Re: [R] Generate multivariate normal data with a random correlation matrix
Thanks for the response, Rex. This is an interesting approach. The
Choleski decomposition approach that John suggested seems to be an
obvious and direct approach to this problem. Your approach is less
obvious to me but may be equal or superior to the Choleski
decomposition.
Are all possible correlation matrices of size k equally likely using
your approach? It would seem so based on your description. If so, it
is a way cool solution.
Rick
On Thu, Feb 10, 2011 at 12:18 PM, rex.dw...@syngenta.com wrote:
If you want a random correlation matrix, why not just generate random data
and accept the correlation matrix that you get? The standard normal
distribution in k dimensions is (hyper)spherically symmetric. If you
generate k standard normal N(0,1) variates, you have a point in k-space with
direction uniformly distributed on the (k-1)sphere and Gaussian magnitude.
If you generate k such, you have a random linear transformation with all
sorts of desirable symmetries. So, if you generate a kxk matrix of standard
normal variates, and another nxk standard normal variates, and multiply the
two matrices to get n points in k space, that seems to be a pretty good
definition of random correlation to me. I'm sure you can decompose the kxk
matrix to get the theoretical distribution, maybe by multiplying it by its
transpose and doing an SVD; I'd have to think about that part.
... unless you have a particular distribution of correlation matrices in mind
to begin with, which doesn't seem to be the case.
-Original Message-
From: r-help-boun...@r-project.org [mailto:r-help-boun...@r-project.org] On
Behalf Of Szumiloski, John
Sent: Wednesday, February 09, 2011 11:30 AM
To: r-help@r-project.org
Cc: Rick DeShon
Subject: Re: [R] Generate multivariate normal data with a random correlation
matrix
The knee jerk thought I had was to express the correlation matrix as a
generic Choleski decomposition, then randomly populate the triangular
decomposed matrix. When you remultiply, you can simply rescale to 1s on the
diagonals. Then rmnorm as usual.
In R, see ?chol
If you want to get fancy, you could look at the random distribution you would
use for the triangular matrix and play with that, including different
distributions for different elements, elements' distributions being
conditional on values of previously randomized elements, etc.
John
-Original Message-
From: r-help-boun...@r-project.org [mailto:r-help-boun...@r-project.org] On
Behalf Of Rick DeShon
Sent: Wednesday, 09 February, 2011 11:06 AM
To: r-h...@stat.math.ethz.ch
Subject: [R] Generate multivariate normal data with a random correlation
matrix
Hi All.
I'd like to generate a sample of n observations from a k dimensional
multivariate normal distribution with a random correlation matrix.
My solution:
The lower (or upper) triangle of the correlation matrix has
n.tri=(d/2)(d+1)-d entries.
Take a uniform sample of n.tri possible correlations (runi(n.tr,-.99,.99)
Populate a triangle of the matrix with the sampled correlations Mirror the
triangle to populate the other triangle forming a symmetric matrix, cormat
Sample n observations from a multivariate normal distribution with mean
vector=0 and varcov=cormat
Problem:
This approach violates the triangle inequality property of correlation
matrices. So, the matrix I've constructed is certainly a valid matrix but it
is not a valid correlation matrix and it blows up when you submit it to a
random number generator such as rmnorm. With a small matrix you sometimes
get lucky and generate a valid correlation matrix but as you increase d the
probability of obtaining a valid correlation matrix drops off quickly.
So, any ideas on how to construct a correlation matrix with random entries
that cover the range (or most of the range) or the correlation [-1,1]?
Here's the code I've used that won't work.
library(mnormt)
n - 1000
d - 50
n.tri - ((d*(d+1))/2)-d
r - runif(n.tri, min=-.5, max=.5)
cormat - diag(c)
count1=1
for (i in 1:c){
for (j in 1:c){
if (ij) {
cormat[i,j]=r[count1]
cormat[j,i]=cormat[i,j]
count1=count1+1
}
}
}
eigen(cormat) # if negative eigenvalue, then the matrix violates the
triangle inequality
x - rmnorm(n, rep(0, c), cormat) # Sample the data
Thanks in advance,
Rick DeShon
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Re: [R] Generate multivariate normal data with a random correlation matrix
If you want a random correlation matrix, why not just generate random data and
accept the correlation matrix that you get? The standard normal distribution
in k dimensions is (hyper)spherically symmetric. If you generate k standard
normal N(0,1) variates, you have a point in k-space with direction uniformly
distributed on the (k-1)sphere and Gaussian magnitude. If you generate k such,
you have a random linear transformation with all sorts of desirable symmetries.
