icajade {ica}

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.

Details

ICA Model The ICA model can be written as X = tcrossprod(S, M) + E, where S contains the source signals, M is the mixing matrix, and E contains 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.

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.

converged

Logical indicating if algorithm converged.

Author(s)

Nathaniel E. Helwig <[e-mail]>

References

Cardoso, J.F., & Souloumiac, A. (1993). Blind beamforming for non-Gaussian signals. IEE Proceedings-F, 140(6), 362-370. doi:10.1049/ip-f-2.1993.0054

Cardoso, J.F., & Souloumiac, A. (1996). Jacobi angles for simultaneous diagonalization. SIAM Journal on Matrix Analysis and Applications, 17(1), 161-164. doi:10.1137/S0895479893259546

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(2), 263-273. doi:10.1016/j.jneumeth.2012.12.009

See Also

icafast for FastICA

icaimax for ICA via Infomax

Examples

##########   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), nrow = 2, ncol = 2)
Xmat <- tcrossprod(Amat, Bmat)

# ICA via JADE with 2 components
imod <- icajade(Xmat, nc = 2)
acy(Bmat, imod$M)
cor(Amat, imod$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), nrow = 100, ncol = 2)
Xmat <- tcrossprod(Amat, Bmat)

# ICA via JADE with 2 components
imod <- icajade(Xmat, nc = 2)
cor(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), nrow = 1000, ncol = 100)
Xmat <- tcrossprod(Amat,Bmat) + Emat

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