R: Minimum Distance index MD

MD {JADE} | R Documentation

## 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.

### Details

MD(W.hat,A) = 1/sqrt(p-1) inf_(P D) ||P D W.hat 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.

### Value

The value of the MD index.

### Author(s)

Klaus Nordhausen

### 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.

Nordhausen, K., Ollila, E. and Oja, H. (2011), On the Performance Indices of ICA and Blind Source Separation. In the Proceedings of 2011 IEEE 12th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC 2011), 486–490.

Miettinen, J., Nordhausen, K. and Taskinen, S. (2017), Blind Source Separation Based on Joint Diagonalization in R: The Packages JADE and BSSasymp, Journal of Statistical Software, 76, 1–31, <doi:10.18637/jss.v076.i02>.

### See Also

`ComonGAP`, `SIR`, `amari.error`, `solve_LSAP`

### 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)

[Package JADE version 2.0-0 Index]