EE 6885 Statistical Pattern Recognition
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1 EE 6885 Sascal Paer Recogo Fall 005 Prof. Shh-Fu Chag hp://.ee.columba.edu/~sfchag Lecure 8 (/8/05 8- Readg Feaure Dmeso Reduco PCA, ICA, LDA, Chaper 3.8, 0.3 ICA Tuoral: Fal Exam Aapo Hyväre ad Erkk Oja, Idepede Compoe Aalyss: Algorhms ad Applcaos, Neural Neorks, 3(4-5:4-430, 000 Dec. 6 h Frday :0-3 pm, Mudd Rm
2 Reve Lecure #3: Mul-varae Gaussa Mulvarae Gaussa, T ( x μ Σ ( x μ N( μ, Σ p( x = e D / ( π Σ here x, μ are D-dmesoal vecors Σ : D D marx Σ s he deerma of Σ x x σ Σ = 0 x Geeral Σ 0 σ ( σ =Σ (, = cov( (, ( j j x x j = E[( x ( μ( ( x( j μ( j] x 8-3 Effec of Lear Trasformao Lear rasformao of Gaussa y= A x y N( A μ, AΣA y: k, A: d k, x: d Wheg rasform Σ=ΦΛΦ Φ :[ φ φ... φ ] colums are egevecors (SVD, Egevecors A Σ A= AΦΛΦ A= I d Wheg Tras. also PCA Trasform y = A x N( A μ, I 8-4 A =ΦΛ /
3 Malalaobs dsace from po x o he mea of a Gaussa x u p( x = exp( ( x μ Σ ( x μ D / ( π Σ = exp( / ( x μ ΦΛ Φ ( x μ D ( π Σ = exp( ( A / ( x μ ( A ( x μ D π Σ ( r r s he Mahalaobs dsace s also he Eucldea ds he PCA space 8-5 PCA for feaure dmeso reduco Approxmae daa h reduced dmesos -D approxmao xˆ = m+ae, m: mea J ˆ ( e = x k x k = ( m + a e k x k k= k= ak e ake ( xk m xk m = ( e xk m + xk m k= k= k= k= k= e ( k ( k k x m x m e x m = ese+ xk m k= k= k= Approxmao Error = + = + S: scaer marx Opmal e mmzg error J = ( sample covarace -- egevecor of S h he larges egevalue Mul-Dm. approxmao d x m a e ha are he opmal e? = + = 8-6 3
4 Exeso o No-Lear Compoes PCA bases ca be foud by backpropagao of mul-layer Neural Neork (deals Chap 6 PCA ca be exeded o Nolear Compoe Aalyss (NLCA by addg usg a mul-layer NN (Chap 6 olear compoes useful for approxmag daa Bu may o be good for classfcao 8-7 Egeface (Pelad e al 9 Trea each mage as a -D vecor Fd he dmesos h he larges varao Ho o classfy? Pros ad Cos? Samples from DARPA FERET Daa Se Sample Egefaces 8-8 4
5 Idepede Compoe Aalyss Seek mos depede drecos, sead of mmze represeao errors (sum-squared-error as PCA Bld source separao speech mxure f : sgmod f( x=/( + e x 8-9 Fd he bes eghs o make he oupu compoes depede Ho o measure depedece? Lear combao of radom varables leads o Normal dsrbuo Use he hgh-order sascs o measure he dvergece from Gaussa (from Ells Use grade dece mehod o refe eghs o reduce Gaussay FasICA Malab package : hp://.cs.hu.f/projecs/ca/fasca/ 8-0 5
![Or maxmze he depedece measure Jo eropy H( y = E l p y ( y = E l J E[ l p s ( s ] δ y δ y d depede of δs δs ps ( s p y ( y = Jacoba J = J δ y δ y d δsd δs d Grade dece Pros : δh ( y ΔW δw o-orhogoal,](/docs-images/95/125781658/images/6-0.jpg "varable umber, combed h olear fuco Cos: o order of mporace, sesve o ose, emporal dmeso o cosdered 8- LDA: Lear Dscrma Aalyss Gve a se of daa x, x,, x, ad her class labels Fd he bes projeco dmeso, y =")
6 Or maxmze he depedece measure Jo eropy H( y = E l p y ( y = E l J E[ l p s ( s ] δ y δ y d depede of δs δs ps ( s p y ( y = Jacoba J = J δ y δ y d δsd δs d Grade dece Pros : δh ( y ΔW δw o-orhogoal, varable umber, combed h olear fuco Cos: o order of mporace, sesve o ose, emporal dmeso o cosdered 8- LDA: Lear Dscrma Aalyss Gve a se of daa x, x,, x, ad her class labels Fd he bes projeco dmeso, y = x so ha are mos separable y m = x= m x D y Y m : sample meas s = ( y m s + s : sample meas of projeced pos m : h-class scaer LDA maxmzes crero fuco: ( = m J m s + s s s 8- m m y 6
7 LDA Scaer Marces before projeco: S = ( x m ( x m afer projeco: = x D s S s + s = ( S + S = S S = S + S : h-class scaer marx Smlarly, beee-class scaer marx S = ( ( B m m m m SB J ( = S S: usually osgular SB : rak = arg max J ( op = S m m ( Remember he Gaussa Cases 8-3 7
EE 6885 Statistical Pattern Recognition
EE 6885 Sascal Paer Recogo Fall 005 Prof. Shh-Fu Chag hp://.ee.columba.edu/~sfchag Reve: Fal Exam (//005) Reve-Fal- Fal Exam Dec. 6 h Frday :0-3 pm, Mudd Rm 644 Reve Fal- Chap 5: Lear Dscrma Fucos Reve
EE 6885 Statistical Pattern Recognition
EE 6885 Sascal Paer Recogo Fall 005 Prof. Shh-Fu Chag hp://www.ee.columba.edu/~sfchag Lecure 5 (9//05 4- Readg Model Parameer Esmao ML Esmao, Chap. 3. Mure of Gaussa ad EM Referece Boo, HTF Chap. 8.5 Teboo,
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