Re: [R] generate random numbers that sum up to 1
So, Alberto, you didn't see my post? If Y has d independent components that are gamma distributed with common rate and shapes A_1, A_2, ..., A_d, then X, given by the components of Y divided by their sum has distribution Dirichlet(A_1, A_2, ..., A_d). If you want Uniform on the d-simplex, then use A_1 = A_2 = ... = A_d = 1 (just as Duncan said) original message: Duncan Murdoch's definition is _the_ only one that I know. X is Uniform on A means E phi(X) = \int_A phi(x) dx / \int_A dx, so that the probability density is equal to 1/ \int_A dx everwhere on the set A. By the way, another way to simulate X ~ Dirichlet(A1, A2, ..., Ad) is to generate d independent gamma variables having equal rate parameter (doesn't matter, so why not 1) and shape parameters A1, A2, ..., Ad Then the vector of components divided by their sum is the desired Dirichlet: n - 10 d - 3 # for three numbers that add to one ( the unit simplex in R^3) A - rep(1, 3) # for uniform X - matrix(0, n, d) for (k in 1:3) X[,k] - rgamma(n, shape=A[k], rate=1) S - X %*% rep(1, d) Y - X/S Present example will simulate n independant 3 vectors, each having non-negative components summing to 1, and having a distribution assigning equal mass to every possible value. Changing d and the components of A will provide an arbitrary Dirichlet on the unit simplex in R^d Grant Izmirlian NCI Duncan Murdoch wrote Another definition of uniform is to have equal density for all possible vectors; the Dirichlet distribution with parameters (1,1,1) would give you that. __ R-help@stat.math.ethz.ch 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.
Re: [R] generate random numbers that sum up to 1
Grant Izmirlian wrote: So, Alberto, you didn't see my post? I think I didn't - but you are demanding too much from my memory; I can hardly remember what I saw yesterday! If Y has d independent components that are gamma distributed with common rate and shapes A_1, A_2, ..., A_d, then X, given by the components of Y divided by their sum has distribution Dirichlet(A_1, A_2, ..., A_d). If you want Uniform on the d-simplex, then use A_1 = A_2 = ... = A_d = 1 (just as Duncan said) The problem is that I wasn't aware that this was the Dirichlet distribution [R does not have this distribution, AFAIK, but I should have consulted the Borg of All Wisdom, the Wikipedia]. Alberto Monteiro __ R-help@stat.math.ethz.ch 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.
[R] generate random numbers that sum up to 1
I found another method to generate this, in the Borg of all Wisdom: http://en.wikipedia.org/wiki/Simplex Alberto Monteiro --- Section Random Sampling: The second method [[the first method is based on the Dirichlet distribution -- avfm]] to generate a random point on the unit simplex is based on the order statistics of the uniform distribution on the unit interval, and was popularized by Horst Kraemer. The algorithm is as follows: Set p0 = 0 and pK=1. Generate K-1 uniform random draws pi from the open interval (0,1). Sort into ascending order the K+1 points p0, ..., pK. The K coordinates t1, ..., tK of the final point on the unit simplex are given by ti=pi-pi-1. It has been pointed out by Smith and Tromble that the second method is technically only valid if none of the differences pi-p(i-1) are equal to zero. In practice, it is sufficient to merely re-run the algorithm to generate a new set of points if this happens. __ R-help@stat.math.ethz.ch 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.
