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# ica - Independent Component Analysis Mixture Data Data that...

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Independent Component Analysis Independent Component Analysis

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2 Mixture Data x Data that are mingled from multiple sources b May not know how many sources b May not know the mixing mechanism x Good Representation b Uncorrelated, information-bearing components h PCA and Fisher’s linear discriminant b De-mixing or separation h ICA (Independent component analysis) x How do they differ?
3 PCA vs. ICA x Independent events vs. Uncorrelated events Knowing X1 doesn’t tell anything about X2 Knowing X1 does tell something about X2 x1 x2 x2

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4 Uncorrelated vs. Independence x Uncorrelated b Global property b Not valid under nonlinear transform b PCA requires uncorrelation x Independence b Local property b Valid for nonlinear transform b ICA assumes independence 0 )) )( (( : )) ( ( )) ( ( )) ( , ), ( ), ( ( : 2 2 1 1 1 1 2 2 1 1 = - - = Ex x Ex x E ed uncorrelat g x g E x g E x g x g x g E ce independen n n n n L L
5 Uncorrelated vs. Independence x Independence is stronger, requiring every possible function of x1 to be uncorrelated with x2 x E((y1-E(y1))(y2-E(y2))=0 -> uncorrelated x y2= y1 2 -> not independent

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6 Uncorrelated vs. Independence x Discrete variables X1 and X2 x (0,1), (0,-1),(1,0),(-1,0) all with ¼ probability x X1 and X2 are uncorrelated x E(x1 2 x2 2 )=0!=1/4=E(x1 2 )E(x2 2 )
7 ICA Limitation x Any symmetrical distribution of x1 and x2 around origin (centered at Ex1 and Ex2) is uncorrelated x Corollary: ICA does not apply to Gaussian

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ica - Independent Component Analysis Mixture Data Data that...

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