Linear Models for Classification
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1 3/8/ Chaer 4 Liear Models for Classifiaio Classifiaio he goal i lassifiaio is o ae a iu veor ad o assig i o oe of K disree lasses C here K he iu sae is divided io K deisio regios hose boudaries are alled deisio boudaries or deisio surfaes ah regio rereses oe lass ad eah iu is lassified io oe of regios lasses Liear Classifiaio Liear lassifiaio meas ha he deisio surfaes are liear fuios of he iu veor ad hee are defied b D - dimesioal herlaes ihi he D- dimesioal iu sae Daa ses hose lasses a be searaed eal b liear deisio surfaes are said o be liearl searable Liear Regressio versus Liear Classifiaio Liear regressio M j jφ j Liear lassifiaio M Regressio j j φ j j here j is j he jh eleme of Classifiaio f here f is a fuio aivaio fuio ha mas io differe lasses C
2 3/8/ Aivaio Fuios he simles aivi fuio is jus iself ie f he mos ommol used aivi fuio is he sigmoid S-shae fuio f e hih mas squashes he real value io a fiie iervals lasses Classifiaio ih sigmoid aivaio fuio is ofe alled logisi regressio Se hreshold fuio Disrimia Fuios A disrimia is a fuio ha aes a iu veor ad assigs i o oe of K lasses deoed C Biar disrimia fuio lassifies iu io o lasses Muli-lass disrimia fuio lassifies iu io K lasses K> hree aroahes o lassifiaio Use disrimia fuios direl ihou robabiliies: Cover he iu veor io oe or more real values so ha a simle oeraio lie hreshholdig a be alied o ge he lass he real values should be hose o maimize he useable iformaio abou he lass label ha is i he real value Disrimiaive aroah: lass C Comue he odiioal robabili of eah lass he mae a deisio ha miimizes some loss fuio Geeraive models -omare he robabili of he iu uder searae lass-seifi g fi a mulivariae Gaussia o he iu veors of eah lass ad see hih Gaussia maes a es daa veor mos robable Is his he bes be? Biar Disrimia he simles rereseaio of a liear disrimia fuio is obaied b aig a liear ombiaio of he iu veor so ha here is alled a eigh veor ad is a bias Iu veor is assiged o lass C if > ad o lass C oherise
3 3/8/ Biar Disrimia o d orresods o a D -dimesioal herlae ihi he D-dimesioal iu sae deermies he orieaio of he deisio surfae Disrimia fuios for > lasses Oe ossibili is o use o-a disrimia fuios ah fuio disrimiaes oe lass from he res Aoher ossibili is o use -/ o-a disrimia fuios ah fuio disrimiaes beee o ariular lasses Boh hese mehods have roblems Problems ih muli-lass disrimia fuios A simle soluio More ha oe good aser o-a referees eed o be rasiive! Use disrimia fuios i j ad i he ma his is guaraeed o give osise ad ove deisio regios if is liear A > α α > α α A j imlies for osiive α A ad B ha j B > A j B B 3
4 3/8/ Mulile Class Disrimia Mulile Class Disrimia A sigle K-lass disrimia osiss of K liear fuios of he form for K assigig o lass C if > j for all j * arg ma he deisio boudar beee lass C ad lass C j is herefore give b j ad hee orresods o a D -dimesioal herlae defied b j j he deisio regios of suh a disrimia are alas sigl oeed ad ove Disrimia Parameer Learig Lear he eighs give iu a se of iu ad ouu Learig mehods Leas-squares mehod Fisher disrimia learig Leas-squares mehod ML mehod simae suh ha he model rediios as lose as ossible o a se of arge values ah lass C is desribed b is o liear model so ha ] K 4
5 3/8/ 5 Leas-squares mehod I veor forma e have Y Y K K K D K K Leas-squares mehod o d Give samles ad heir orresodig lass labels here is a K veor ha follos -of-k odig ie he elemes are all zeros ee for he lass ha belogs o Leas-squares soluio is o fid ha miimizes he sum-of-squares errors ie ] ] ] ] Y Y Leas-squares mehod o d D X ae he derivaives of r o ad seig i o zero e a solve as follos here X X X Leas-squares mehod o d LSQ mehod is suseible o ouliers If he righ aser is ad he model sas 5 i loses so i hages he boudar o avoid beig oo orre
