suffcet data each state ae to be avalable fo estmato of state obabltes. ISSN Pt ISSN Ole 55

17 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK A at fom the oblem of zeos states o the otetal of fa obsevatos o whch to base the estmato of state obabltes the ssue that fo a gve state a zeo fom mght mea somethg etely dffeet fom a zeo comg fom f the otmal ocedue s used. Moeove esecally f samles of dsootoate szes ae avalable eo ates geeated by the somewhat foced allocato caused by a zeo say state fom ca be otetally msleadg. It s because of these dffcultes that some eseaches have bee eluctat to aly the full multomal ocedues stuatos whee the data dffes fom sevee saseess. 5. The Neaest Neghbou Pocedue The kth eaest eghbou method K-NN s aothe tool that s used wheeve the class desty fuctos f ae kow. I fact ths was the fst o-aametc method fo classfcato ad was toduced by F ad Hodges 95. The dea behd the method s elatvely smle. Clak 978 defe a adom obsevato Xm... } as the eaest m { eghbou to f: M d=dm = 5. whee d s a dstace fucto. The eaest eghbou ule decdes that belogs to the class of ts eghbou Xm. The above s the sgle eaest eghbou ule that s k = I ad oly ales to the sgle eaest eghbou to. All othe obsevatos ae goed. The dea s eteded atually to the k-eaest eghbous of. Lachebuch 975 descbes the geeal K-NN ules as follows: Suose thee ae ad samle obsevatos fom ad esectvely. Suose that the obectve s to classfy a obsevato to oe of o. Usg a dstace fucto d ode the values. Let k be the umbe of obsevatos fom amog the k closet obsevatos to. The ule s to assg to f: k k 5. othewse to. I othe wods the ocedue volves the elatvely smle cocet of assgg a adom obsevato to the class havg the geate ooto of obsevatos closet to. As t has bee foud that teds to the mamum lkelhood ISSN Pt ISSN Ole 56

18 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK ule. Thee ae seveal vaatos of the dscete aalogue to the above estmato each wth the ow oeatoal dffcultes. See Had 993 fo detals. Hlls 967 toduced ehas the smlest eaest eghbou estmato fo bay data whch classfes a atcula esose vecto based o the umbe of cells esose vectos y that dffe fom. Secfcally let k be the umbe of cells whch ad y dffe. The defe y y y k by o moe tha k comoets. That s classfy to f: ad to othewse. Advatages ad dsadvatages to be a ule whch classfes f each of ts cells dffes y y Howeve these methods ae ot wthout the lmtatos ad ae based o some assumtos. Although eaest eghbou s dstbuto fee ad the classfe the has o elct fuctoal fom t s vey dffcult to check the assumto that the dstbuto s locally costat ea. Also the choce of the dstace fucto must be take to cosdeato. It must be aoate ad meagful. Fo eamle Eucldea dstace s usually the default choce but may ot be aoate as such cases whee the vaables ae of vey dffeet magtudes ad must be stadadzed fst. Also dstaces hgh dmesos becomes comlcated ad assgg oe obect to be eae tha othe gets blued because as gets ceasgly lage the ato of eaest to futhest eghbous aoaches. 6. Smulato Eemets ad esults The fou classfcato ocedues ae evaluated at each of the 8 cofguatos of ad d. The 8 cofguatos of ad d ae all ossble combatos of = =3 4 5 ad d = ad 0.4. A smulato eemet whch geeates the data ad evaluates the ocedues s ow descbed. A tag data set of sze s geeated va -ogam whee obsevatos ae samled fom whch has multvaate eoull dstbuto wth ut aamete ad obsevatos samled fom whch s multvaate 5.3 ISSN Pt ISSN Ole 57