Re: [R] generate random numbers that sum up to 1
Thanks for the answers. I am sorry if the question is too simple to make everyone thought it is a homework, but it is not. I am setting up a simulation with expected utility theory which use EU = sum(Pi*Ui) form and I have to randomly generate these Pis, I do not really know which distribution these Pis has to follow but just know they have to be equally random hence the question. I thought Dirichlet(1,1, ..., 1) could do the trick for me. Sun - Original Message - From: Duncan Murdoch [EMAIL PROTECTED] To: Alberto Monteiro [EMAIL PROTECTED] Cc: r-help@stat.math.ethz.ch Sent: Wednesday, October 11, 2006 11:39 PM Subject: Re: [R] generate random numbers that sum up to 1 On 10/11/2006 5:04 PM, Alberto Monteiro wrote: I don't have the previous messages, but it seems to me that the solutions didn't quite get the requirements of the problem. For example, it's obvious that for n = 2, the random variables X1 and X2 should be X1 - runif(1) and X2 - 1 - X1; while the solution X - runif(2); X1 - X[1] / sum(X); X2 - X[2] / sum(X) will give different distributions, as shown in this test case: N - 1000 M - matrix(runif(2 * N), 2, N) X - M[1,] / (M[1,] + M[2,]) hist(X) It's obvious that for a generic n-th dimensional set of uniform variables X1, ... X_n subject to the restriction X1 + ... + X_n = 1, the solution is to take a uniform distribution in the (n-1)-th dimensional hyperpyramid generated by the above relation and the restrictions that each X_i = 0. For example, for n = 3, we should sample from the equilateral triangle with vertices c(1,0,0), c(0,1,0) and c(0,0,1). For n = 4, we should sample from the pyramid whose vertices are c(1,0,0,0), c(0,1,0,0), c(0,0,1,0) and c(0,0,0,1). I don't know if there is a simple formula to do this sampling. That's exactly what the Dirichlet(1,1, ..., 1) distribution does. Duncan Murdoch __ R-help@stat.math.ethz.ch 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. __ R-help@stat.math.ethz.ch 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.
Re: [R] generate random numbers that sum up to 1
Duncan Murdoch's definition is _the_ only one that I know. X is Uniform on A means E phi(X) = \int_A phi(x) dx / \int_A dx, so that the probability density is equal to 1/ \int_A dx everwhere on the set A. By the way, another way to simulate X ~ Dirichlet(A1, A2, ..., Ad) is to generate d independent gamma variables having equal rate parameter (doesn't matter, so why not 1) and shape parameters A1, A2, ..., Ad Then the vector of components divided by their sum is the desired Dirichlet: n - 10 d - 3 # for three numbers that add to one ( the unit simplex in R^3) A - rep(1, 3) # for uniform X - matrix(0, n, d) for (k in 1:3) X[,k] - rgamma(n, shape=A[k], rate=1) S - X %*% rep(1, d) Y - X/S Present example will simulate n independant 3 vectors, each having non-negative components summing to 1, and having a distribution assigning equal mass to every possible value. Changing d and the components of A will provide an arbitrary Dirichlet on the unit simplex in R^d Grant Izmirlian NCI Duncan Murdoch wrote Another definition of uniform is to have equal density for all possible vectors; the Dirichlet distribution with parameters (1,1,1) would give you that. __ R-help@stat.math.ethz.ch 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.
Re: [R] generate random numbers that sum up to 1
I don't have the previous messages, but it seems to me that the solutions didn't quite get the requirements of the problem. For example, it's obvious that for n = 2, the random variables X1 and X2 should be X1 - runif(1) and X2 - 1 - X1; while the solution X - runif(2); X1 - X[1] / sum(X); X2 - X[2] / sum(X) will give different distributions, as shown in this test case: N - 1000 M - matrix(runif(2 * N), 2, N) X - M[1,] / (M[1,] + M[2,]) hist(X) It's obvious that for a generic n-th dimensional set of uniform variables X1, ... X_n subject to the restriction X1 + ... + X_n = 1, the solution is to take a uniform distribution in the (n-1)-th dimensional hyperpyramid generated by the above relation and the restrictions that each X_i = 0. For example, for n = 3, we should sample from the equilateral triangle with vertices c(1,0,0), c(0,1,0) and c(0,0,1). For n = 4, we should sample from the pyramid whose vertices are c(1,0,0,0), c(0,1,0,0), c(0,0,1,0) and c(0,0,0,1). I don't know if there is a simple formula to do this sampling. Alberto Monteiro __ R-help@stat.math.ethz.ch 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.
Re: [R] generate random numbers that sum up to 1
On 10/11/2006 5:04 PM, Alberto Monteiro wrote: I don't have the previous messages, but it seems to me that the solutions didn't quite get the requirements of the problem. For example, it's obvious that for n = 2, the random variables X1 and X2 should be X1 - runif(1) and X2 - 1 - X1; while the solution X - runif(2); X1 - X[1] / sum(X); X2 - X[2] / sum(X) will give different distributions, as shown in this test case: N - 1000 M - matrix(runif(2 * N), 2, N) X - M[1,] / (M[1,] + M[2,]) hist(X) It's obvious that for a generic n-th dimensional set of uniform variables X1, ... X_n subject to the restriction X1 + ... + X_n = 1, the solution is to take a uniform distribution in the (n-1)-th dimensional hyperpyramid generated by the above relation and the restrictions that each X_i = 0. For example, for n = 3, we should sample from the equilateral triangle with vertices c(1,0,0), c(0,1,0) and c(0,0,1). For n = 4, we should sample from the pyramid whose vertices are c(1,0,0,0), c(0,1,0,0), c(0,0,1,0) and c(0,0,0,1). I don't know if there is a simple formula to do this sampling. That's exactly what the Dirichlet(1,1, ..., 1) distribution does. Duncan Murdoch __ R-help@stat.math.ethz.ch 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.
