icafast {ica}

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 = "par", fun = "logcosh", alpha = 1)

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: fun="logcosh" for log of hyperbolic cosine, fun="exp" for Gaussian kernel, and fun="kur" for kurtosis.

alpha

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

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).

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) or Helwig and Hong (2013) for specifics of fixed-point 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.

alg

Algorithm used (same as input).

fun

Contrast function (same as input).

alpha

Tuning parameter (same as input).

converged

Logical indicating if algorithm converged.

Author(s)

Nathaniel E. Helwig <[e-mail]>

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

Hyvarinen, A. (1999). Fast and robust fixed-point algorithms for independent component analysis. IEEE Transactions on Neural Networks, 10(3), 626-634. doi:10.1109/72.761722

See Also

icaimax for ICA via Infomax

icajade for ICA via JADE

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 FastICA with 2 components
imod <- icafast(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 FastICA with 2 components
imod <- icafast(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 FastICA with 2 components
imod <- icafast(Xmat, nc = 2)
cor(Amat, imod$S)