rjd {JADE} | R Documentation

## Joint Diagonalization of Real Matrices

### Description

This is an R version of Cardoso's rjd matlab function for joint diagonalization of k real-valued square matrices. A version written in C is also available and preferrable.

### Usage

    rjd(X, eps = 1e-06, maxiter = 100, na.action = na.fail)
    frjd(X, weight = NULL, maxiter = 100, eps = 1e-06, na.action = na.fail)
    frjd.int(X, maxiter = 100, eps = 1e-06)
    rjd.fortran(X, weight = NULL, maxiter = 100, eps = 1e-06, na.action = na.fail)

### Arguments

`X` | A matrix of k stacked pxp matrices with dimension c(kp,p) or an array with dimension c(p,p,k). In case of `frjd_int` it has to be an array.
`weight` | A vector of length k to give weight to the different matrices, if NULL, all matrices have equal weight
`eps` | Convergence tolerance.
`maxiter` | Maximum number of iterations.
`na.action` | A function which indicates what should happen when the data contain 'NA's. Default is to fail.

### Details

Denote the square matrices as A_i, i=1,...,k. The algorithm searches an orthogonal matrix V so that D_i=V'A_iV is diagonal for all i. If the A_i commute then there is an exact solution. Otherwise, the function will perform an approximate joint diagonalization by trying to make the D_i as diagonal as possible.

Cardoso points out that notion of approximate joint diagonalization is ad hoc and very small values of `eps` make in that case not much sense since the diagonality criterion is ad hoc itself.

`rjd`, `frjd` and `rjd.fortran` terminate with an error in case maxiter is reach without convergence whereas `frjd_int` returns the current state at when `maxiter` is reached and does not warn about convergence problems.

### Value

A list with the components

`V` | An orthogonal matrix.
`D` | A stacked matrix with the diagonal matrices or an array with the diagonal matrices. The form of the output depends on the form of the input.
`iter` | The `frjd` function returns also the number of iterations.

### Author(s)

Jean-Francois Cardoso. Ported to R by Klaus Nordhausen. C code by Jari Miettinen

### References

Cardoso, J.-F. and Souloumiac, A., (1996), Jacobi angles for simultaneous diagonalization, SIAM J. Mat. Anal. Appl., 17, 161–164.

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>.

### Examples

    Z <- matrix(runif(9), ncol = 3)
    U <- eigen(Z %*% t(Z))$vectors
    D1 <- diag(runif(3))
    D2 <- diag(runif(3))
    D3 <- diag(runif(3))
    D4 <- diag(runif(3))

    X.matrix <- rbind(t(U) %*% D1 %*% U, t(U) %*% D2 %*% U,
                      t(U) %*% D3 %*% U, t(U) %*% D4 %*% U)
    res.matrix <- rjd(X.matrix)
    res.matrix$V
    round(U %*% res.matrix$V, 4) # should be a signed permutation
                                 # matrix if V is correct.

    round(res.matrix$D, 4)

    # compare to C version

    #res.matrix.C <- frjd(X.matrix)
    #res.matrix.C$V
    #round(U %*% res.matrix.C$V, 4)
    #round(res.matrix.C$D, 4)

    X.array <- aperm(array(t(X.matrix), dim = c(3,3,4)), c(2,1,3))

    res.array <- rjd(X.array)
    round(res.array$D, 4)

    res.array.C <- frjd(X.array)
    round(res.array.C$D, 4)

    res.array.C2 <- frjd.int(X.array)
    round(res.array.C2$D, 4)

[Package JADE version 2.0-2 Index]