Introduction to logistic regression
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1 Itroducto to logstc rgrsso Gv: datast D {... } whr s a k-dmsoal vctor of ral-valud faturs or attrbuts ad s a bar class labl or targt. hus w ca sa that R k ad {0 }. For ampl f k 4 a datast of 3 data pots or ampls would b: I ths datast thr s o postv data pot ad two gatv data pots 3. All vctors ar 4-dmsoal. Each of th four dmsos s calld a fatur sa blood prssur sugar lvl bo dst tc.. Each class labl dots o of th two groups.g. ma ma that patt wth faturs has a partcular dsas or trat whl 0 would th ma that patt dscrbd wth faturs dos ot hav that dsas. I summar th datast D ths ampl cossts of th 3 vctors of faturs ach vctor bg assocatd wth a corrspodg class labl. h task of our data mg sstm s to costruct a prdctor that would for a us data pot fr ts class labl. For ampl f a w patt coms ad w prform four chap tsts w mght b abl to fr a dsas wthout rug vr psv tsts whch would ffctvl labl that data pot. Evr frc s assocatd wth a qualt of frc. W wll masur qualt of frc b th umbr of mstaks th prdctor maks prct ol for prvousl us data pots. Formall th qualt of frc wll b prssd through prdctor accurac. You ca s a prdctor as a smpl mathmatcal fucto. hus t vrtuall maps a k- dmsoal vctor to a zro or o. hs tp of a prdctor s calld a bar classfr; classfr bcaus ca tak dscrt valus ad bar bcaus thr ar ol two such valus that ca tak. Of cours ou ca also mag gralzatos to ths cas. If was takg valus from th st of ral umbrs w would call ths procss rgrsso stad of classfcato A cor of a statstcal or mach larg approach s to assum that vctors ad thr labls obsrvd D wr gratd b som sourc that outputs vctors ad labls accordg to som probablt dstrbuto p. I such a cas w ca prss class mmbrshp probablstcall. h basc da for logstc rgrsso approach s to tr to stablsh a smpl possbl lar closd-form dpdc mag thr s a formula btw th probablt of a /6
2 class mmbrshp calld postror probablt ad th st of faturs. O such form could b whr R k s a vctor of k ral-valud umbrs gral ad s a traspos of th vctor. hrfor a dot product rsults a sgl umbr. For ampl f ad th h problm wth quato s that R whl th probablt ds to b lmtd to th trval [0 ]. hus w caot us closd-form from quato. Aothr approach to modl th postror probablt s to tr to prss th odds fucto as a lar combato of th paramtr vctor ad fatur vctor. hat s odds fucto Closr look at ths fucto plas vrf ths! rvals that ths fucto s co-doma s trval [0 whl R. hrfor ths s ot a approprat paramtrc dpdc thr. Our thrd tr wll b to tak a logarthm of th odds fucto ad rprst t as a lar combato of ad. hat s log 3 I ths cas both ad logodds blog to th trval -. h logarthm th prsso abov s to th bas whr A rorgazato of th prsso 3 gvs th followg /6
3 3/6 4 hrfor a probablt that th class of th vctor s ca ths cas b modld accordg to th prsso 4. h fucto ft t s calld th sgmod fucto or th logstc fucto s th plot all th wa at th bottom. Dtrmg optmal coffcts For a gv datast D ad assumd dpdc from prsso 4 th optmal st of coffcts s dtrmd b mamzg th followg prsso calld lklhood fucto 5 or a formal mathmatcal otato ma arg whch sas that vctor * s th o for whch prsso 5 s mamal. Now lt s us th followg rprstato ou ca asl vrf t s tru b substtutg
4 4/6 0 for for Now w ca mamz th followg prsso ot that f th frst part dsappars whl f th scod part dsappars whch follows from prsso 4 or aftr takg a logarthm to th bas log log I assumd hr ou ar famlar wth som basc logarthm mapulatos. hs quato s mamzd usg tratv optmzato procss. Dtrmg optmal coffcts * s calld trag or larg hc th am mach larg. Formall ths whol optmzato procss s calld mamum lklhood optmzato ad thr ar varous toolbos o th Itrt that ca do ths for ou ma b prtt compl. h trag procss s ow ovr. W just calculatd *. How do w prdct or fr class for a w data pot? For a us data pot ad optmal or somtms arl optmal st of coffcts * dtrmd from a trag datast D w smpl calculat th followg prsso If 5 0. w smpl coclud that th data pot should b labld as postv. O th othr had f 5 0. < w labl th us data pot as gatv. h prdcto ca b mad v wthout calculatg th logstc prsso: f 0 w prdct postv ad othrws w prdct gatv ou ca asl vrf ths b substtutg th prsso abov
5 What s th shap of th sgmod fucto? Fal pot Although th assumpto that ma ot look tutv logstc rgrsso has b show to work surprsgl wll practc!!! 5/6
6 K-arst ghbor algorthm K-NEARES NEIGHBOR s a smpl algorthm that stors all avalabl data pots ampls ad classfs w data pots basd o a smlart masur. A varat of ths algorthm addrsss th task of fucto appromato. I dtal: Eampls ar dscrbd b umrcal attrbut-valus. hat s 3... k whr k s th dmsoalt of th stac spac h complt ampl or data st D s smpl stord th "trag phas". h data st D s dfd as D {..} R k {0 } Calculatos ar dlad utl qurs occur o trad modl th usual ss s rturd! rad modls ar mplctl dfd b th stord ampl st ad th ruls w stacs ar classfd b. If thr ar k attrbuts all vctors ca b trprtd as stacs of R k. h dstac d of two ampl vctors ad s dfd as thr usual vctor dstac Euclda dstac. hat s: d 3 3 K k k h dstac btw two ampl vctors s rgardd as a masur for thr smlart. h smallr th dstac th mor smlar th vctors. o classf a w stac from th st of stord ampls th K ampls most smlar to ar dtrmd. h w stac s assgd th class labl most of ths K ampls blog to. hs approach s sutd for fucto appromato as wll. Istad of assgg th most frqut classfcato amog th K ampls most smlar to a stac a avrag of th fucto valus of th K ampls s calculatd as th prdcto for th fucto valu. A varat of ths approach calculats a wghtd avrag of th arst ghbors. Gv a spcfc stac that shall b classfd th wght of a ampl crass wth crasg smlart to. A major problm of th smpl approach of K-NEARES NEIGHBOR s that th vctor dstac wll ot cssarl b sutd for fdg tutvl smlar ampls spcall f rrlvat attrbuts ar prst. Sourc modfd: 6/6
Introduction to logistic regression
Itroducto to logstc rgrsso Gv: datast D { 2 2... } whr s a k-dmsoal vctor of ral-valud faturs or attrbuts ad s a bar class labl or targt. hus w ca sa that R k ad {0 }. For ampl f k 4 a datast of 3 data
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