From: Robert A LaBudde <ral_at_lcfltd.com>

Date: Thu, 27 Mar 2008 04:02:07 -0500

Robert A. LaBudde, PhD, PAS, Dpl. ACAFS e-mail: ral_at_lcfltd.com

R-help_at_r-project.org mailing list

https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. Received on Thu 27 Mar 2008 - 08:05:24 GMT

Date: Thu, 27 Mar 2008 04:02:07 -0500

At 05:06 PM 3/26/2008, Ted Harding wrote:

>On 26-Mar-08 21:26:59, Ala' Jaouni wrote:

*> > X1,X2,X3,X4 should have independent distributions. They should be
**> > between 0 and 1 and all add up to 1. Is this still possible with
**> > Robert's method?
**> >
**> > Thanks
**>
**>I don't think so. A whileago you wrote
**>"The numbers should be uniformly distributed" (but in the
**>context of an example where you had 5 variable; now you
**>are back to 4 variables). Let's take the 4-case first.
**>
**>The two linear constraints confine the point (X1,X2,X3,X4)
**>to a triangular region within the 4-dimensional unit cube.
**>Say it has vertices A, B, C.
**>You could then start by generating points uniformly distributed
**>over a specific triangle in 2 dimentions, say the one with
**>vertices at A0=(0,0), B0=(0,1), C0=(1,0). This is easy.
**>
**>Then you need to find a linear transformation which will
**>map this triangle (A0,B0,C0) onto the triangle (A,B,C).
**>Then the points you have sampled in (A0,B0,C0) will map
**>into points which are uniformly distributed over the
**>triangle (A,B,C).
**>
**>More generally, you will be seeking to generate points
**>uniformly distributed over a simplex.
**>
**>For example, the case (your earlier post) of 5 points
**>with 2 linear constraints requires a tetrahedron with
**>vertices (A,B,C,D) in 5 dimensions whose coordinates you
**>will have to find. Then take an "easy" tetrahedron with
**>vertices (A0,B0,C0,D0) and sample uniformly within this.
**>Then find a linear mapping from (A0,B0,C0,D0) to (A,B,C,D)
**>and apply this to the sampled points.
**>
**>This raises a general question: Does anyone know of
**>an R function to sample uniformly in the interior
**>of a general (k-r)-dimensional simplex embedded in
**>k dimensions, with (k+1) given vertices?
**><snip>
*

The method of "rejection":

- Generate numbers randomly in the hypercube.
- Test to see if the point falls within the prescribed area.
- Accept the point if it does.
- Repeat if it doesn't.

Efficiency depends upon the ratio of volumes involved.

Robert A. LaBudde, PhD, PAS, Dpl. ACAFS e-mail: ral_at_lcfltd.com

Least Cost Formulations, Ltd. URL: http://lcfltd.com/ 824 Timberlake Drive Tel: 757-467-0954 Virginia Beach, VA 23464-3239 Fax: 757-467-2947

"Vere scire est per causas scire"

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https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. Received on Thu 27 Mar 2008 - 08:05:24 GMT

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