# 2 ICA. (ICA, Independent Component Analysis) (PCA, Principal Compoenent Analysis) x(t) =(x 1 (t),...,x m (t)) T t =0, 1, 2,... PCA 2 ICA.

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1 s T (ICA, Independent Component Analysis) (PCA, Principal Compoenent Analysis) x =(x,...,x m ) T t =,,,... PCA ICA m m n s x PCA x =As, () ICA ICA ICA Blind Source Separation A m n BSS s A x n n m n m W y =W x, () y WA = I (I n n ) y s ICA y permutation amplitude ICA n>m ICA [] [] x =As+n, (3) ICA ICA ICA Blind Source Separation (BSS) Blind Source Deconvolution BSS Blind Source Deconvolution BSS s =(s,...,s n ) T t =,,,... n () m = n Blind Source Deconvolution x =A s x i = a ik s k, (4) k a ik s k = a ik (τ)s k (t τ), τ=

2 y W x x x y =W x (5) x BSS m n A 3 ICA ICA PCA (Principal Component Analysis, ) ICA (3.) BSS (3.) gradient (3.3) Blind Source Deconvolution (3.4) 3. PCA ICA : Observed signals R R R T R = R V V x x = V x x N x x T = V N V V T = I, (7) t= PCA x, x, x x s s : Source signals s s 3: Applying PCA on observed signals PCA 3 ICA ICA 4 PCA s () x ( ) V = N xx T, (6) N t= y y y 4: Result of solving ICA problem y ICA ICA PCA ICA Whitening Sphering

3 ICA 3. s i y p(y) =p(y,,y n ) W x y y i p(y i ) y i p(y) p(y) = n i= p(y i) p(y) n i= p(y i) W p(y) n i= p(y i) Kullback-Leibler divergence ( {Y i } (i =,...,n) ) W K-L divergence p(y) KL(W) = p(y) log n i= p(y i) dy = H(Y ; W )+ n H(Y i ; W ). (8) i= H(Y ; W ) H(Y i ; W ) p(y)dy = p(x)dx p(y) = p(x)/ W ( W W ) (8) H(Y ; W ) H(Y i ; W ) p(x) W H(Y ; W ) = p(y) log p(y)dy = p(x)(log p(x) log W )dx = H(X) + log W, (9) H(Y i ; W ) = p(y) log p(y i )dy = p(x) log p(y i )dx () KL(W) p(y i ) W KL(W) W W KL(W) W = ( (W T ) E x [ϕ(y)x T ] ) = ( I E x [ϕ(y)y T ] ) (W T ) () ( log p(y ) ϕ(y) =,..., log p(y ) T n) y y n () W () (W T ) W T W [5] W ( I E x [ϕ(y)y T ] ) W (3) [4] η (4) W W t+ = W t + η ( I ϕ(y)y T ) W t. (4) ϕ(y) W [3] Cardoso Souloumiac[9] Jutten Herault[8] Bell Sejnowski [7] sigmoid Common [] Edgeworth Gram-Charlier [5] [6] (sub-gaussian) (super-gaussian) Sigmoid 3.3 { Q = n } log E[yi ] log det E[yy T ], (5) i= 3

4 (E[ ] ) FIR BSS [, 9] K-L divergence Molgedey Schuster [3] (5) [9] [3] ICA [7, 6] (5) x, s, A Fourier ω xx(t + τ) T = A ss(t + τ) T A T ˆx(ω) R s (τ) =Â(ω)ŝ(ω), = A... AT, (6) msec R sn (τ) x R si (τ) s i W y yy(t + τ) T = (WAs) ˆx(ω, t s )=Â(ω)ŝ(ω, t s ), (9) (WAs(t + τ)) T λ R s (τ) =..., (7) λ n R s n (τ),,...,n,,...,n λ i W τ y ((7) ) x τ i W W xx(t + τ i ) T W T = Λ i, i =,...,r, (8) ˆx(ω), ŝ(ω), Â(ω) ˆx(ω, t s ) ŝ(ω, t s ) x, s windowed Fourier BSS BSS perumutation amplitude Blind Source Deconvolution [7] Λ i W 4 [3] τ i 4. [6, 3] [] ICA EEG (Electroencephalogram, ), MEG (Magnetoencephalography, ), MRI (Magnetic Resonance Imaging, ) [7,, 8, 6, 8] optical recording Photo-detector 3.4 Blind Source Deconvolution (Multi-unit Recording) ICA EEG, MEG MRI 3.,3.3 FIR BSS EEG, MEG EEG MEG EEG MEG 4

5 ( ) EEG, MEG BSS ch ch ch 3 ch 4 ch 5 [ft] Time(msec) (Source 4) MRI ICA [] MRI x x t (t =,..., ) x x i (i =,..., ) EEG, MEG ICA MRI ICA 5: 4. ICA Source Source ICA [] MRI Source 3 ICA sparce Source 4 coding[5] ICA [5] Source Time(msec) Cocktail Party 6: [4] 3 Blind Source Deconvolution BSS MEG [6] MEG [4] URL ICA ICA 99 MEG 9channel 5 3 ICA ICA 6 (Source ) (Source,3) shiro/ 3 5

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