CHAPTER 5: MULTIVARIATE METHODS
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1 CHAPER 5: MULIVARIAE MEHODS
2 Mulvarae Daa 3 Mulple measuremens (sensors) npus/feaures/arbues: -varae N nsances/observaons/eamples Each row s an eample Each column represens a feaure X a b correspons o he a-h arbue of he b-h eample N N N X X X X X X X X X X
3 Mulvarae Parameers 4 Mean Mean: E X E[ X,..., X ] [ E[ X],..., E[ X ]] μ,..., hs s a vecor of componens Is -h componen correspons o mean of he -h feaure For nfne sample (populaon) we use epecaon o represen mean, for fne sample we use average Covarance beween -h an -h arbue s Covarance: Cov X, X E[( X )( X )] E[ X X ] When =, becomes he varance of he -h arbue Correlaon I s normalze beween + an - Covarance mar Is (I,)-h enry s Varance: Cov X, X E[( X ) ] E[ X ] X X E X X Cov, μ μ Correlaon: Corr X, X Cov X E X μx μ
4 Parameer Esmaon N N s s s r N m m s N m : :,...,, : R S m Correlaon mar Covarancemar Samplemean 5
5 Esmaon of Mssng Values 6 Wha o o f ceran nsances have mssng arbues? Ignore hose nsances: no a goo ea f he sample s small Impuaon: Fll n he mssng value by esmang her values Mean mpuaon: Use he mos lkely value (e.g., mean) Average value of he same arbue Average value of same arbue comng from same class Impuaon by regresson: Prec base on oher arbues as a soluon o a regresson problem
6 Mulvarae Normal Dsrbuon 7 X s srbue as a - mensonal normal srbuon ~ N μ, Σ I has wo parameers Mean μ, hs s a vecor wh componens Covarance mar Σ, hs s a mar Is ensy funcon s gven as p ep / / μ μ Σ Σ
7 Mulvarae Normal Dsrbuon 8 Mahalanobs sance: ( μ) ( μ) measures he sance from o μ n erms of (normalzes for fference n varances an correlaons) Wha happens f s eny mar? Bvarae: = p, ep z / z z z z
8 Bvarae Normal 9 Suppose here are wo feaures X=(X, X ) In he pcure, X s represene along as an X s represene along y as Relave varance of X an X eermnes he shape of he normal srbuon Crcular vs ellpcal Whenever covarance, Cov(X,X ) s nonzero, s ncae roaon of he shape Posve covarance means roaon n one recon (say clockwse) an negave covarance means roaon n he oher recon When here are more han wo feaures (>) Same hng as above hols for every par of feaures
9 0
10 Inepenen Inpus: Nave Bayes For a -mensonal ranom varable, s covarance mar s Wha happens o hs mar f all parwse covarances are zero? I becomes a agonal mar Why o we care? Each feaure becomes nepenen of anoher Jon probably srbuon can be wren as prouc of margnal srbuons nvolvng separae nvual feaures If are nepenen, offagonals of are 0, Mahalanobs sance reuces o weghe (by /σ ) Euclean sance: p ep / ( ) ep p ( ) hs s eacly Naïve Bayes assumpon f each class cononal probably (lkelhoo) s moele hs way If varances are also equal, reuces o Euclean sance
11 Properes of agonal mar Deermnan Deermnan of a agonal mar s equal o prouc of s agonal enres Inverse Inverse of a agonal mar s a agonal mar whose -h agonal enry s he one over he -h agonal enry of he orgnal mar Why?
