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

From: Florent Bresson <f_bresson_at_yahoo.fr>
Date: Mon 16 Oct 2006 - 16:17:51 GMT


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

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@math.unb.ca

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