16 5 25

1(Independent Component Analysis:ICA) 198090

ICA ICA ICA

1

ICA 1980 Jutten Herault 2

Comon[3], Amari & Cardoso[4] ICA Comon

(PCA) projection persuit projection persuit ICA ICA

ICA

1

2[1] [2] ICA

ICAEEG [5] fMRI [6]

[7, 8]

[9, 10]

J-F. Cardoso JADE [11] A. Hyvarinen FastICA [12] ICA central web page [13] Matlab ()

ICA

ICA

ICAICA

1

ICA

2

2.1 ICA

s(t) = (s1(t), . . . , sn(t))T

t = 0, 1, 2, . . .

s(t) 0T

x(t) = (x1(t), . . . , xm(t))T

t = 0, 1, 2, . . .

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

x

x

1

2

1

2

11a

a21

a 22

a 12

1: 2 2 ICA

n m nm W

y(t) = Wx(t), (2)

y(t)WA = I(I n n ) y(t) s(t) y(t)

WA = PD (P 1 n n D n n )

2.2

ICA

3 [14, 15]

JADE FastICA ICA

si ()

W

x y

y

p(y) = p(y1, , yn)

W

y yi p(yi) yi

p(y) =n

i=1

p(yi)

p(y)n

i=1 p(yi) W

p(y) n

i=1 p(yi) Kullback-Leibler ()KL

KL(W ) =

p(y) logp(y)n

i=1 p(yi)dy (3)

= H(Y ;W ) +n

i=1

H(Yi;W ).

H(Y ;W ) H(Yi;W ) p(y)dy = p(x)dxp(y) = p(x)/|W | (|W | W )H(Y ;W ) H(Yi;W ) p(x) W

2

H(Y ;W ) =

p(y) logp(y)dy

=

p(x)(log p(x) log |W |)dx

= H(X) + log |W |,

H(Yi;W ) =

p(y) logp(yi)dy

=

p(x) log p(yi)dx

KL(W ) p(yi) 0 KL(W )

W KL(W )W

=((WT )1 Ex[(y)xT ]

)=(I Ex[(y)yT ]

)(WT )1 (4)

(y) = (

log p(y1)y1

, . . . , log p(yn)

yn

)T

W (4) (WT )1

WTW [16]

W (I Ex[(y)yT ])W (5)

[17]

W

Wt+1 = Wt + (I (y)yT)Wt.

(y)

W [4]W (y) ICA

x(t)

x(t)x(t + )T

= A

s(t)s(t+ )T

AT

= A

Rs1() 0. . .

0 Rsn()

AT ,

x(t) Rsi() si(t) s(t)

si(t) W y(t)

y(t)y(t+ )T

=(WAs(t)) (WAs(t+ ))T

=

21Rs1 () 0. . .

0 2nRsn ()

,

1, 2, . . . , n 1, 2, . . ., n i

W

y(t) x(t) i W

[18]

W x(t)x(t+ i)T WT = i, i = 1, . . . , r, (6)

i 2

W [18]2 i

[15]

KL [19]

3

si(t) [20]

Q(t) =12

{n

i=1

logE[y2i (t)] log detE[y(t)y(t)T ]}

,

(7)

(E[] ) Q ICA

Q E[y(t)y(t)T ] 0 W

W Q 0

Q

W

3

ICA

ICA

3.1

MEG [21, 22]MEG 100200MEG (1msec) ( mm)

(1014T)MEG () ( 1)

1

T Hz

1014 a few 20

1045 a few 10() 1011 1013 50 or 60

1014

1:

Low-Pass

50100

505

fT

x1

505

x2

100 0 100 200 300 4005

05

time(msec)

x3

2: MEG : 100 120 3 0 100msec 412msec

100 2 0msec

ICA x

x = As

1 s

4

ICA

x = As+

x = As+ = As = (A, I)

(s

)

ICA

[22] ICA A

ICA 3

[23, 24]

3.2

3

ICA MEG

(artifact) artifact

3 y9 180Hz

y11

412msec

y1

y2

y3

y4

y5

y6

y7

y8

y9

y10

y11

y12

y13

y14

y15

y16

100 0 100 200 300 400time(msec)

y17

3:

3

505

fT

x1

505

x2

100 0 100 200 300 4005

05

time(msec)

x3

4:

5

MEG 4 2 (100msec0msec) ( 80msec150msec)

MEG [25] 5 ICA 4

fMRI

5

()ICA

2 150msec 80150msec

V1

ICA

ICA

ICA

ICA

5: MEG

4

ICA

ICA

MEG, EEG, fMRI

6

x(t) = A(t) s(t)xi(t) =

k

aik(t) sk(t),

aik(t) sk(t) =

=0

aik()sk(t ),

( 6)

