A Comparison of the Optimal Classification Rule and Maximum Likelihood Rule for Binary Variables

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1 Joual of Mathematcs Reseach; Vol. 6 No. 4; 04 ISSN E-ISSN Publshed by Caada Cete of Scece ad Educato A Comaso of the Otmal Classfcato Rule ad Mamum Lelhood Rule fo ay Vaables I. Egbo S. I. Oyeagu D. D. Eeze 3 &. Uzoma Pete O. 4 Deatmet of Mathematcs Alva Iou Fedeal College of Educato Owe Ngea. Deatmet of Statstcs Namd Azwe Uvesty Awa Ngea. 3 Deatmet of Statstcs Imo State Uvesty Owe Ngea. 4 Deatmet of Comute Scece Alva Iou Fedeal College of Educato Owe Ngea Coesodece: I. Egbo Deatmet of Mathematcs Alva Iou Fedeal College of Educato Owe Ngea. E-mal: egboe@gmal.com Receved: Setembe 6 04 Acceted: Octobe 9 04 Ole Publshed: Novembe 04 do:0.5539/m.v644 URL: htt://d.do.og/0.5539/m.v644 Abstact Otmal classfcato ule ad mamum lelhood ules have the lagest ossble osteo obablty of coect allocato wth esect to the o. They have a ce otmal oety ad aoate fo the develomet of lea classfcato models. I ths ae we cosde the oblem of choosg betwee the two methods ad set some gudeles fo oe choce. The comaso betwee the methods s based o seveal measues of edctve accuacy. The efomace of the methods s studed by smulatos. Keywods: otmal classfcato ule mamum lelhood ule ay vaables.. Itoducto Otmal classfcato ules ad mamum lelhood ule ae wdely used multvaate statstcal methods fo aalyss of data wth categocal outcome vaables. oth of them ae aoate fo the develomet of lea classfcato models.e. models assocated wth lea boudaes betwee the gous. ay classfcato s the tas of classfyg the elemets of a gve set to two gous o the bass of a classfcato ule. Classfcato s of boad teest scece because t emeates may scetfc studes ad also ases the cotets of may alcatos Pael o Dscmat Aalyss Classfcato ad Clusteg 989. Eamles the educatoal socal ad behavoual sceces clude detfyg chlde degate at s fo futue eadg dffcultes Catts Fey Zhag ad Tombl 00 detfyg dvduals at s fo addcto Robso 00 ad edctg the cmes that male uvele offedes may commt accodg to the esoalty chaactestcs Glase Calhou ad Petocell 00. I the bologcal ad medcal sceces alcato of classfcato ocedues clude detfyg atets wth choc heat falue Uds 00 detectg lug cace Phls 003 ad detemg whethe ceta beast masses ae malgat o beg Sahe 004. I the maagemet sceces methods fo classfcato have bee used fo such uoses as edctg bautcy Jo Ha ad Lee 997; Dchotomous classfcato of Foeg Asssted Poect mlemetato status Nwouh ad Ayam 00. I ths ae we shall be coceed wth = oulato classfcato oblems. Ou teest s devg a ule that ca be used to otmally assg a tem to oe of the oulatos. The otmalty cteo s to mmze the s assocated wth the ule Oyeagu & Osu 00. The goal of ths ae s to set some gudeles as to whe the choce of ethe oe of the methods s stll aoate. Whle otmal s much moe geeal ad has a umbe of theoetcal oetes mamum lelhood must be the bette choce f we ow the oulato s omally dstbuted. Howeve actce the assumtos ae ealy always volated ad we have theefoe ted to chec the efomace of both methods wth smulatos. Ths d of eseach demads a caeful cotol so we have decded to study ust a few chose stuatos tyg to fd a logc the behavou ad the th about the easo oto moe geeal cases. We have cofed ouselves to comae oly the edctve owe of the methods. Secto ad 3 befly descbes the algothms; secto 4 descbes the ocess of the smulatos. The esults 4

2 Joual of Mathematcs Reseach Vol. 6 No. 4; 04 obtaed ae eseted ad dscussed secto 5 ad coclusos ad ecommedatos ae gve secto 6.. The Otmal Classfcato Rule Ideedet Radom Vaables: Let ad be ay two multvaate eoull oulatos. 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. Suose that we assg a tem wth measuemet vecto to f t s some ego R R ad to f s some ego R R whee R R R ad R R 0. The eected cost of msclassfcato s gve by: ECM c / f / c/ f /. R R whee f / =classfyg to / =/ whee R classfed as. The otmal ule s the oe that attos R such that ECM f / = classfyg to / =/ s a mmum. R ECM c / c / f / c/ f / R R c/ f / c / f / R / whe obsevato s coectly..3 ECM s mmzed f the secod tem s mmzed. ECM s mmzed f R s chose such that c / f / c / f / 0.4 / f / c / f / c.5 R / f / c/ f / 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 to 0 f f c/ R : f c/ f c/ R :.7 f 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 mc ude these assumtos the otmal ule educes to classfyg a tem wth measuemet 0 to f f 0 / R 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 5