6 3/8/ Aoher eamle here leas squares regressio gives oor deisio surfaes Fisher Liear Disrimia Aoher a o fid a ha a bes liearl searae he lasses ie a ha miimizes he mislassifiaio errors his a be ahieved b rojeig he iu veor ie oo a e subsae here he lasses are bes searaed his is also alled dimesioali reduio sie he dimesio of is iall muh smaller ha ha of Fisher Liear Disrimia o d Fisher Liear Disrimia for Biar Classifiaio he lass searaio i he e feaure sae a be quaiaivel haraerized b he small ihi-lass variae ad large beeelass variae he Fisher rierio is defied o be he raio of he beee-lass variae o he ihilass variae Fisher disrimia aalsis is o fid ha he Fisher rierio is maimized 6
7 3/8/ Fisher Liear Disrimia for Biar Classifiaio o d he Fisher rierio a be defied as S B J S here S B is he beee-lass ovariae mari ad is give b m ad m are samle meas for o lasses ad S is he oal ihi-lass ovariae mari give b Fisher Liear Disrimia for Biar Classifiaio o d Differeiaig Log J ih rese o e fid ha J is maimized he S m m here oe is deermied u o a sale faor Bu ha is o? Fisher Crierio v Leas-square Crierio he leas-squares aroah is based o he goal of maig he model rediios as lose as ossible o a se of arge values he Fisher rierio is derived b requirig maimum lass searaio i he ouu sae More robus ha he LSQ mehod For he o-lass roblem he Fisher rierio a be obaied as a seial ase of leas squares Fisher s disrimia for mulile lasses Cosider he geeralizaio of he Fisher disrimia o K > lasses he eigh veors { } a be osidered o be he olums of a eigh mari so ha he ihi ad beee lass variaes are here S ad m are he variae ad mea of h lass ad m is he oal mea 7
8 3/8/ Fisher s disrimia for mulile lasses o d Oe rierio ha measures he raio of beee-lass variae ad ihi lass variae is Maimizig J leads o hose olums are eigeveors of S S orresodig o he D larges eigevalues Ḅ Geeraive models vs Disrimiaive Model Geeraive models I models he joi robabili disribuio of he lass C ad iu daa ie C Give PC lassifiaio is he erformed b esimaig PC from he joi robabili disribuio Disrimiaive models I direl models he oserior lass disribuio ie PC ad lassifiaio is he doe usig PC Geeraive Models PCPC PC -geeraive model osiss of lass rior P C ad he lielihood PC PC PC Geeraive Models o d Give PC usig Baes rule PC a be omued as C C C C C o use PC for lassifiaio e eed firs esimae PC hih meas e esimae he rior P C ad he lielihood PC C C C C C 8
9 3/8/ 9 ML simae of PC Give iid raiig daa C here Assume o lass ie or ad Pπ P P ] ] π π ML simae of PC o d aig he log of he equaio above ieldig aig he derivaive of LP r o π ad seig i o zero e a solve π as here deoes he oal umber of daa ois for C ad he oal umbers ois for C ] l l ] l l P L π π π ML simae of oiuous PC o d For he lielihood fuio P is arameers deed o he e of disribuio Assume is oiuous ad P follos Gaussia disribuio ad o lasses share he same ovariae mari Σ } e{ / / D π Σ Σ ML simae of PC o d Subsiuig he lielihood fuio ba o he L P disribuio ad aig derivaive ih r ad Σ ieldig heir ML esimaes as C C S S S S 9 Σ
10 3/8/ ML simae of disree PC o d Assume is disree ad i oais D ideede values for eah lass ie for lass ad Hee Subsiuig P ba o L P ad aig derivaive r o i e a obai i D i i i D ] i i Biar Disrimiaive Models Logisi Regressio e model P direl from he daa P is iall modeled as a logisi regressio fuio Give a daa se here ] ad e } { } { his is he so-alled logisi regressio lassifiaio Biar Disrimiaive Models o d aig log of he above L ields ae derivaives of r o ields a solved ieraivel as here a is he learig rae a be radoml iiialized ad i should overge o he global oimal Give P a be obaied for lassifiaio he umber of arameers for disrimiaive models is iall muh smaller ha for he geeraive model ] l l L α ] Ses for Biar Logisi Regressio Give he raiig daa here ad {} Iiialize he veor radoml Comue usig for eah Comue Ieraivel udae usig ih differe learig raes α eg uil he hage of is small Plug io eq o omue for quer daa Classif io lass if P is larger ha 5 oherise o lass e ] α e
11 3/8/ Mulilass Disrimiaive Models Poserior disribuio for lass K> e C e here PC is modeled b a mulilass sigmoid fuio For he joi oserior robabili here is he h lass value for h samle ih folloig -of-k odig K K ] Mulilass Disrimiaive Models ae he log of he joi rob disribuio L L K L aig he derivaive of L r o ields L K L Mulilass Disrimiaive Models a be solved ieraivel as η Give he label of es samle is obaied b * argma
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