dcovICA: ICA via distance covariance for 2 components

Description

This algorithm finds the rotation which minimizes the distance covariance between two orthogonal components via the angular parameterization of a 2x2 orthogonal matrix with the function stats::optimize. The results will be (approximately) equivalent to steadyICA but this function is much faster (but does not extend to higher dimensions).

Usage

dcovICA(Z, theta.0 = 0)

Arguments

The whitened n x d data matrix, where n is the number of observations and d the number of components.

theta.0

Determines the interval to be searched by the optimizer: lower bound = theta.0, upper bound = pi/2. Changing theta.0 affects the initial value, where the initial value = theta.0+(1/2+sqrt(5)/2)*pi/2, see optimize.

Value

theta.hatEstimated minimum.
WW = t(theta2W(theta.hat))
SEstimated independent components.
objThe distance covariance of S.

References

Matteson, D. S. & Tsay, R. Independent component analysis via U-Statistics.

Examples

Run this code

library(JADE)
library(ProDenICA)
set.seed(123)
simS = cbind(rjordan(letter='j',n=1024),rjordan(letter='m',n=1024))
simM = mixmat(p=2)
xData = simS%*%simM
xWhitened = whitener(xData)

#Define true unmixing matrix as true M multiplied by the estimated whitener:
#Call this the target matrix:
W.true <- solve(simM%*%xWhitened$whitener) 


a=Sys.time()
est.dCovICA = dcovICA(Z = xWhitened$Z,theta.0=0)
Sys.time()-a

#See the example with steadyICA for an explanation
#of the parameterization used in amari.error:
amari.error(t(est.dCovICA$W),W.true)

##NOTE: also try theta.0 = pi/4 since there may be local minima
  est.dcovICA = dcovICA(Z = xWhitened$Z,theta.0=pi/4)
  amari.error(t(est.dcovICA$W),W.true)

a=Sys.time()
est.steadyICA = steadyICA(X=xWhitened$Z,verbose=TRUE)
Sys.time()-a
amari.error(t(est.steadyICA$W),W.true)
##theta parameterization with optimize is much faster