(FIR)

s

s

x

x

1

2

1

2

21

22

11

12

a (t)a

(t)

a (t)

a (t)

6: 2 2

W y(t) W (t)

y(t) = W (t) x(t)W (t) = {wij(t)}

FIR

(4) (7) [10, 26]

[27, 15, 28]x(t), s(t), A(t) Fourier

x(), s(), A()

x() = A()s(),

msec

x(, ts) = A()s(, ts),

x(, ts) s(, ts) x(t), s(t) windowed Fourier

(1) (1)

ICA

W () W ()1 A() W () W ()1

[28, 15]

ICA

5

ICA

34 ICAICA

7

ICA

ICA

s A

s

si

2

2 2

[29]

A

A [30, 8]

ICA

[31]KL [32, 33]

[34]

6

ICA web

[35, 36, 37]3 [38, 39]

ICA ICA 1999 1

1 3

ICAICA2001 [40]

2003 4 4 ICA [41]

7

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sources, Part I, Signal Processing, Vol. 24, No. 1,pp. 110.

[2] Jutten, C. and Taleb, A.: Source separation:from dusk till dawn, in Proceedings of Interna-tional Workshop on Independent Component Analy-sis and Blind Signal Separation (ICA2000), pp. 1526 (2000).

[3] Comon, P. (1994): Independent component analysis,A new concept?, Signal Processing, Vol. 36, No. 3,pp. 287314.

[4] Amari, S. and Cardoso, J.-F. (1997): Blind SourceSeparation Semiparametric Statistical Approach,

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IEEE Transactions on Signal Processing, Vol. 45,No. 11, pp. 26922700.

[5] Makeig, S., Jung, T.-P., Bell, A. J., Ghahremani, D.and Sejnowski, T. J. (1997): Blind Separation ofAuditory Event-related Brain Responses into Inde-pendent Components, Proc. Natl. Acad. Sci. USA,pp. 1097910984.

[6] Mckeown, M. J., Jung, T.-P., Makeig, S., Brown, G.,Kindermann, S. S., Lee, T.-W. and Sejnowski, T. J.:Spatially independent activity patterns in functionalmagnetic resonance imaging data during the Stroopcolor-naming task, in Proceedings of the NationalAcademy of Sciences, Vol. 95, pp. 803810 (1998).

[7] Bell, A. J. and Sejnowski, T. J. (1997): The inde-pendent component of natural scenes are edge l-ters, Vison Research, Vol. 37, pp. 33273338.

[8] Olshausen, B. A. and Field, D. J. (1996): Emergenceof simple-cell receptive eld properties by learninga sparce code for natural images, Nature, Vol. 381,pp. 607609.

[9] Bell, A. J. and Sejnowski, T. J. (1995): An informa-tion maximization approach to blind separation andblind deconvolution, Neural Computation, Vol. 7,No. 6, pp. 11291159.

[10] Douglas, S. C. and Cichocki, A. (1997): NeuralNetworks for Blind Decorrelation of Signals, IEEETransactions on Signal Processing, Vol. 45, No. 11,pp. 28292842.

[11] http://www.tsi.enst.fr/icacentral/Algos/cardoso/.

[12] http://www.cis.hut./projects/ica/fastica/.

[13] http://www.tsi.enst.fr/icacentral/.

[14] Cardoso, J.-F.: The three easy routes to indepen-dent component analysis; contrasts and geometry,in Proceedings of International Workshop on Inde-pendent Component Analysis and Blind Signal Sep-aration (ICA2001), pp. 16 (2001).

[15] Murata, N., Ikeda, S. and Ziehe, A. (2001): An Ap-proach to Blind Source Separation Based on Tem-poral Structure of Speech Signals, Neurocomputing,Vol. 41, No. 1-4, pp. 124.

[16] Amari, S., Cichocki, A. and Yang, H. H.: A NewLearning Algorightm for Blind Signal Separation, inTouretzky, D. S., Mozer, M. C. and Hasselmo, M. E.eds., Advances in Neural Information ProcessingSystems, Vol. 8, pp. 757763, The MIT Press, Cam-bridge MA (1996).

[17] Amari, S., Chen, T. and Cichocki, A. (1997): Stabil-ity Analysis of Learning Algorithms for Blind SourceSeparation, Neural Networks, Vol. 10, No. 8, pp.13451351.

[18] Molgedey, L. and Schuster, H. G. (1994): Separationof a mixture of independent signals using time de-layed correlations, Phys. Rev. Lett., Vol. 72, No. 23,pp. 36343637.