19 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK eoull wth ut aamete.... These samles ae used to costuct the ule fo each ocedue ad estmate the obablty of msclassfcato fo each ocedue s obtaed by the lug- ule o the cofuso mat the sese of the full multomal. The lkelhood atos ae used to defe classfcato ules. The lug- estmates of eo ates ae detemed fo each of the classfcato ules. Ste ad ae eeated 000 tmes ad the mea lug- eo ad vaaces fo the 000 tals ae ecoded. The method of estmato used hee s called the esubsttuto method. The followg table cotas a dslay of oe of the esults obtaed Table 6a Aaet eo ates fo classfcato ules ude dffeet aamete values samle szes ad elcatos P = P = ISSN Pt ISSN Ole 58

20 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK Samle szes Otmal Full M. NN ML mc = Table 6b Actual Eo ate fo the classfcato ules ude dffeet aamete values samle szes ad elcatos. P = P = mc mc Samle sze Otmal Full M. NN ML Tables 6a ad b eset the mea aaet eo ates ad stadad devato actual eo ates fo classfcato ules ude dffeet aamete values. The mea aaet eo ates ISSN Pt ISSN Ole 59

21 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK ceases wth the cease samle szes ad stadad devato deceases wth the cease samle szes. Pedctve ad Dllo-Goldste moved wth the cease the umbe of vaables whle mamum decease efomaces. Fom the aalyss otmal s aked fst followed by lea dscmat aalyss edctve ule Dllo-Goldste lkelhood ato mamum lkelhood full multomal ad eaest eghbou occued the last osto as show below. Classfcato ule Pefomace Otmal OP Mamum Lkelhood ML Full Multomal FM 3 Neaest Neghbou NN 4 Cocluso We obtaed two mao esults fom ths study. Fstly usg smulato eemets we aked the ocedues as follows: Otmal Mamum lkelhood Full multomal ad Neaest Neghbou. The best method was the otmal classfcato ule. Secodly we cocluded that t s bette to cease the umbe of vaables because accuacy ceases wth ceasg umbe of vaables. efeeces Clak G.M. 978.Pedctg the Suvval ued Patets Usg Dscmat Aalyss UNS Cocha W.G. & Hoks C.E. 96. Some classfcato oblems wth multvaate Qualtatve Data ometcs Glbet S.E O Dscmato usg Qualtatve Vaables Joual of the Ameca Statstcal Assocato Glck N. 97. Samle based classfcato ocedues deved fom desty estmatos. Joual of Ameca Statstcal Assocato Goldste M. & abowtz 975. Selecto of vaables fo the two gou multomal classfcato oblem. JASA Goldste M. & Wolf 977. O the oblem of as multomal classfcato. ometcs ISSN Pt ISSN Ole 60

22 Iteatoal Joual of Mathematcs ad Statstcs Studes Vol.No Setembe 04 Publshed by Euoea Cete fo eseach Tag ad Develomet UK Had D.J New stumets fo detfyg good ad bad cedt sks: a feasblty study eot Tustee Savgs ak Lodo Hlls M Dscmato ad allocato wth dscete data Aled Statstcs Kzaowsk W.J Pcles of Multvaate Aalyss: Uses esectve. Joh Wlley ad sos Ic. ew Yok.. Lachebuch P.A Dscmat Aalyss Hafe ess New Yok. Mat D.C. & adley.a. 97. Pobablty Models Estmato ad Classfcato fo Multvaate Dchotomous Poulatos ometcs Mooe D.H Evaluato of fve dscmato ocedues fo bay vaables. Joual of the Ameca Statstcal Assocato Oludae S. 0. obust Lea classfe fo eual Cost atos of msclassfcato CN Joual of Aled Statstcs. Oyeagu S Devato of a otmal classfcato ule fo dscete vaables. Joual of Ngea Statstcal Assocato vol Oyeagu S.I. & Osu G.A. 03. Evaluato of seve classfcato ocedues fo bay vaables. Joual of the Ngea Statstcal Assocato vol 0. ISSN Ott J. & Komal.A Some classfcato ocedues fo multvaate bay data usg othogoal fuctos. Joual of Ameca Statstcal Assocato chad A.J. & Dea W.W Aled Multvaate Statstcal Aalyss. 4th edto Petce Hall Ic. New Jessey. ISSN Pt ISSN Ole 6