So, if you generate a kxk matrix of standard normal variates, and another nxk
standard normal variates, and multiply the two matrices to get n points in k
space, that seems to be a pretty good definition of random correlation to me.
I'm sure you can decompose the kxk matrix to get the theoretical distribution,
maybe by multiplying it by its transpose and doing an SVD; I'd have to think
about that part.
... unless you have a particular distribution of correlation matrices in mind
to begin with, which doesn't seem to be the case.
-Original Message-
From: r-help-boun...@r-project.org [mailto:r-help-boun...@r-project.org] On
Behalf Of Szumiloski, John
Sent: Wednesday, February 09, 2011 11:30 AM
To: r-help@r-project.org
Cc: Rick DeShon
Subject: Re: [R] Generate multivariate normal data with a random correlation
matrix
The knee jerk thought I had was to express the correlation matrix as a generic
Choleski decomposition, then randomly populate the triangular decomposed
matrix. When you remultiply, you can simply rescale to 1s on the diagonals.
Then rmnorm as usual.
In R, see ?chol
If you want to get fancy, you could look at the random distribution you would
use for the triangular matrix and play with that, including different
distributions for different elements, elements' distributions being conditional
on values of previously randomized elements, etc.
John
-Original Message-
From: r-help-boun...@r-project.org [mailto:r-help-boun...@r-project.org] On
Behalf Of Rick DeShon
Sent: Wednesday, 09 February, 2011 11:06 AM
To: r-h...@stat.math.ethz.ch
Subject: [R] Generate multivariate normal data with a random correlation matrix
Hi All.
I'd like to generate a sample of n observations from a k dimensional
multivariate normal distribution with a random correlation matrix.
My solution:
The lower (or upper) triangle of the correlation matrix has n.tri=(d/2)(d+1)-d
entries.
Take a uniform sample of n.tri possible correlations (runi(n.tr,-.99,.99)
Populate a triangle of the matrix with the sampled correlations Mirror the
triangle to populate the other triangle forming a symmetric matrix, cormat
Sample n observations from a multivariate normal distribution with mean
vector=0 and varcov=cormat
Problem:
This approach violates the triangle inequality property of correlation
matrices. So, the matrix I've constructed is certainly a valid matrix but it
is not a valid correlation matrix and it blows up when you submit it to a
random number generator such as rmnorm. With a small matrix you sometimes get
lucky and generate a valid correlation matrix but as you increase d the
probability of obtaining a valid correlation matrix drops off quickly.
So, any ideas on how to construct a correlation matrix with random entries that
cover the range (or most of the range) or the correlation [-1,1]?
Here's the code I've used that won't work.
library(mnormt)
n - 1000
d - 50
n.tri - ((d*(d+1))/2)-d
r - runif(n.tri, min=-.5, max=.5)
cormat - diag(c)
count1=1
for (i in 1:c){
for (j in 1:c){
if (ij) {
cormat[i,j]=r[count1]
cormat[j,i]=cormat[i,j]
count1=count1+1
}
}
}
eigen(cormat) # if negative eigenvalue, then the matrix violates the
triangle inequality
x - rmnorm(n, rep(0, c), cormat) # Sample the data
Thanks in advance,
Rick DeShon
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Re: [R] Generate multivariate normal data with a random correlation matrix
Hi Rick,
you can generate an arbitrary matrix X and calculate X'X, which is at
least positive semi-definite (and definite, when X has full rank)
X-matrix(rnorm(16),4)
covmat-t(X)%*%X
#check
!any(eigen(covmat)$value0)
#corresponding correlation matrix
cov2cor(covmat)
Random multivariate can be generated with 'mvrnorm' from the MASS-package.
hth.
Am 09.02.2011 17:19, schrieb Rick DeShon:
Hi All.
I'd like to generate a sample of n observations from a k dimensional
multivariate normal distribution with a random correlation matrix.
My solution:
1) The lower (or upper) triangle of the correlation matrix has
n.tri=(d/2)(d+1)-d entries.
