Re: Sample uniformly from a simplex
- To: mathgroup at smc.vnet.net
- Subject: [mg94469] Re: Sample uniformly from a simplex
- From: Asim <maa48 at columbia.edu>
- Date: Sun, 14 Dec 2008 07:37:59 -0500 (EST)
- References: <ghtjg8$rb3$1@smc.vnet.net>
On Dec 12, 6:55 am, Andreas <aa... at ix.netcom.com> wrote: > I need to develop Mathematica code to sample uniformly from a unit n-dime= nsional simplex. > > I came across a description of the problem at:http://geomblog.blogspot.co= m/2005/10/sampling-from-simplex.html > > Specifically, I would like a uniform sample from the set > > X = { (x1, x2, ..., xD) | 0 <= xi <= 1, x1 + x2 + ... + xD = 1}. > > D is the dimension of the simplex. > > So, the coordinates of any point on the simplex would sum to 1 and I need= to sample points on the simplex. > > geomblog's solution suggested: > > Generating IID random samples from an exponential distribution by samplin= g X from [0,1] uniformly, and returning -log(X)). > > Take n samples, then normalize. > > This should result in a list of numbers which is a uniform sample from th= e simplex. > > I've searched extensively for a Mathematica implementation of something l= ike this, to no avail. > > I keep trying different things but haven't made much headway. > > Any suggestions for how to develop this (or an equivelant) in Mathematica= much appreciated > > A The following is code from Mathematica tutorials for sampling from the Dirichlet distribution. DirichletDistribution /: Random`DistributionVector[ DirichletDistribution[alpha_?(VectorQ[#, Positive] &)], n_Integer, prec_?Positive] := Block[{gammas}, gammas = Map[RandomReal[GammaDistribution[#, 1], n, WorkingPrecision -> prec] &, alpha]; Transpose[gammas]/Total[gammas]] Asim