Support vector machines for regression

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1 S 75 Mchne ernng ecture 5 Support vector mchnes for regresson Mos Huskrecht mos@cs.ptt.edu 539 Sennott Squre S 75 Mchne ernng he decson oundr: ˆ he decson: Support vector mchnes ˆ α SV ˆ sgn αˆ SV!!: Decson on ne requres to compute the nner product eteen the empes Smr the optmzton depends on j n n J α α α α j j j j S 75 Mchne ernng

2 Nonner cse he ner cse requres to compute he non-ner cse cn e hnded usng set of fetures. Essent e mp nput vectors to rger feture vectors φ It s posse to use SVM formsm on feture vectors Kerne functon φ φ ' ruc de: If e choose the kerne functon se e cn compute ner seprton n the feture spce mpct such tht e keep orkng n the orgn nput spce!!!! K ' φ φ ' S 75 Mchne ernng Kerne functon empe Assume [ nd feture mppng tht mps the nput ] nto qudrtc feture set φ [ ] Kerne functon for the feture spce: K ' φ ' φ ' ' ' ' ' ' ' ' ' he computton of the ner seprton n the hgher dmenson spce s performed mpct n the orgn nput spce S 75 Mchne ernng

3 Kerne functon empe ner seprtor n the feture spce Non-ner seprtor n the nput spce S 75 Mchne ernng Ponom kerne Kerne functons ner kerne K ' ' [ ] ' k K ' Rd ss kerne K ' ep ' S 75 Mchne ernng

4 Kernes he dot product s dstnce mesure Kernes cn e seen s dstnce mesures Or converse epress degree of smrt Desgn crter - e nt kernes to e vd Stsf Mercer condton of postve semdefnteness good emod the true smrt eteen ojects pproprte generze e effcent the computton of k s fese NP-hrd proems ound th grphs S 75 Mchne ernng Kernes Reserch hve proposed kernes for comprson of vret of ojects: Strngs rees Grphs oo thng: SVM gorthm cn e no pped to cssf vret of ojects S 75 Mchne ernng

5 Support vector mchne SVM SVM mmze the mrgn round the seprtng hperpne. he decson functon s fu specfed suset of the trnng dt the support vectors. S 75 Mchne ernng Regresson fnd functon tht fts the dt. A dt pont m e rong due to the nose Ide: Error from ponts hch re cose shoud count s vd nose ne shoud e nfuenced the re dt not the nose. Support vector mchne for regresson S 75 Mchne ernng

6 ner mode rnng dt: n {... } R R Our go s to fnd functon f tht hs t most devton from the ctu otned trget for the trnng dt. f S 75 Mchne ernng ner mode ner functon: f We nt functon tht s: ft: mens tht one seeks sm dt ponts re thn ts neghorhood he proem cn e formuted s conve optmzton proem: mnmze suject to A dt ponts re ssumed to e n the neghorhood S 75 Mchne ernng

7 S 75 Mchne ernng f ner mode Re dt: not dt ponts s f nto the neghorhood Ide: penze ponts tht f outsde the neghorhood S 75 Mchne ernng f ner mode ner functon: Ide: penze ponts tht f outsde the neghorhood suject to mnmze

8 S 75 Mchne ernng -ntensve oss functon ntensve oss functon otherse for ner mode S 75 Mchne ernng grngn tht soves the optmzton proem Optmzton Suject to Prm vres

9 S 75 Mchne ernng Optmzton Dervtves th respect to prm vres S 75 Mchne ernng ω ω Optmzton

10 S 75 Mchne ernng ω ω Optmzton S 75 Mchne ernng ] [ : suject to - j j j - Optmzton Mmze the du

11 Souton We cn get: f t the optm souton the grnge mutpers re non-zero on for ponts outsde the nd. S 75 Mchne ernng Nonner etenson Kerne trck Repce the nner product th kerne A e chosen kerne eds to effcent computton S 75 Mchne ernng

12 Evuton frmeork Dt set rnng set est set cse contro cse contro ern on the trnng set he mode Evute on the test set S 75 Mchne ernng Evuton metrcs onfuson mtr: Records the percentges of empes n the testng set tht f nto ech group Predcton se ontro se P.3 FN. Actu ontro FP. N.4 Mscssfcton error: E FP FN Senstvt: P SN P FN Specfct: N SP N FP S 75 Mchne ernng

13 Evuton Proem: f the smpe sze s retve sm one spt m e uck or unuck hence sng the sttstcs Souton: use mutpe trn/test spts nd verge ther resuts Rndom resmpng vdton technques: rndom su-smpng k-fod cross-vdton ootstrp-sed vdton S 75 Mchne ernng Rndom su-smpng Spt the dt nto trn nd test set th some spt rto tpc 7:3 Repet ths k tmes for dfferent rndom spts Averge the resuts of sttstcs rn rn Dt Spt rndom nto 7% rn 3% est est est est ernng ssf/evute Averge Stts S 75 Mchne ernng

14 K-fod cross-vdton Spt the dt nto k equ sze groups Use ech group once s test set nd the remnng groups s the trnng set Repet ths k tmes for k groups Averge the resuts of sttstcs Dt Spt nto k groups of equ sze est th group rn on the rest rn rn est est est ernng ssf/evute Averge Stts S 75 Mchne ernng Bootstrp-sed vdton Bootstrp technque used prmr to estmte the smpng dstruton of n estmtor Generte rndom th repcement trnng dtset of sze n tht equs the orgn dt sze Some empes re repeted n the trnng set some re mssng Bud test set from empes not used n the trnng set. rn rn Dt of sze n Generte rndom th repcement trn dtset of sze n ernng est est est ssf/evute Averge Stts S 75 Mchne ernng

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