ICA [6] ICA) [7, 8] ICA ICA ICA [9, 10] J-F. Cardoso. [13] Matlab ICA. Comon[3], Amari & Cardoso[4] ICA ICA

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1 16 1 (Independent Component Analysis: ICA) ICA ICA ICA 1 ICA 198 Jutten Herault Comon[3], Amari & Cardoso[4] ICA Comon (PCA) projection persuit projection persuit ICA ICA ICA 1 [1] [] ICA ICA EEG [] fmri [6] [7, 8] [9, 1] J-F. Cardoso JADE [11] A. Hyvärinen FastICA [1] ICA central web page [13] Matlab ( ) ICA ICA ICA ICA 1

2 ICA.1 ICA. ICA 3 [14, 1] s(t) =(s 1 (t),...,s n (t)) T t =, 1,,... s(t) T x(t) =(x 1 (t),...,x m (t)) T t =, 1,,... x m n s(t) x(t) x(t) =As(t), (1) ( 1) A m n ICA s(t) A x(t) n s s 1 a 1 a 1 a 11 a 1: ICA n m n m W y(t) =W x(t), () y(t) WA= I (I n n ) y(t) s(t) y(t) WA = PD (P 1 n n D n n ) x x 1 JADE FastICA ICA s i () W x y y p(y)=p(y 1,,y n ) W y y i p(y i ) y i p(y) = n p(y i ) i=1 p(y) n i=1 p(y i) W p(y) n i=1 p(y i) Kullback-Leibler ( ) KL KL(W) = p(y) p(y) log n i=1 p(y dy i) (3) n = H(Y ; W )+ H(Y i ; W ). i=1 H(Y ; W ) H(Y i ; W ) p(y)dy = p(x)dx p(y) =p(x)/ W ( W W ) H(Y ; W ) H(Y i ; W ) p(x) W

3 H(Y ; W )= = p(y) log p(y)dy p(x)(log p(x) log W )dx x(t) = H(X) + log W, H(Y i ; W )= p(y) log p(y i )dy = p(x) log p(y i )dx KL(W) p(y i ) KL(W) W KL(W) W = ( (W T ) 1 E x [ϕ(y)x T ] ) = ( I E x [ϕ(y)y T ] ) (W T ) 1 (4) ( log p(y1 ) ϕ(y) =,..., log p(y ) T n) y 1 y n W (4) (W T ) 1 W T W [16] W ( I E x [ϕ(y)y T ] ) W () [17] η W W t+1 = W t + η ( I ϕ(y)y T) W t. ϕ(y) W [4] W ϕ(y) ICA x(t)x(t + τ) T = A s(t)s(t + τ) T A T R s1 (τ) = A... AT, R sn (τ) x(t) R si (τ) s i (t) s(t) τ s i (t) W y(t) y(t)y(t + τ) T = (WAs(t)) (WAs(t + τ)) T λ 1 R s 1 (τ) =..., λ n R s n (τ) 1,,...,n 1,,...,n λ i W τ y(t) x(t) τ i W [18] W x(t)x(t + τ i ) T W T = Λ i, i =1,...,r, (6) Λ i W [18] τ i [1] KL [19] 3

4 s i (t) [] { Q(t) = 1 n } log E[yi (t)] log det E[y(t)y(t) T ], i=1 (7) (E[ ] ) Q ICA Q E[y(t)y(t) T ] W W Q Q W 3 ICA ICA 3.1 MEG [1, ] MEG 1 MEG (1msec) ( mm) (1 14 T) MEG ( ) ( 1) 1 T Hz 1 14 a few 1 4 a few 1 () or : Low-Pass 1 ft x 1 x x time(msec) : MEG : msec 41msec 1 msec ICA x x = As 1 s 4

5 ICA x = As + ɛ ɛ ( ) x = As + ɛ = A s s =(A, I) ɛ ICA ɛ [] ICA A ɛ ICA 3 [3, 4] y 1 y y 3 y 4 y y 6 y 7 y 8 y 9 y 1 y 11 y 1 y 13 y 14 y 1 y 16 y time(msec) 3: 3. 3 ICA MEG (artifact) artifact 3 y 9 18Hz y 11 41msec 3 ft x 1 x x time(msec) 4:

6 MEG 4 ( 1msec msec) ( 8msec 1msec) MEG [] ICA 4 fmri ICA ( ) ICA ICA : MEG 1msec 8 1msec V1 ICA ICA 4 ICA ICA MEG, EEG, fmri 6

7 x(t) =A(t) s(t) x i (t) = a ik (t) s k (t), k a ik (t) s k (t) = a ik (τ)s k (t τ), τ= ( 6) (FIR ) s s 1 a (t) 1 1 a (t) a (t) 11 a (t) 6: W y(t) W (t) y(t) =W (t) x(t) W (t) ={w ij (t)} FIR (4) (7) [1, 6] [7, 1, 8] x(t), s(t), A(t) Fourier ω ˆx(ω), ŝ(ω), Â(ω) ˆx(ω) =Â(ω)ŝ(ω), msec x x 1 ˆx(ω, t s )=Â(ω)ŝ(ω, t s), ˆx(ω, t s ) ŝ(ω, t s ) x(t), s(t) windowed Fourier ω (1) (1) ICA W (ω) W (ω) 1 A(ω) W (ω) W (ω) 1 [8, 1] ICA ICA 3 4 ICA ICA 7

8 ICA ICA s A s s i [9] A A [3, 8] ICA [31] KL [3, 33] [34] 6 ICA web [3, 36, 37] 3 [38, 39] ICA ICA ICA ICA1 [4] ICA [41] 7 [1] Jutten, C. and Herault, J. (1991): Separation of sources, Part I, Signal Processing, Vol. 4, No. 1, pp [] Jutten, C. and Taleb, A.: Source separation: from dusk till dawn, in Proceedings of International Workshop on Independent Component Analysis and Blind Signal Separation (ICA), pp. 1 6 (). [3] Comon, P. (1994): Independent component analysis, A new concept?, Signal Processing, Vol. 36, No. 3, pp [4] Amari, S. and Cardoso, J.-F. (1997): Blind Source Separation Semiparametric Statistical Approach, 3 8

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