BSSasymp (version 1.2-3)

CRB: Cramer-Rao bound for the unmixing matrix estimate in the independent component model.

Description

Cramer-Rao bound for the unmixing matrix estimate in the independent component model.

Usage

CRB(sdf,supp=NULL,A=NULL,eps=1e-04,...)

Arguments

sdf

a list of density functions of the sources scaled so that the mean is 0 and variance is 1.

supp

a two column matrix, where each row gives the lower and the upper limit used in numerical integration for the corresponding source component which is done using integrate. Can be NULL if each support is the whole real line.

A

the mixing matrix, identity by default.

eps

a value which is used when the derivative functions of the density functions are approximated.

…

arguments to be passed to integrate.

Value

A list containing the following components:

CRLB

A matrix whose elements give the Cramer-Rao lower bounds for the asymptotic variances of the corresponding elements of \(\hat{G}\).

FIM

The Fisher information matrix.

EMD

The sum of the Cramer-Rao lower bounds of the off-diagonal elements of \(\hat{G}\).

Details

Let \(\hat{W}\) denote an unmixing matrix estimate. If the estimate is affine equivariant, then the matrix \(\hat{G}=\hat{W}A\) does not depend on the mixing matrix \(A\) and the estimated independent components are \(\hat{S}=\hat{G}S\), where \(S\) is the matrix of the true independent components.

References

Ollila, E., Kim, H. J. and Koivunen, V. (2008), Compact Cramer-Rao bound expression for independent component analysis. IEEE Transactions on Signal Processing, 56(4), 1421--1428.

Examples

# NOT RUN {
# source components have t(9)- and Gaussian distribution  

f1<-function(x)
{
 gamma(5)*(1+(x*sqrt(9/7))^2/9)^(-5)/
 (sqrt(9*pi/(9/7))*gamma(9/2))
}

f2<-function(x)
{
 exp(-(x)^2/2)/sqrt(2*pi)
}

CRB(sdf=c(f1,f2))

# }