16 1 (Independent Component Analysis: ICA) 198 9 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
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
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
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 () 1 11 1 13 or 6 1 14 1: Low-Pass 1 ft x 1 x x 3 1 1 3 4 time(msec) : MEG : 1 1 3 1msec 41msec 1 msec ICA x x = As 1 s 4
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 17 1 1 3 4 time(msec) 3: 3. 3 ICA MEG (artifact) artifact 3 y 9 18Hz y 11 41msec 3 ft x 1 x x 3 1 1 3 4 time(msec) 4:
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
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
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 1999 1 1 3 ICA ICA1 [4] 3 4 4 ICA [41] 7 [1] Jutten, C. and Herault, J. (1991): Separation of sources, Part I, Signal Processing, Vol. 4, No. 1, pp. 1 1. [] 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. 87 314. [4] Amari, S. and Cardoso, J.-F. (1997): Blind Source Separation Semiparametric Statistical Approach, 3 8
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[31] Bach, F. R. and Jordan, M. I.: Kernel Independent Component Analysis, Technical Report UCB//CSD-1-1166, University of California, Berkeley, (1). [3] Matsuyama, Y., Katsumata, N. and Imahara, S.: Convex Divergence as a Surrogate Function for Independence: The f-divergence ICA, in Proceedings of International Workshop on Independent Component Analysis and Blind Signal Separation (ICA1), pp. 31 36 (1). [33] Minami, M. and Eguchi, S.: Robust Blind Source Separation by β-divergence, to appear in Neural Computation (). [34] Akuzawa, T.: New Fast Factorization Method for Multivariate Optimization and its Realization as ICA Algorithm, in Proceedings of International Workshop on Independent Component Analysis and Blind Signal Separation (ICA1), pp. 114 119 (1). [3] Lee, T.-W. (1998): Independent Component Analysis, Kluwer Academic Publishers. [36] Hyvärinen, A., Karhunen, J. and Oja, E. (1): Independent Component Analysis, John Wiley & Sons, Inc. [37] Cichocki, A. and Amari, S. (): Adaptive Blind Signal and Image Processing, John Wiley & Sons, Inc. [38] Girolami, M. ed. (): Advances in Independent Component Analysis, Springer-Verlag London Ltd. [39] Haykin, S. ed. (): Unsupervised Adaptive Filtering, Volume 1, Blind Source Separation, Vol. 1, John Wiley & Sons, Inc. [4] http://www.ica1.org. [41] http://ica3.jp. 1