EE 583 PATTERN RECOGNITION
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1 METU EE583 Lecue Noes by A.Aydn ALATAN 4 EE 583 ATTEN EOGNITION Sascal aen econon Bayes Decson Theoy Suevsed Leann Lnea Dscmnan Funcons Unsuevsed Leann METU EE583 Lecue Noes by A.Aydn ALATAN 4 Bayes Decson Theoy Fundamenal sascal aoach o Assumons: Decson oblem s obablsc All elevan obably values ae knon The decson ules ae omal n he sense ha ehe mnmzes aveae obably of eo o oveall sk.
2 METU EE583 Lecue Noes by A.Aydn ALATAN 4 Eamle : Bayes Decson / lassfcaon oblem of ale and each by colo Assume nal obsevaon obables ae no equal.e. assume ale each If you do no have a chance o see he fu evey me decde/edc as ale! If you ae able o obseve he colo of he fu Queson : alecolo? eachcolo? Inuvely choose he class h hhe condonal obably. Ho o fnd hese obables? Ty usn Bayes ule : alecolo coloale*ale / colo eachcolo coloeach*each / colo METU EE583 Lecue Noes by A.Aydn ALATAN 4 Eamle : Bayes Decson / df coloeach coloale eachcolo alecolo colo colo Bayes Decson ule : If alecolo each colo hen choose ale Lkelhood ao coloale/coloeach each/ale onsan Noe ha he evdence colo s only necessay fo nomalzaon uoses; does no affec he decson ule
3 METU EE583 Lecue Noes by A.Aydn ALATAN 4 Bayes Decson Theoy Geneal[/4] Genealze Bayes Decson Theoy by allon o use mul feaues allon o use moe ha o saes allon acons ahe han choosn saes noducn a loss funcon ahe han obably of eo : feaue veco d Ω { L s }: saes classes A { L a }: acons allos ossbly of eecon : loss fo akn acon fo sae A oseo obably : s METU EE583 Lecue Noes by A.Aydn ALATAN 4 Bayes Decson Theoy Geneal [/4] Mnmum sk lassfe We obseve hen should ake one of he acons Bayes decson ule should mnmze he oveall sk : hee eeced loss condonal sk by akn acon : d s : loss fo akn acon fo sae ule : omue condonal sk fo evey acon and selec he acon h mnmum condonal sk. mn { } mn { } 3
4 4 METU EE583 Lecue Noes by A.Aydn ALATAN 4 Bayes Decson Theoy Geneal [3/4] Take acon- : decde on class- f < Mnmum sk lassfe : To aeoy ase Assume hee ae only classes : loss fo akn acon fo sae < onsan Lkelhood ao a funcon of METU EE583 Lecue Noes by A.Aydn ALATAN 4 Bayes Decson Theoy Geneal [4/4] Mnmum Eo-ae lassfe : Secal case fo Mnmum sk lassfe oec acons : zeo loss ; on acons : equal un loss If eos ae o be avoded decson ule should mnmze aveae obably of eo.e. eo-ae Loss funcon : s ule : Mamze oseo obably n ode o mnmze sk.e. aveae obably of eo oblem class - a Fo all fo f on Decde
5 5 METU EE583 Lecue Noes by A.Aydn ALATAN 4 Mnmzn lassfcaon Eo d eo d eo eo decde f decde f eo Q : Afe obsevn ha s obably of eo f e decde on one of he classes? Assume class oblem A : obably of obann he ohe class { } mn eo ule Decson Bayes f decde f decde METU EE583 Lecue Noes by A.Aydn ALATAN 4 Anohe ay o sho mnmum eo : The aveae obably of eo ll be smalle snce eo s foced o be mnmum by Bayes decson ule fo evey. Mnmzn lassfcaon Eo d eo d d d d d d eo Snce ule Decson Bayes f decde f decde { } mn eo
6 6 METU EE583 Lecue Noes by A.Aydn ALATAN 4 emembe he elaon fo eo fo classes and d d eo Movn fom * o B he eo obably should decease Mnmzn lassfcaon Eo METU EE583 Lecue Noes by A.Aydn ALATAN 4 eceve Oean haacesc O : coec eecon : mss :false alam :h * * * * < < false alam h Fo a -class oblem & s measued h nose Le * denoe a deecon heshold of he classfe Dscmnably ndeenden of * : d - / false alam h O These obables can be esmaed eemenally hane * and deemne h and false alam O
7 METU EE583 Lecue Noes by A.Aydn ALATAN 4 Dscmnan Funcons & Bayes lassfe Dscmnan funcon s one of he ays o oban a aen classfe. A classfe based on a dscmnan funcon assns a feaue o class- f fo all Bayes classfes can be eesened by hs aoach: : Mnmum condonal sk : Mnmum eo-ae Selecon of a dscmnan funcon s no unque Fo mnmum eo-ae classfe: o o ln ln METU EE583 Lecue Noes by A.Aydn ALATAN 4 Dscmnan Funcons & Bayes lassfe Dscmnan funcons mh be n dffeen foms bu he effec of he decson ules s he same : Decson boundaes ae obaned df The elaon deemnn he decson bounday beeen class- and class : 7
8 8 METU EE583 Lecue Noes by A.Aydn ALATAN 4 Dscmnan Funcons fo Nomal obably Densy /7 Unvaae Nomal Densy : π e Mulvaae Nomal Densy : / / π d e s called Mahalanobs dsance : ovaance ma Σ deemnes shae of Gaussan cuve } { T E Σ Nomal Gaussan obably Densy Funcon df E{} : mean value } { E : vaance METU EE583 Lecue Noes by A.Aydn ALATAN 4 Eenvalues and eenvecos of } { T E Σ u u Σ Dscmnan Funcons fo Nomal obably Densy /7 oees of Nomal df Manal df of a mulvaae nomal dsbuon s also Gaussan Lnea ansfoms also yeld Gaussans N y df N df a y A A A N y df N df A y T T T Σ Σ Σ I s ossble o ansfom an abay shaed covaance ma no oban a ccula one
9 9 METU EE583 Lecue Noes by A.Aydn ALATAN 4 Dscmnan Funcons fo Nomal obably Densy /7 Fo mnmum-eo-ae classfcaon one can choose dscmnan funcon as : lo lo Fo mulvaae nomal condonal densy dscmnan funcon s : lo lo lo d π / / d e π METU EE583 Lecue Noes by A.Aydn ALATAN 4 Dscmnan Funcons fo Nomal obably Densy 3/7 [ ] lo lo lo hee noe: s a lnea funcon lnea dscmnan funcon Decson bounday : lo hee a hyelane hu o ase : I ndeendence equal
10 METU EE583 Lecue Noes by A.Aydn ALATAN 4 Dscmnan Funcons fo Nomal obably Densy 4/7 ase : I ndeendence equal METU EE583 Lecue Noes by A.Aydn ALATAN 4 Dscmnan Funcons fo Nomal obably Densy 5/7 [ ] lo lo hee A hyelane hu o ase : abay & dencal Σ Decson bounday :
11 METU EE583 Lecue Noes by A.Aydn ALATAN 4 Dscmnan Funcons fo Nomal obably Densy 6/7 ase : abay & dencal Σ METU EE583 Lecue Noes by A.Aydn ALATAN 4 Dscmnan Funcons fo Nomal obably Densy 7/7 ase 3 : abay Σ W Decson bounday s a hyequadc
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