R: ICA via Infomax Algorithm

icaimax {ica}

R Documentation

ICA via Infomax Algorithm

Description

Computes ICA decomposition using Bell and Sejnowski's (1995) Information-Maximization (Infomax) approach with various options.

Usage

icaimax(X,nc,center=TRUE,maxit=100,tol=1e-6,Rmat=diag(nc),
        alg=c("newton","gradient"),fun=c("tanh","log","ext"),
        signs=rep(1,nc),signswitch=TRUE,rate=1,rateanneal=NULL)

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="newton"` for Newton iteration, and `alg="gradient"` for gradient descent.
`fun`	Nonlinear (squashing) function to use for algorithm: `fun="tanh"` for hyperbolic tangent, `fun="log"` for logistic, and `fun="ext"` for extended Infomax.
`signs`	Vector of length `nc` such that `signs[j]==1` if j-th component is super-Gaussian and `signs[j]==-1` if j-th component is sub-Gaussian. Only used if `fun="ext"`. Ignored if `signswitch=TRUE`.
`signswitch`	If `TRUE`, the `signs` vector is automatically determined from the data; otherwise a confirmatory ICA decomposition is calculated using input `signs` vector. Only used if `fun="ext"`.
`rate`	Learing rate for gradient descent algorithm. Ignored if `alg="newton"`.
`rateanneal`	Annealing angle and proportion for gradient descent learing rate (see Details). Ignored if `alg="newton"`.

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).

Infomax The Infomax approach finds the orthogonal rotation matrix R that (approximately) maximizes the joint entropy of a nonlinear function of the estimated source signals. See Bell and Sejnowski (1995) and Helwig (in prep) for specifics of algorithms.

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).
`signs`	Component signs (same as input).
`rate`	Learning rate (same as input).

Author(s)

Nathaniel E. Helwig <helwig@umn.edu>

References

Bell, A.J. & Sejnowski, T.J. (1995). An information-maximization approach to blind separation and blind deconvolution. Neural Computation, 7, 1129-1159.

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 Infomax with 2 components
imod <- icaimax(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 Infomax with 2 components
imod <- icaimax(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 Infomax with 2 components
imod <- icaimax(Xmat,2)
congru(Amat,imod$S)

[Package ica version 1.0-1 Index]