[R] generate random numbers that sum up to 1
I am trying to generate a vector of random numbers with the constraint that
they have to sum up to one with uniform distribution.
eg. {0.1,0.7,0.2 }
any function to do this? Thanks.
__
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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.
Re: [R] generate random numbers that sum up to 1
As I have previously asked, in response to a similar question: Is this a homework problem? cheers, Rolf Turner [EMAIL PROTECTED] __ R-help@stat.math.ethz.ch 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.
Re: [R] generate random numbers that sum up to 1
Rolf Turner [EMAIL PROTECTED] writes: As I have previously asked, in response to a similar question: Is this a homework problem? If so, he has a pretty nasty teacher... -- O__ Peter Dalgaard Øster Farimagsgade 5, Entr.B c/ /'_ --- Dept. of Biostatistics PO Box 2099, 1014 Cph. K (*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918 ~~ - ([EMAIL PROTECTED]) FAX: (+45) 35327907 __ R-help@stat.math.ethz.ch 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.
Re: [R] generate random numbers that sum up to 1
sun [EMAIL PROTECTED] writes:
I am trying to generate a vector of random numbers with the constraint that
they have to sum up to one with uniform distribution.
eg. {0.1,0.7,0.2 }
any function to do this? Thanks.
Depending on what you mean by uniform, this may be a solution
x-runif(3)
x/sum(x)
[1] 0.1130642 0.4098608 0.4770750
What you can't have is the individual components uniform and
identically distributed because the sum of the means would be 1.5 and
equal to the mean of the sum which is 1...
--
O__ Peter Dalgaard Øster Farimagsgade 5, Entr.B
c/ /'_ --- Dept. of Biostatistics PO Box 2099, 1014 Cph. K
(*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918
~~ - ([EMAIL PROTECTED]) FAX: (+45) 35327907
__
R-help@stat.math.ethz.ch mailing list
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PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
Re: [R] generate random numbers that sum up to 1
The trick is in defining random in the face of the three apparently
incompatible constraints that the numbers be random, that they sum to 1,
and that they be uniformly distributed. Tell us more. Perhaps extend
Peter D's solution by first deciding (at random) how many components
each solution will have.
Ben Fairbank
-Original Message-
From: [EMAIL PROTECTED]
[mailto:[EMAIL PROTECTED] On Behalf Of sun
Sent: Tuesday, October 10, 2006 8:28 AM
To: r-help@stat.math.ethz.ch
Subject: [R] generate random numbers that sum up to 1
I am trying to generate a vector of random numbers with the constraint
that they have to sum up to one with uniform distribution.
eg. {0.1,0.7,0.2 }
any function to do this? Thanks.
__
R-help@stat.math.ethz.ch 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.
__
R-help@stat.math.ethz.ch 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.
Re: [R] generate random numbers that sum up to 1
On 10/10/2006 9:53 AM, Peter Dalgaard wrote:
sun [EMAIL PROTECTED] writes:
I am trying to generate a vector of random numbers with the constraint that
they have to sum up to one with uniform distribution.
eg. {0.1,0.7,0.2 }
any function to do this? Thanks.
Depending on what you mean by uniform, this may be a solution
x-runif(3)
x/sum(x)
[1] 0.1130642 0.4098608 0.4770750
What you can't have is the individual components uniform and
identically distributed because the sum of the means would be 1.5 and
equal to the mean of the sum which is 1...
Another definition of uniform is to have equal density for all possible
vectors; the Dirichlet distribution with parameters (1,1,1) would give
you that. RSiteSearch finds at least a couple of packages (VGAM,
MCMCpack) that provide simulations from the Dirichlet.
Duncan Murdoch
__
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and provide commented, minimal, self-contained, reproducible code.