3 Joual of Mathematcs Reseach Vol. 6 No. 4; 04 Othewse classfy the tem to. Ths ule smlfes to: Classfy a tem wth esose atte to f I I.0 Othewse classfy to. Fo ay ule the aveage o eected cost of msclassfcato ECM s ovded by the oduct of the off-etes by the obabltes of occuece. A good classfcato ule should have a ECM as small as ossble. The egos R ad R that mmze the ECM ae defed by the values fo whch the eualtes ae defed. If the aametes ae uow the they ae estmated by the mamum lelhood estmatos gve by.9. whee s eual to the umbe of obsevato fom wth th vaable. The ule fo uow aametes s: classfy a tem wth esose atte to f I I othewse classfy the tem to. The Otmal Rule fo a Case of Two Vaables Two Gou Classfcatos. Suose we have oly two deedet eoull vaables. The the ule becomes: classfy a tem wth esose atte to f: R : I I I I.. Othewse classfy the tem to. Wtte aothe fom the ule smlfes to: classfy a tem wth esose atte to f: R : w w c.. Othewse classfy the tem to whee To fd the dstbuto of z we ote that w I I I..3 w I I..4 I c I..5 { [ / ]..6 0 othewse 6

4 Joual of Mathematcs Reseach Vol. 6 No. 4; 04 Sce The age of z s z w w w..7 R { 0 w w w w} [ z 0/ ] 0 0/..8 z w / 0/..9 z w w / ] /..0 f 0 z z / If w w the dstbuto fucto of z s gve by 0 f z =0 z / f z w.. f z w.. f z w w..3 f 0 z w..4 f w z w f w z w w..5. Otmal Rule fo a Case of Thee Vaables Two Gou Classfcatos. f w w z..6 Suose we have thee deedet vaables accodg to Oyeagu 003 the ule s: classfy a tem wth esose atte to f: : I I I R I.. othewse classfy the tem to. Wtte aothe fom the ule smlfes to: classfy a tem wth esose atte to f: R 3 : w 3 3 w w c.. othewse classfy the tem to. w I w I w 3 I c I Otmal Rules fo a Case of Fou Vaables Two Gou Classfcatos Suose we have fou deedet eoull vaables the ule s classfy a tem wth esose atte to f R 4 : I I I 4 I I I I I

5 Joual of Mathematcs Reseach Vol. 6 No. 4; 04 othewse classfy the tem to. Wtte aothe fom the ule smlfes to: classfy a tem wth esose atte to f: R w w 4 w3 3 w4 4 c c c3 c4 othewse classfy the tem to. Fo the case of fou vaables let 3 3 w I w I w3 I 3 3 w 4 I The dstbuto fucto s deved ust the same way as the case of thee vaables. Usg the same method the obablty mass fucto of z ad the dstbuto fucto fo the case of fve vaables could be deved..4 Pobablty of Msclassfcato I costuctg a ocedue of classfcato t s desed to mmze o the aveage the bad effects of msclassfcato Oyeagu 003 Rchad ad Dea 988 Oludae 0. Suose we have a tem wth esose atte fom ethe o. We th of a tem as a ot a -dmesoal sace. We atto the sace R to two egos R ad R whch ae mutually eclusve. If the tem falls R we classfy t as comg fom ad f t falls R we classfy t as comg fom. I followg a gve classfcato ocedue the eseache ca mae two ds 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 ow the elatve udesablty of these two ds 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 R ad to be R. The the obablty of coectly classfyg a obsevato that s actually fom to s / f R ad the obablty of msclassfyg such a obsevato to s / f R.4. Smlaly the obablty of coectly classfyg a obsevato fom to s / f ad the obablty of msclassfyg a tem fom to s The total obablty of msclassfcato usg the ule s / f.4. R TPMC R f f.4.3 R R I ode to deteme the efomace of a classfcato ule R the classfcato of futue tems we comute the total obablty of msclassfcato ow as the eo ate. Lachebuch 975 defed the followg tyes of eo ates.. Eo ate fo the otmum classfcato ule R ot. Whe the aametes of the dstbutos ae ow the eo ate s TPMC R f f whch s otmum fo these dstbuto. R R 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 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. R 8

