Lecture Slides for INTRODUCTION TO. Machine Learning. ETHEM ALPAYDIN The MIT Press,

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1 Lecure ldes for INRODUCION O Machne Learnng EHEM ALPAYDIN he MI Press, 004 alpaydn@boun.edu.r hp://.cpe.boun.edu.r/~ehe/l

2 CHAPER 6: Densonaly Reducon

3 Why Reduce Densonaly?. Reduces e copley: Less copuaon. Reduces space copley: Less paraeers 3. aves he cos of observng he feaure 4. pler odels are ore robus on sall daases 5. More nerpreable; spler eplanaon 6. Daa vsualzaon (srucure, groups, oulers, ec) f ploed n or 3 densons Lecure Noes for E Alpaydın 004 Inroducon o Machne Learnng he MI Press (V.) 3

4 Feaure elecon vs Eracon Feaure selecon: Choosng k<d poran feaures, gnorng he reanng d k ubse selecon algorhs Feaure eracon: Projec he orgnal,,...,d densons o ne k<d densons, z j, j,...,k Prncpal coponens analyss (PCA), lnear dscrnan analyss (LDA), facor analyss (FA) Lecure Noes for E Alpaydın 004 Inroducon o Machne Learnng he MI Press (V.) 4

5 ubse elecon here are d subses of d feaures Forard search: Add he bes feaure a each sep e of feaures F nally Ø. A each eraon, fnd he bes ne feaure j argn E ( F ) Add j o F f E ( F j ) < E ( F ) Hll-clbng O(d ) algorh Backard search: ar h all feaures and reove one a a e, f possble. Floang search (Add k, reove l) Lecure Noes for E Alpaydın 004 Inroducon o Machne Learnng he MI Press (V.) 5

6 Prncpal Coponens Analyss (PCA) Fnd a lo-densonal space such ha hen s projeced here, nforaon loss s nzed. he projecon of on he drecon of s: z Fnd such ha Var(z) s azed Var(z) Var( ) E[( µ) ] E[( µ)( µ)] E[ ( µ)( µ) ] E[( µ)( µ) ] here Var() E[( µ)( µ) ] Lecure Noes for E Alpaydın 004 Inroducon o Machne Learnng he MI Press (V.) 6

7 Maze Var(z) subjec o ( ) a Σ α α ha s, s an egenvecor of Choose he one h he larges egenvalue for Var(z) o be a econd prncpal coponen: Ma Var(z ), s.., and orhogonal o ( ) ( β 0) a Σ α α ha s, s anoher egenvecor of and so on. Lecure Noes for E Alpaydın 004 Inroducon o Machne Learnng he MI Press (V.) 7

8 Wha PCA does z W ( ) here he coluns of W are he egenvecors of, and s saple ean Ceners he daa a he orgn and roaes he aes Lecure Noes for E Alpaydın 004 Inroducon o Machne Learnng he MI Press (V.) 8

9 Ho o choose k? Proporon of Varance (PoV) eplaned λ λ + λ + λ + L + λk + L + λ + L + λ hen λ are sored n descendng order ypcally, sop a PoV>0.9 cree graph plos of PoV vs k, sop a elbo k d Lecure Noes for E Alpaydın 004 Inroducon o Machne Learnng he MI Press (V.) 9

10 Lecure Noes for E Alpaydın 004 Inroducon o Machne Learnng he MI Press (V.) 0

11 Lecure Noes for E Alpaydın 004 Inroducon o Machne Learnng he MI Press (V.)

12 Facor Analyss Fnd a sall nuber of facors z, hch hen cobned generae : µ v z + v z v k z k + ε here z j, j,...,k are he laen facors h E[ z j ]0, Var(z j ), Cov(z,, z j )0, j, ε are he nose sources E[ ε ] ψ, Cov(ε, ε j ) 0, j, Cov(ε, z j ) 0, and v j are he facor loadngs Lecure Noes for E Alpaydın 004 Inroducon o Machne Learnng he MI Press (V.)

13 PCA vs FA PCA Fro o z z W ( µ) FA Fro z o µ Vz + ε z z Lecure Noes for E Alpaydın 004 Inroducon o Machne Learnng he MI Press (V.) 3

14 Facor Analyss In FA, facors z j are sreched, roaed and ranslaed o generae Lecure Noes for E Alpaydın 004 Inroducon o Machne Learnng he MI Press (V.) 4

15 Muldensonal calng Gven parse dsances beeen N pons, d j,,j,...,n place on a lo-d ap s.. dsances are preserved. z g ( θ ) Fnd θ ha n aon sress E ( θ X ) r,s ( ) r s r s z z r s r,s ( ( θ) ( θ) ) r s r s g g r s Lecure Noes for E Alpaydın 004 Inroducon o Machne Learnng he MI Press (V.) 5

16 Map of Europe by MD Map fro CIA he World Facbook: hp://.ca.gov/ Lecure Noes for E Alpaydın 004 Inroducon o Machne Learnng he MI Press (V.) 6

17 Lecure Noes for E Alpaydın 004 Inroducon o Machne Learnng he MI Press (V.) 7 Lnear Dscrnan Analyss Fnd a lo-densonal space such ha hen s projeced, classes are ell-separaed. Fnd ha azes ( ) ( ) ( ) + r s r r s s J

18 Lecure Noes for E Alpaydın 004 Inroducon o Machne Learnng he MI Press (V.) 8 Beeen-class scaer: Whn-class scaer: ( ) ( ) ( )( ) ( )( ) B B here ( ) ( )( ) ( )( ) here here + + W W s s r r r s

19 Fsher s Lnear Dscrnan Fnd ha a J ( ) LDA soln: Paraerc soln: B W c W ( ) ( ) W ( µ ) µ p( C ) ~ N ( Σ) Σ hen, µ Lecure Noes for E Alpaydın 004 Inroducon o Machne Learnng he MI Press (V.) 9

20 Lecure Noes for E Alpaydın 004 Inroducon o Machne Learnng he MI Press (V.) 0 K> Classes Whn-class scaer: Beeen-class scaer: Fnd W ha a ( )( ) K W r ( )( ) K K B K N ( ) W W W W W W B J he larges egenvecors of W - B Mau rank of K-

21 Lecure Noes for E Alpaydın 004 Inroducon o Machne Learnng he MI Press (V.)

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