2) Take a uniform sample of n.tri possible correlations (runi(n.tr,-.99,.99)
3) Populate a triangle of the matrix with the sampled correlations
4) Mirror the triangle to populate the other triangle forming a
symmetric matrix, cormat
5) Sample n observations from a multivariate normal distribution with
mean vector=0 and varcov=cormat
Problem:
This approach violates the triangle inequality property of correlation
matrices. So, the matrix I've constructed is certainly a valid matrix
but it is not a valid correlation matrix and it blows up when you
submit it to a random number generator such as rmnorm. With a small
matrix you sometimes get lucky and generate a valid correlation matrix
but as you increase d the probability of obtaining a valid correlation
matrix drops off quickly.
So, any ideas on how to construct a correlation matrix with random
entries that cover the range (or most of the range) or the correlation
[-1,1]?
Here's the code I've used that won't work.
library(mnormt)
n - 1000
d - 50
n.tri - ((d*(d+1))/2)-d
r - runif(n.tri, min=-.5, max=.5)
cormat - diag(c)
count1=1
for (i in 1:c){
for (j in 1:c){
if (ij) {
cormat[i,j]=r[count1]
cormat[j,i]=cormat[i,j]
count1=count1+1
}
}
}
eigen(cormat) # if negative eigenvalue, then the matrix violates
the triangle inequality
x - rmnorm(n, rep(0, c), cormat) # Sample the data
Thanks in advance,
Rick DeShon
__
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https://stat.ethz.ch/mailman/listinfo/r-help
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and provide commented, minimal, self-contained, reproducible code.
--
Eik Vettorazzi
Institut für Medizinische Biometrie und Epidemiologie
Universitätsklinikum Hamburg-Eppendorf
Martinistr. 52
20246 Hamburg
T ++49/40/7410-58243
F ++49/40/7410-57790
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Re: [R] Generate multivariate normal data with a random correlation matrix
The knee jerk thought I had was to express the correlation matrix as a generic
Choleski decomposition, then randomly populate the triangular decomposed
matrix. When you remultiply, you can simply rescale to 1s on the diagonals.
Then rmnorm as usual.
In R, see ?chol
If you want to get fancy, you could look at the random distribution you would
use for the triangular matrix and play with that, including different
distributions for different elements, elements' distributions being conditional
on values of previously randomized elements, etc.
John
-Original Message-
From: r-help-boun...@r-project.org [mailto:r-help-boun...@r-project.org] On
Behalf Of Rick DeShon
Sent: Wednesday, 09 February, 2011 11:06 AM
To: r-h...@stat.math.ethz.ch
Subject: [R] Generate multivariate normal data with a random correlation matrix
Hi All.
I'd like to generate a sample of n observations from a k dimensional
multivariate normal distribution with a random correlation matrix.
My solution:
The lower (or upper) triangle of the correlation matrix has n.tri=(d/2)(d+1)-d
entries.
Take a uniform sample of n.tri possible correlations (runi(n.tr,-.99,.99)
Populate a triangle of the matrix with the sampled correlations Mirror the
triangle to populate the other triangle forming a symmetric matrix, cormat
Sample n observations from a multivariate normal distribution with mean
vector=0 and varcov=cormat
Problem:
This approach violates the triangle inequality property of correlation
matrices. So, the matrix I've constructed is certainly a valid matrix but it
is not a valid correlation matrix and it blows up when you submit it to a
random number generator such as rmnorm. With a small matrix you sometimes get
lucky and generate a valid correlation matrix but as you increase d the
probability of obtaining a valid correlation matrix drops off quickly.
So, any ideas on how to construct a correlation matrix with random entries that
cover the range (or most of the range) or the correlation [-1,1]?
Here's the code I've used that won't work.
library(mnormt)
n - 1000
d - 50
n.tri - ((d*(d+1))/2)-d
r - runif(n.tri, min=-.5, max=.5)
cormat - diag(c)
count1=1
for (i in 1:c){
for (j in 1:c){
if (ij) {
cormat[i,j]=r[count1]
cormat[j,i]=cormat[i,j]
count1=count1+1
}
}
}
eigen(cormat) # if negative eigenvalue, then the matrix violates the
triangle inequality
x - rmnorm(n, rep(0, c), cormat) # Sample the data
Thanks in advance,
Rick DeShon
__
R-help@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.
Notice: This e-mail message, together with any attachme...{{dropped:11}}
__
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