ProDenICA (version 1.0)

ProDenICA: Product Density Independent Component Analysis

Description

Fits an ICA model by directly estimating the densities of the independent components using Poisson GAMs. The densities have the form of tilted Gaussians, and hense directly estimate the contrast functions that lead to negentropy measures. This function supports Section 14.7.4 of 'Elements of Statistical Learning (Hastie, Tibshirani and Friedman, 2009, 2nd Edition)'. Models include 'FastICA'.

Usage

ProDenICA(x, k = p, W0 = NULL, whiten = FALSE, maxit = 20, thresh = 1e-07,
restarts = 0, trace = FALSE, Gfunc = GPois, eps.rank = 1e-07, ...)

Arguments

x

input matrix

k

Number of components required, less than or equal to the number of columns of x

W0

Optional initial matrix (for comparing algorithms)

whiten

Logical variable - should x be whitened. If TRUE, the SVD of X=UDV' is computed, and U is used (up to rank(X) columns). Also k is reduced to min(k,rank(X)). If FALSE (default), it is assumed that the user has pre-whitened x (and if not, the function may not perform properly)

maxit

Maximum number of iterations; default is 20

thresh

Convergence threshold, in terms of relative change in Amari metric; dfault is 1e-7

restarts

Number of random restarts; default is 0

trace

Trace iterations; default is FALSE

Gfunc

Contrast functional which is basis for negentropy measure. Default is 'GPois' which fits a tilted Gaussian density using a Poisson GAM. Other options are 'G1' (cosh negentropy) and 'G0' (kurtosis negentropy)

eps.rank

Threshold for deciding rank of x if option whiten=TRUE. Any singular value less than eps.thresh smaller than the largest is treated as zero

…

Additional arguments for Gfunc areguments

Value

An object of S3 class "ProDenICA" is returned, with the following components:

W

Orthonormal matrix that takes the whitened version of x to the independent components

negentropy

The total negentropy measure of this solution

s

the matrix of k independent components

whitner

if whiten=TRUE, the matrix that whitens x, else NULL

call

the call that produced this object

density

If Gfunc=GPois, an list of length k with the density estimates for each component

Details

See Section 14.7.4 of Elements of Statistical Learning (Hastie, Tibshirani and Friedman, 2009, 2nd Edition)

References

Hastie, T. and Tibshirani, R. (2003) Independent Component Analysis through Product Density Estimation in Advances in Neural Information Processing Systems 15 (Becker, S. and Obermayer, K., eds), MIT Press, Cambridge, MA. pp 649-656 Hastie, T., Tibshirani, R. and Friedman, J. (2009) Elements of Statistical Learning (2nd edition), Springer. http://www-stat.stanford.edu/~hastie/Papers/ESLII.pdf

See Also

GPois, G1 and plot method.

Examples

# NOT RUN {
p=2
### Can use letters a-r below for dist
dist="n" 
N=1024
A0<-mixmat(p)
s<-scale(cbind(rjordan(dist,N),rjordan(dist,N)))
x <- s %*% A0
###Whiten the data
x <- scale(x, TRUE, FALSE)
sx <- svd(x)	### orthogonalization function
x <- sqrt(N) * sx$u
target <- solve(A0)
target <- diag(sx$d) %*% t(sx$v) %*% target/sqrt(N)
W0 <- matrix(rnorm(2*2), 2, 2)
W0 <- ICAorthW(W0)
W1 <- ProDenICA(x, W0=W0,trace=TRUE,Gfunc=G1)$W
fit=ProDenICA(x, W0=W0,Gfunc=GPois,trace=TRUE, density=TRUE)
W2 <- fit$W
#distance of FastICA from target
amari(W1,target)
#distance of ProDenICA from target
amari(W2,target)
par(mfrow=c(2,1))
plot(fit)
# }

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