ica (version 1.0-2)

icafast: ICA via FastICA Algorithm

Description

Computes ICA decomposition using Hyvarinen's (1999) FastICA algorithm with various options.

Usage

icafast(X, nc, center = TRUE, maxit = 100, tol = 1e-6,
        Rmat = diag(nc), alg = c("par", "def"),
        fun = c("logcosh", "exp", "kur"), alpha = 1)

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.

alg

Algorithm used (same as input).

fun

Contrast function (same as input).

alpha

Tuning parameter (same as input).

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.

alg

Algorithm to use: alg="par" to estimate all nc components in parallel (default) or alg="def" for deflation estimation (i.e., projection pursuit).

fun

Contrast function to use for negentropy approximation.

alpha

Tuning parameter for "logcosh" contrast function (1 <= alpha <= 2).

Author

Nathaniel E. Helwig <[e-mail]>

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).

FastICA The FastICA algorithm finds the orthogonal rotation matrix R that (approximately) maximizes the negentropy of the estimated source signals. Negentropy is approximated using $$J(s) = [E\{G(s)\}-E\{G(z)\} ]^2$$ where E denotes the expectation, G is the contrast function, and z is a standard normal variable. See Hyvarinen (1999) for specifics of fixed-point algorithm.

References

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.

Hyvarinen, A. (1999). Fast and robust fixed-point algorithms for independent component analysis. IEEE Transactions on Neural Networks, 10, 626-634.

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),2,2)
Xmat <- tcrossprod(Amat,Bmat)

# ICA via FastICA with 2 components
imod <- icafast(Xmat,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),100,2)
Xmat <- tcrossprod(Amat,Bmat)

# ICA via FastICA with 2 components
imod <- icafast(Xmat,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),1000,100)
Xmat <- tcrossprod(Amat,Bmat)+Emat

# ICA via FastICA with 2 components
imod <- icafast(Xmat,2)
cor(Amat,imod$S)