Linear models for classification
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1 CS 75 Mache Lear Lecture 9 Lear modes for cassfcato Mos Hausrecht mos@cs.ptt.edu 539 Seott Square ata: { d d.. d} d Cassfcato represets a dscrete cass vaue Goa: ear f : X Y Bar cassfcato A speca case he Y {} Frst step: e eed to devse a mode of the fucto f
2 scrmat fuctos A commo a to represet a cassfer s b us scrmat fuctos Wors for both the bar ad mut-a cassfcato Idea: For ever cass = defe a fucto mapp X Whe the decso o put shoud be made choose the cass th the hhest vaue of * ar ma scrmat fuctos
3 scrmat fuctos scrmat fuctos
4 scrmat fuctos ecso boudar: dscrmat fuctos are equa Quadratc decso boudar 3 ecso boudar
5 Ho to des dscrmat fuctos? Assume to ear modes for casses Cass decso: * ar ma ra va reresso: f ra th vaue ra th vaue f ra th vaue ra th vaue Use east squares error to fd both Ho to des dscrmat fuctos? Prevous des used to dscrmat fuctos oe for each cass Bar cassfcato s smper: We ca use oe set of ehts ra va reresso: f * ar ma ra th vaue f ra th vaue - Ho to mae a decso o cass? 5
6 Ho to des dscrmat fuctos? Prevous des used to dscrmat fuctos oe for each cass Bar cassfcato s smper o to casses: We ca use oe set of shared ehts ra va reresso: f ra th vaue f ra th vaue - Ho to mae a decso o cass? Cass Cass scrmat fuctos ad decso boudar Lear decso boudar
7 Ho to des dscrmat fuctos? Propert of the above mode: efes a ear decso boudar Lmtatos of the above mode Reresso s reated to a Gaussa But e have o to dfferet vaues e tr to ft We oud e to have a probabstc mode for cassfcato Is t possbe to proper defe p= ad p=? Lostc reresso mode efes a ear decso boudar scrmat fuctos: here / e f - s a ostc fucto Iput vector f d Lostc fucto d 7
8 Fucto: Lostc fucto e Is aso referred to as a smod fucto taes a rea umber ad outputs the umber the terva [] Modes a smooth stch fucto; repaces hard threshod fucto Lostc smooth stch hreshod hard stch Lostc reresso mode scrmat fuctos: Vaues of dscrmat fuctos var terva [] Probabstc terpretato f p p Iput vector d d 8
9 Lostc reresso We ear a probabstc fucto f : X [] here f descrbes the probabt of cass ve f p Note that: p p Ma decsos th the ostc reresso mode:? Lostc reresso We ear a probabstc fucto f : X [] here f descrbes the probabt of cass ve f p Note that: p p Ma decsos th the ostc reresso mode: If p / the choose Ese choose 9
10 Lear decso boudar Lostc reresso mode defes a ear decso boudar Wh? Aser: Compare to dscrmat fuctos. ecso boudar: For the boudar t must hod: o o o ep ep o o o o ep ep Lostc reresso mode. ecso boudar LR defes a ear decso boudar Eampe: casses bue ad red pots ecso boudar =
11 Lehood of outputs Let he Fd ehts that mame the ehood of outputs App the o-ehood trc. he optma ehts are the same for both the ehood ad the o-ehood Lostc reresso: parameter ear o o P L p o o Lostc reresso: parameter ear Notato: Lo ehood ervatves of the oehood Gradet descet: o o f ] [ Noear ehts!! f ] [ j j p
12 ervato of the radet Lo ehood ervatves of the oehood o o f j j o o o o ervatve of a ostc fucto j j Lostc reresso. Oe radet descet O-e compoet of the oehood O-e ear update for eht th update for the ostc reresso ad ] [ oe J J oe f ] [ o o oe J
13 Oe ostc reresso aorthm Oe-ostc-reresso stopp_crtero tae ehts d he stopp_crtero = FALSE do seect et data pot set update ehts parae ed retur ehts [ f ] Oe aorthm. Eampe. 3
14 Oe aorthm. Eampe. Oe aorthm. Eampe. 4
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