JADE (version 1.1-0)

MD: Minimum Distance index MD

Description

Computes the Minimum Distance index MD to evaluate the performance of an ICA algorithm.

Usage

MD(W.hat, A)

Arguments

W.hat

The estimated square unmixing matrix W.

A

The true square mixing matrix A.

Value

• The value of the MD index.

Details

$$MD(\hat{W},A)=\frac{1}{\sqrt{p-1}} \inf_{P D}{||PD \hat{W} A-I||,}$$ where $P$ is a permutation matrix and $D$ a diagonal matrix with nonzero diagonal entries. The step that minimizes the index of the set over all permutation matrix can be expressed as a linear sum assignment problem (LSAP) for which we use as solver the Hungarian method implemented as solve_LASP in the clue package. Note that this function assumes the ICA model is $X = S A'$, as is assumed by JADE and ics. However fastICA and PearsonICA assume $X = S A$. Therefore matrices from those functions have to be transposed first. The MD index is scaled in such a way, that it takes a value between 0 and 1. And 0 corresponds to an optimal separation.

References

Ilmonen, P., Nordhausen, K., Oja, H. and Ollila, E. (2010): A New Performance Index for ICA: Properties, Computation and Asymptotic Analysis. In Vigneron, V., Zarzoso, V., Moreau, E., Gribonval, R. and Vincent, E. (editors) Latent Variable Analysis and Signal Separation, 229--236, Springer.

See Also

ComonGAP, SIR, amari.error, solve_LASP

Examples

S <- cbind(rt(1000, 4), rnorm(1000), runif(1000))
A <- matrix(rnorm(9), ncol = 3)
X <- S %*% t(A)

W.hat <- JADE(X, 3)$W
MD(W.hat, A)