ica (version 1.0-1)

icajade: ICA via JADE Algorithm

Description

Computes ICA decomposition using Cardoso and Souloumiac's (1993, 1996) Joint Approximate Diagonalization of Eigenmatrices (JADE) approach.

Usage

icajade(X,nc,center=TRUE,maxit=100,tol=1e-6,Rmat=diag(nc))

Arguments

X

Data matrix with n rows (samples) and p columns (variables).

nc

Number of components to extract.

center

If TRUE, columns of X are mean-centered before ICA decomposition.

maxit

Maximum number of algorithm iterations to allow.

tol

Convergence tolerance.

Rmat

Initial estimate of the nc-by-nc orthogonal rotation matrix.

Value

S

Matrix of source signal estimates (S=Y%*%R).

M

Estimated mixing matrix.

W

Estimated unmixing matrix (W=crossprod(R,Q)).

Y

Whitened data matrix.

Q

Whitening matrix.

R

Orthogonal rotation matrix.

vafs

Variance-accounted-for by each component.

iter

Number of algorithm iterations.

Details

ICA Model The ICA model can be written as X=tcrossprod(S,M)+E, where columns of S contain the source signals, M is the mixing matrix, and columns of E contain the noise signals. Columns of X are assumed to have zero mean. The goal is to find the unmixing matrix W such that columns of S=tcrossprod(X,W) are independent as possible.

Whitening Without loss of generality, we can write M=P%*%R where P is a tall matrix and R is an orthogonal rotation matrix. Letting Q denote the pseudoinverse of P, we can whiten the data using Y=tcrossprod(X,Q). The goal is to find the orthongal rotation matrix R such that the source signal estimates S=Y%*%R are as independent as possible. Note that W=crossprod(R,Q).

JADE The JADE approach finds the orthogonal rotation matrix R that (approximately) diagonalizes the cumulant array of the source signals. See Cardoso and Souloumiac (1993,1996) and Helwig and Hong (2013) for specifics of the JADE algorithm.

References

Cardoso, J.F., & Souloumiac, A. (1993). Blind beamforming for non-Gaussian signals. IEE Proceedings-F, 140, 362-370.

Cardoso, J.F., & Souloumiac, A. (1996). Jacobi angles for simultaneous diagonalization. SIAM Journal on Matrix Analysis and Applications, 17, 161-164.

Helwig, N.E. & Hong, S. (2013). A critique of Tensor Probabilistic Independent Component Analysis: Implications and recommendations for multi-subject fMRI data analysis. Journal of Neuroscience Methods, 213, 263-273.

Examples

# NOT RUN {
##########   EXAMPLE 1   ##########

# generate noiseless data (p==r)
set.seed(123)
nobs <- 1000
Amat <- cbind(icasamp("a","rnd",nobs),icasamp("b","rnd",nobs))
Bmat <- matrix(2*runif(4),2,2)
Xmat <- tcrossprod(Amat,Bmat)

# ICA via JADE with 2 components
imod <- icajade(Xmat,2)
acy(Bmat,imod$M)
congru(Amat,imod$S)



##########   EXAMPLE 2   ##########

# generate noiseless data (p!=r)
set.seed(123)
nobs <- 1000
Amat <- cbind(icasamp("a","rnd",nobs),icasamp("b","rnd",nobs))
Bmat <- matrix(2*runif(200),100,2)
Xmat <- tcrossprod(Amat,Bmat)

# ICA via JADE with 2 components
imod <- icajade(Xmat,2)
congru(Amat,imod$S)



##########   EXAMPLE 3   ##########

# generate noisy data (p!=r)
set.seed(123)
nobs <- 1000
Amat <- cbind(icasamp("a","rnd",nobs),icasamp("b","rnd",nobs))
Bmat <- matrix(2*runif(200),100,2)
Emat <- matrix(rnorm(10^5),1000,100)
Xmat <- tcrossprod(Amat,Bmat)+Emat

# ICA via JADE with 2 components
imod <- icajade(Xmat,2)
congru(Amat,imod$S)

# }