From: Ted Harding <Ted.Harding_at_manchester.ac.uk>

Date: Wed, 26 Mar 2008 23:10:59 +0000 (GMT)

*>
*

> I don't think so. A whileago you wrote

*> "The numbers should be uniformly distributed" (but in the
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*> context of an example where you had 5 variable; now you
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*> are back to 4 variables). Let's take the 4-case first.
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*>
*

*> The two linear constraints confine the point (X1,X2,X3,X4)
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*> to a triangular region within the 4-dimensional unit cube.
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*> Say it has vertices A, B, C.
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*> You could then start by generating points uniformly distributed
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*> over a specific triangle in 2 dimentions, say the one with
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*> vertices at A0=(0,0), B0=(0,1), C0=(1,0). This is easy.
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*>
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*> Then you need to find a linear transformation which will
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*> map this triangle (A0,B0,C0) onto the triangle (A,B,C).
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*> Then the points you have sampled in (A0,B0,C0) will map
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*> into points which are uniformly distributed over the
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*> triangle (A,B,C).
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*>
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*> More generally, you will be seeking to generate points
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*> uniformly distributed over a simplex.
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*>
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*> For example, the case (your earlier post) of 5 points
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*> with 2 linear constraints requires a tetrahedron with
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*> vertices (A,B,C,D) in 5 dimensions whose coordinates you
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*> will have to find. Then take an "easy" tetrahedron with
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*> vertices (A0,B0,C0,D0) and sample uniformly within this.
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*> Then find a linear mapping from (A0,B0,C0,D0) to (A,B,C,D)
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*> and apply this to the sampled points.
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*>
*

*> This raises a general question: Does anyone know of
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*> an R function to sample uniformly in the interior
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*> of a general (k-r)-dimensional simplex embedded in
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*> k dimensions, with (k+1) given vertices?
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*>
*

*> Best wishes to all,
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*> Ted.
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*>
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*>
*

*>
*

*> --------------------------------------------------------------------
*

*> E-Mail: (Ted Harding) <Ted.Harding_at_manchester.ac.uk>
*

*> Fax-to-email: +44 (0)870 094 0861
*

*> Date: 26-Mar-08 Time: 22:06:38
*

> ------------------------------ XFMail ------------------------------

*>
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*> PLEASE do read the posting guide
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*> http://www.R-project.org/posting-guide.html
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*> and provide commented, minimal, self-contained, reproducible code.
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E-Mail: (Ted Harding) <Ted.Harding_at_manchester.ac.uk> Fax-to-email: +44 (0)870 094 0861

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 - 02:30:03 GMT

Date: Wed, 26 Mar 2008 23:10:59 +0000 (GMT)

OOPS! A mistake below. I should have written:

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-r+1) given vertices?

On 26-Mar-08 22:06:54, 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

>> On Wed, Mar 26, 2008 at 12:52 PM, Ted Harding >> <Ted.Harding_at_manchester.ac.uk> wrote: >>> On 26-Mar-08 20:13:50, Robert A LaBudde wrote: >>> > At 01:13 PM 3/26/2008, Ala' Jaouni wrote: >>> >>I am trying to generate a set of random numbers that fulfill >>> >>the following constraints: >>> >> >>> >>X1 + X2 + X3 + X4 = 1 >>> >> >>> >>aX1 + bX2 + cX3 + dX4 = n >>> >> >>> >>where a, b, c, d, and n are known. >>> >> >>> >>Any function to do this? >>> > >>> > 1. Generate random variates for X1, X2, based upon whatever >>> > unspecified distribution you wish. >>> > >>> > 2. Solve the two equations for X3 and X4. >>> >>> The trouble is that the original problem is not well >>> specified. Your suggestion, Robert, gives a solution >>> to one version of the problem -- enabling Ala' Jaouni >>> to say "I have generated 4 random numbers X1,X2,X3,X4 >>> such that X1 and X2 have specified distributions, >>> and X1,X2,X3,X4 satisfy the two equations ... ". >>> >>> However, suppose the real problem was: let X2,X2,X3,X4 >>> have independent distributions F1,F2,F3,F4. Now sample >>> X1,X2,X3,X4 conditional on the two equations (i.e. from >>> the coditional density). That is a different problem. >>> >>> As a slightly simpler example, suppose we have just X1,X2,X3 >>> and they are independently uniform on (0,1). Now sample >>> from the conditional distribution, conditional on >>> X1 + X2 + X3 = 1. >>> >>> The result is a random point uniformly distributed on the >>> planar triangle whose vertices are at (1,0,0),(0,1,0),(0,0,1). >>> >>> Then none of X1,X2,X3 is uniformly distributed (in fact >>> the marginal density of each is 2*(1-x)). >>> >>> However, your solution would work from either point of >>> view if the distributions were Normal. >>> >>> If X1,X2,X3,X4 were neither Normally nor uniformly >>> distributed, then finding or simulating the conditional >>> distribution would in general be difficult. >>> >>> Ala' Jaouni needs to tell us whether what he precisely >>> wants is as you stated the problem, Robert, or whether >>> he wants a conditional distribution for given distributions >>> if X1,X2,X3,X4, or whether he wants something else. >>> >>> Best wishes to all, >>> Ted. >>> >>> -------------------------------------------------------------------- >>> E-Mail: (Ted Harding) <Ted.Harding_at_manchester.ac.uk> >>> Fax-to-email: +44 (0)870 094 0861 >>> Date: 26-Mar-08 Time: 19:52:16 >>> ------------------------------ XFMail ------------------------------ >>> >> >> ______________________________________________ >> 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.

> ------------------------------ XFMail ------------------------------

E-Mail: (Ted Harding) <Ted.Harding_at_manchester.ac.uk> Fax-to-email: +44 (0)870 094 0861

Date: 26-Mar-08 Time: 23:04:56 ------------------------------ XFMail ------------------------------ ______________________________________________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 - 02:30:03 GMT

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