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

From: Rolf Turner <rolf_at_erdos.math.unb.ca>
Date: Mon 16 Oct 2006 - 15:50:24 GMT


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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