6 Joual of Mathematcs Reseach Vol. 6 No. 4; 04 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. Fuuaga ad Kessel 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 Pobablty of Msclassfcato fo the Otmal Rule R ot The otmal classfcato ule R ot fo... whch s dstbuted multvaate eoull s: classfy a tem wth esose atte to f R Othewse classfy to We ca obta the obablty of msclassfcato fo two cases Case I Kow aametes a Geeal case whee... ot : I I b Secal case whee wth the assumto c Secal case b wth addtoal assumto that 0 Fo case a the otmal classfcato ule R ot fo... whch s dstbuted multvaate eoull s: Classfy a tem wth esose atte f R ot : I I.5. Othewse classfy to Case b: Secal case whee... wth the assumto that the otmal classfcato ule R ot fo the -vaate eoull models becomes: classfy a tem wth esose atte to f othewse classfy to. The obablty of msclassfcato usg the secal case of R ot s R ot : I I.5.3 / I I I.5.4 I y y y y0 whee have bomal dstbuto wth aametes ad.5.5 / I I I.5.6 I 9

7 Joual of Mathematcs Reseach Vol. 6 No. 4; 04 I I mc.5.7 I I Case c: Secal case b wth addtoal assumto that ad ad. The otmal classfcato ule R ot fo... dstbuted multvaate eoull s: classfy the tem wth esose atte to f ad to othewse. R ot : I I P The obablty of msclassfcato usg the secal case of R ot whe s mc.5.8 I /.5.9 I / I I I I Fo the fed values of ad dffeet values of ad Case : Uow aametes I I.5.0 a Geeal case... I ode to estmate ad we tae tag samles of sze ad fom ad esectvely. I we have the samle... The mamum lelhood estmate of s Smlaly the mamum lelhood of estmate of s.5.3 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 R : I I

8 Joual of Mathematcs Reseach Vol. 6 No. 4; 04 3 othewse classfy to b Secal case of b whee... wth the assumto that I ths secal case s dstbuted s dstbuted The mamum lelhood estmate of s.5.6 Lewse the mamum lelhood estmate of s.5.7 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.5.8 Othewse classfy to The obablty of msclassfcato s gve by I I I I mc.5.9 mc mc = Estmate of omal tems of dffeece..5.0 Whee I I y y y y 0 whee s omal fucto.5. c Secal case of b wth 0 we tae tag samles of sze fom ad estmate by

9 Joual of Mathematcs Reseach Vol. 6 No. 4; Mamum Lelhood Rule ML-Rule The mamum lelhood dscmat ule fo allocatg a obsevato to oe of the oulatos.. s to allocate to the oulato whch gves the lagest lelhood to. That s the mamum lelhood ule says oe should allocate to whe f L ma L Adeso s the N oulato... g ad 0 the the mamum lelhood dscmat ule allocate to whee {... } s that value of whch mmzed the Mahalaobs dstace whee g the ule allocate to. If a 0 ad a { } 0 whee a ad ad to othewse. Alteatvely classfy f w / / w o to f w / w / 3. whee w / s the osteo obablty whch ca be foud by the ayes Rule. ut ths s the same as: classfy to f / w w / w w Smulato Eemets ad Results The two 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 whee = umbe of vaables d = effect sze = samle sze. A smulato eemet whch geeates the data ad evaluates the ocedues s ow descbed. A tag data set of sze s geeated va R-ogam whee obsevatos ae samled fom whch has multvaate eoull dstbuto wth ut aamete ad obsevatos samled fom whch s multvaate 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 lelhood 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 some of the esults obtaed Table 4.a Effect of ut aametes P ad P o classfcato ules at vaous values of samle sze ad Relcatos mea aaet eo ates P = P = Samle szes Otmal ML mc =

10 Joual of Mathematcs Reseach Vol. 6 No. 4; 04 Table 4.b Effect of ut aametes P ad P o classfcato ules at vaous values of samle sze ad Relcatos actual eo ates P = P = mc mc Samle szes Otmal ML Tables 4.a ad b eset the mea aaet eo ate ad stadad devato actual eo ates of two classfcato ules. The aaet eo ates ceases wth the samle sze. Fom the table 4.b the eo ates deceases wth the samle sze. Wth =000 two classfcato ules have the same eo ate. O the aveage mamum lelhood as fst followed by otmal. Classfcato Rule Pefomace Mamum Lelhood ML Otmal OP Table 4.a Aaet eo ates fo classfcato ules ude dffeet aamete values samle szes ad Relcatos P = P = Samle szes Otmal ML mc =