[19] Amari, S.: ICA of temporally correlated signals learning algorithm, in Proceedings of InternationalWorkshop on Independent Component Analysis andBlind Signal Separation (ICA99), pp. 1318 (1999).

[20] Matsuoka, K., Ohya, M. and Kawamoto, M. (1995):A Neural Net for Blind Separation of NonstationarySignals, Neural Networks, Vol. 8, No. 3, pp. 411419.

[21] Ikeda, S.: ICA on Noisy Data: A Factor AnalysisApproach, in Girolami, M. ed., Advances in Inde-pendent Component Analysis, chapter 11, pp. 201215, Springer-Verlag London Ltd. (2000).

[22] Ikeda, S. and Toyama, K. (2000): Independent Com-ponent Analysis for Noisy Data MEG data analy-sis, Neural Networks, Vol. 13, No. 10, pp. 10631074.

[23] Cao, J., Murata, N., Amari, S., Cichocki, A. andTakeda, T.: A Robust ICA Approach for unaver-aged single-trial auditory evoked elds data decom-position, in Proceedings of International Workshopon Independent Component Analysis and Blind Sig-nal Separation (ICA2001), pp. 445450 (2001).

[24] Kawanabe, M. and Murata, N.: Independent Com-ponent Analysis in the Presence of Gaussian noisebased on estimating functions, in Proceedings of In-ternational Workshop on Independent ComponentAnalysis and Blind Signal Separation (ICA2000),pp. 3944 (2000).

[25] Toyama, K., Yoshikawa, K., Yoshida, Y., Kondo, Y.,Tomita, S., Takanashi, Y., Ejima, Y. andYoshizawa, S. (1999): A new method for magnetoen-cephalography: A three dimensional magnetometer-spatial lter system, Neuroscience, Vol. 91, No. 2,pp. 405415.

[26] Kawamoto, M., Matsuoka, K. and Ohnishi, N.(1998): A method of blind separation for convolvednon-stationary signals, Neurocomputing, Vol. 22,No. 1-3, pp. 157171.

[27] Smaragdis, P. (1998): Blind separation of convolvedmixtures in the frequency domain, Neurocomputing,Vol. 22, No. 1-3, pp. 2134.

[28] Ikeda, S. and Murata, N.: A method of blind separa-tion based on temporal structure of signals, in Pro-ceedings of 1998 International Conference on Neu-ral Information Processing (ICONIP98), Vol. 2, pp.737742, Kitakyushu, Japan (1998).

[29] Rickard, S., Balan, R. and Rosca, J.: Real-TimeTime-Frequency Based Blind Source Separation, inProceedings of International Workshop on Indepen-dent Component Analysis and Blind Signal Separa-tion (ICA2001), pp. 651656 (2001).

[30] Hyvarinen, A., Hoyer, P. and Inki, M. (2001): To-pographic Independent Component Analysis, Neu-ral Computation, Vol. 13, No. 7, pp. 15251558.

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[31] Bach, F. R. and Jordan, M. I.: Kernel In-dependent Component Analysis, Technical Re-port UCB//CSD-01-1166, University of California,Berkeley, (2001).

[32] Matsuyama, Y., Katsumata, N. and Imahara, S.:Convex Divergence as a Surrogate Function for Inde-pendence: The f-Divergence ICA, in Proceedings ofInternational Workshop on Independent ComponentAnalysis and Blind Signal Separation (ICA2001),pp. 3136 (2001).

[33] Minami, M. and Eguchi, S.: Robust Blind SourceSeparation by -Divergence, to appear in NeuralComputation (2002).

[34] Akuzawa, T.: New Fast Factorization Method forMultivariate Optimization and its Realization asICA Algorithm, in Proceedings of InternationalWorkshop on Independent Component Analysis andBlind Signal Separation (ICA2001), pp. 114119(2001).

[35] Lee, T.-W. (1998): Independent Component Analy-sis, Kluwer Academic Publishers.

[36] Hyvarinen, A., Karhunen, J. and Oja, E. (2001): In-dependent Component Analysis, John Wiley & Sons,Inc.

[37] Cichocki, A. and Amari, S. (2002): Adaptive BlindSignal and Image Processing, John Wiley & Sons,Inc.

[38] Girolami, M. ed. (2000): Advances in IndependentComponent Analysis, Springer-Verlag London Ltd.

[39] Haykin, S. ed. (2000): Unsupervised Adaptive Fil-tering, Volume 1, Blind Source Separation, Vol. 1,John Wiley & Sons, Inc.

[40] http://www.ica2001.org.

[41] http://ica2003.jp.

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