12 An mporan ha we wll use laer 3 Suppose hen where N(, ) an w s a mensonal column vecor w w w... w N( w, w w) E[ w ] w E[ ] w Var( w ) E ( w w ) E ( w w )( w w ) E w ( )( ) w w E ( )( ) w w w
13 Paramerc Classfcaon If p ( C ) ~ N ( μ, ) Dscrmnan funcons C p μ μ Σ Σ ep / / C P C P C p g log log log log log μ Σ μ Σ 4
14 Esmaon of Parameers r r r r N r C P m m m S ˆ C P g ˆ log log m m S S 5 Pluggng n hese esmaes, he scrmnan funcon becomes hs s a quarac form, ha s here s a vecor-mar-vecor mulplcaon Such a scrmnan funcon s calle quarac scrmnan funcon
15 Dfferen S Epanng he quarac form, he scrmnan funcon becomes C P w w C P g log ˆ log where log ˆ log S S S S W W S S S S 0 0 m m m w w m m m 6 quarac erm lnear erm scalar erm You can relae o he quarac formula n -, for eample, a +b+c
16 lkelhoos scrmnan: P (C ) = 0.5 poseror for C 7 Quarac scrmnan funcon gves rse o a quarac ecson bounary
17 Why such a moel can be unrealsc? 8 oal number of parameers for a K class classfcaon problem s O(K ) Mean for each class has parameers Covarance mar (symmerc) for each class has (+)/ snc parameers Fnng nverse of a mar akes me O( 3 ) hs s problemac for large scale applcaons, e.g., say when =50,000 So wha can we o? ry o reuce moel parameers by makng assumpons
18 Approach #: common covarance mar S 9 Share common sample covarance S across all K classes Dscrmnan reuces o whch s a lnear scrmnan because S - s same for all classes g w w so ecson bounary s now lnear Number of parameers s sll O( ) k+(+)/ Inverng covarance mar s sll O( 3 ) C S Pˆ S m S m log C g Pˆ 0 where, w S m w m S m log Pˆ C 0
19 0 Common Covarance Mar S
20 Approach #: Share agonal S When =,.., are nepenen, s agonal, we can wre, p ( C ) = p ( C ) (Nave Bayes assumpon) Dscrmnan funcon has he form m g logpˆ C s Classfy base on weghe Euclean sance (n s uns) o he neares mean, when are pror class probables are equal Snce s s same for all classes Dscrmnan funcon s lnear, hence so s he ecson bounary So wha happene o number of parameers an mar nverson oal number of parameer s O(K) K mean vecors of lengh Share agonal mar has parameers (agonal enres) Mar nverson Can be one n O() me Inverse s a agonal mar an each enry s over orgnal agonal enry
21 Dagonal S varances may be fferen
22 Approach # 3: Share agonal S, equal varances 3 S=σ I, ha s S s a agonal mar, where each agonal enry s same Neares mean classfer: Classfy base on Euclean sance o he neares mean when Each mean can be consere a prooype or emplae an hs s emplae machng Dscrmnan funcon s lnear an so s ecson bounary So wha happene o number of parameers an mar nverson g logpˆ C oal number of parameer s O(K) K mean vecors of lengh s m s m logpˆ C Share agonal mar has parameers (common agonal enry) Mar nverson Can be one n O() me (consan me) Inverse s a agonal mar an each enry s over orgnal agonal enry
23 Dagonal S, equal varances 4 *?
24 Moel Selecon 5 Assumpon Covarance mar No of parameers Share, Hyperspherc S =S=s I Share, As-algne S =S, wh s =0 Share, Hyperellpsoal S =S (+)/ Dfferen, Hyperellpsoal S K (+)/ As we ncrease compley (less resrce S), bas ecreases an varance ncreases Assume smple moels (allow some bas) o conrol varance (regularzaon)
25 6
26 Dscree Feaures 7 Bnary feaures: g f are nepenen (Nave Bayes ) he scrmnan s lnear logp C logpc p logp log p logpc Esmae parameers p p C C p p pˆ r r
27 Dscree Feaures 8 Mulnomal (-of-n ) feaures: n {v, v,..., v n } k z C p v C p p k f are nepenen p g pˆ C z logp logpc k z k k k k r r n p z k k k k
28 Mulvarae regresson 9 In mulvarae lnear regresson numerc oupu r s assume o be wren as a lnear funcon ha s, a weghe sum of several npu varables (arbues),,, r g w, w,..., w 0 g w, w,..., w w w w w 0 0 We can mamze he log-lkelhoo of p(r,,, ) an show ha hs s equvalen o mnmzng E w0, w,..., w X r w 0 w w
29 Mulvarae regresson 30 We can mnmze epresson from las sle by akng paral ervave wh respec o each parameer an seng o zero hese are normal equaons All normal equaons can be compacly wren as n mar noaon X Xw X r where, w w0, w,..., w We can solve by pre-mulplyng boh se wh mar nverse o ge Here w X X X r X s N (+) aa mar w s (+) column vecor r s N column vecor of N oupu values
30 Mulvarae polynomal regresson 3 By he way, we can also efne Mulvarae polynomal moel: Defne new hgher-orer varables z =, z =, z 3 =, z 4 =, z 5 = an use he lnear moel n hs new z space (bass funcons, kernel rck: Chaper 3)
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