11 Joual of Mathematcs Reseach Vol. 6 No. 4; 04 Table 4.b Actual Eo ate fo the classfcato ules ude dffeet aamete values samle szes ad elcatos. P = P = mc mc Samle sze Otmal ML Tables 4.a ad b eset the mea aaet eo ates ad stadad devato fo the classfcato ules ude dffeet aamete values. The aaet eo ates ceases wth the cease the samle szes. Classfcato Rule Pefomace Mamum Lelhood ML Otmal OP Table 4.3a Aaet eo ates fo classfcato ules ude dffeet aamete values samle szes ad Relcatos P = P = Samle szes Otmal ML mc =

12 Joual of Mathematcs Reseach Vol. 6 No. 4; 04 Table 4.3b Actual eo ate fo the classfcato ules ude dffeet aamete values samle szes ad elcatos. P = P = mc mc Samle sze Otmal ML Table 4.3a ad b show the mea aaet eo ates ad stadad devato actual eo ates fo the classfcato ules ude dffeet aamete values. It s clea to see that the mea aaet eo ate ceases wth the cease the samle szes. The stadad devato deceases wth the cease samle szes. As the umbe of vaables ceases the efomace of the mamum lelhood deceases. Fom the aalyss otmal ule s aed fst followed by mamum lelhood. Classfcato Rule Pefomace Otmal OP Mamum Lelhood ML 5. Cocluso Mamum lelhood ocedue efomed well fo small ad modeate umbe of vaables esectve of the samle sze whle otmal classfcato ule aeas to be moe cosstet fo small modeate ad lage umbe of vaables. Theefoe otmal s moe effectve classfe tha mamum lelhood. Refeeces Adeso T. W. 98. Classfcato by Multvaate Aalyss. Psychometa htt://d.do.og/0.007/f03345 Catts H. W. Fey M. E. Zhag X. & Tombl J Estmatg the s of futue eadg dffcultes degate chlde: A eseach-based model ad ts clcal mlemetato. Laguage Seech & Heag sevces schools htt://d.do.og/0.044/ /004 Fuuaga K. & Kessel D. L. 97. Alcato of otmum eo-eect fucto. IEEE Tas. Ifomato IT Glase. A. Calhou G.. & Petocell J. V. 00. Pesoalty chaactestcs of male uvele offedes by adudcated offeses as dcated by the MMH-A. Cmal Justce ad ehavou htt://d.do.og/0.77/ Goldste M. & Wolf O the oblem of as multomal classfcato. ometcs htt://d.do.og/0.307/5978 Hlls M Dscmato ad allocato wth dscete data. Aled Statstcs htt://d.do.og/0.307/

13 Joual of Mathematcs Reseach Vol. 6 No. 4; 04 Jo H. Ha I. & Lee H autcy edcto usg case-based easog eual etwos ad dscmat aalyss. Eet Systems wth Alcatos htt://d.do.og/0.06/s Lachebuch P. A Dscmat Aalyss Hafe ess New Yo. Nwouh G. E. & Ayam K. E. 00. Alcatos of Dscmat Aalyss o the classfcato of Imlemeted Foeg Asssted oect Ngea. Joual of the Ngea Statstcal Assocato. Oludae S. 0. Robust Lea classfe fo eual Cost Ratos of msclassfcato. CN Joual of Aled Statstcs. Oyeagu S. I Devato of a otmal classfcato ule fo dscete vaables. Joual of Ngea Statstcal Assocato Oyeagu S. I. & Osu G. A. 00. Evaluato of seve classfcato ocedues fo bay vaables. Joual of the Ngea Statstcal Assocato 0. Pael o Dscmat Aalyss Classfcato ad Clusteg. Statstcal Scece Phlls M Detecto of lug cace wth volatle maes the beath. Chest htt://d.do.og/0.378/chest Rchad A. J. & Dea W. W Aled Multvaate Statstcal Aalyss. 4th edto Petce Hall Ic. New Jessey. Robso. 00. A stuctual ad dscmat aalyss of the Wo Addcto Rs Test. Educatoal ad Psychologcal Measuemet Sahe Comutezed chaactezato of beast masses of thee-dmesoal ultasoud volumes. Medcal Physcs htt://d.do.og/0.8/ Uds E. M. 00. Comag methods to detfy geeal teal medce clc atets wth choc heat falue. Ameca Heat Joual htt://d.do.og/0.067/mh Coyghts Coyght fo ths atcle s etaed by the authos wth fst ublcato ghts gated to the oual. Ths s a oe-access atcle dstbuted ude the tems ad codtos of the Ceatve Commos Attbuto lcese htt://ceatvecommos.og/lceses/by/3.0/. 36