[R] Re : Re : Generate a random bistochastic matrix

[Florent Bresson]

Yes, I would like every generated matrix to be drawn from a uniform distribution. Martin Maechler's solution was interesting but when I compute the product of the obtained matrix with any N-vector Y, the resulting vector is most of the time quite close to a vector like mean(Y)*rep(1,N).

Florent Bresson

----- Message d'origine ----
De : Rolf Turner <rolf at erdos.math.unb.ca>
À : f_bresson at yahoo.fr
Cc : r-help at stat.math.ethz.ch
Envoyé le : Lundi, 16 Octobre 2006, 17h50mn 24s
Objet : Re: [R] Re :  Generate a random bistochastic matrix

I don't think this idea has been suggested yet:

    (1) Form all n! n x n permutation matrices,
    say M_1, ..., M_K, K = n!.

    (2) Generate K independent uniform variates
    x_1, ..., x_k.

    (3) Renormalize these to sum to 1,
        x <- x/sum(x)

    (4) Form the convex combination

        M = x_1*M_1 + ... + x_K*M_K

M is a ``random'' doubly stochastic matrix.

The point is that the set of all doubly stochastic matrices
is a convex set in n^2-dimensional space, and the extreme
points are the permutation matrices.  I.e. the set of all
doubly stochastic matrices is the convex hull of the the
permuation matrices.

The resulting M will *not* be uniformly distributed on this
convex hull.  If you want a uniform distribution something
more is required.  It might be possible to effect uniformity
of the distribution, but my guess is that it would be a
hard problem.

                cheers,

                    Rolf Turner
                    rolf at math.unb.ca