R help - Jun 2004 - Simulating from a Multivariate Normal Distribution Using a Correlation Matrix

Hello,
I would like to simulate randomly from a multivariate normal distribution using
a correlation
matrix, rho.  I do not have sigma.  I have searched the help archive and the R
documentation as
well as doing a standard google search.  What I have seen is that one can either
use rmvnorm in
the package: mvtnorm or mvrnorm in the package: MASS.  I believe I read
somewhere that the latter
was more robust.  I have seen conflicting (or at least seemingly conflicting to
me, a relative
statistics novice), views on whether one can use the correlation matrix with
these commands
instead of the covariance matrix.  I thought that if the commands standardized
the covariance
matrix, then it would not matter, but I end up with larger values when I test
the covariance
matrix versus when I test rho.  So, my question is, if one does not know sigma,
can they use rho?
 And, if so, which command (or is there another) is better to use?  I gather
that both use eigen
decomposition?  Thank you so much  in advance for your help.
Best,
Matt

Hi Matt,

the correlation marix can be regarded as a covaraince matrix with unit
variance, so I think you could use it instead of your Sigma matrix,
e.g.,

rho <- cbind(c(1, .3, .1), c(.3, 1, .2), c(.1, .2, 1))
library(MASS)
x <- mvrnorm(5000, mu=1:3, Sigma=rho)
var(x)
          [,1]      [,2]      [,3]
[1,] 1.0234457 0.3101399 0.1146355
[2,] 0.3101399 0.9923358 0.2076328
[3,] 0.1146355 0.2076328 1.0115746

cor(x)
          [,1]      [,2]      [,3]
[1,] 1.0000000 0.3077485 0.1126647
[2,] 0.3077485 1.0000000 0.2072372
[3,] 0.1126647 0.2072372 1.0000000


Moreover, if you want different variance you could use

rho <- cbind(c(1, .3, .1), c(.3, 1, .2), c(.1, .2, 1))
vars <- 5
library(MASS)
x <- mvrnorm(5000, mu=1:3, Sigma=vars*rho)
var(x)
          [,1]      [,2]      [,3]
[1,] 4.9036297 1.4153234 0.4396337
[2,] 1.4153234 4.8801609 0.8947988
[3,] 0.4396337 0.8947988 4.8844880

cor(x)
           [,1]      [,2]       [,3]
[1,] 1.00000000 0.2893209 0.08983026
[2,] 0.28932087 1.0000000 0.18327313
[3,] 0.08983026 0.1832731 1.00000000


I hope this helps.

Best,
Dimitris

----
Dimitris Rizopoulos
Doctoral Student
Biostatistical Centre
School of Public Health
Catholic University of Leuven

Address: Kapucijnenvoer 35, Leuven, Belgium
Tel: +32/16/396887
Fax: +32/16/337015
Web: http://www.med.kuleuven.ac.be/biostat/
     http://www.student.kuleuven.ac.be/~m0390867/dimitris.htm




----- Original Message ----- 
From: "Matthew David Sylvester" <msylvest at
uclink.berkeley.edu>
To: <r-help at stat.math.ethz.ch>
Sent: Friday, June 25, 2004 9:48 AM
Subject: [R] Simulating from a Multivariate Normal Distribution Using
a Correlation Matrix

> Hello,
> I would like to simulate randomly from a multivariate normal
distribution using a correlation
> matrix, rho.  I do not have sigma.  I have searched the help archive
and the R documentation as
> well as doing a standard google search.  What I have seen is that
one can either use rmvnorm in
> the package: mvtnorm or mvrnorm in the package: MASS.  I believe I
read somewhere that the latter
> was more robust.  I have seen conflicting (or at least seemingly
conflicting to me, a relative
> statistics novice), views on whether one can use the correlation
matrix with these commands
> instead of the covariance matrix.  I thought that if the commands
standardized the covariance
> matrix, then it would not matter, but I end up with larger values
when I test the covariance
> matrix versus when I test rho.  So, my question is, if one does not
know sigma, can they use rho?
>  And, if so, which command (or is there another) is better to use?
I gather that both use eigen
> decomposition?  Thank you so much  in advance for your help.
> Best,
> Matt
>
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://www.stat.math.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide!
http://www.R-project.org/posting-guide.html

Short answer: to know a MVN distribution you need to know the mean vector 
and the covariance matrix.  If you don't know a distribution you cannot 
simulate from it.

So you need to know the marginal variances (the diagonal of the covariance 
matrix).  If you have those, you can form the covariance matrix and use 
rmvnorm or mvrnorm.  If you are willing to assume they are one, you have 
the covariance (= correlation matrix).  If you don't know the marginal 
variances the problem is incompletely specified.

On Fri, 25 Jun 2004, Matthew David Sylvester wrote:
> Hello,
> I would like to simulate randomly from a multivariate normal distribution
using a correlation
> matrix, rho.  I do not have sigma.  I have searched the help archive and
the R documentation as
> well as doing a standard google search.  What I have seen is that one can
either use rmvnorm in
> the package: mvtnorm or mvrnorm in the package: MASS.  I believe I read
somewhere that the latter
> was more robust.  I have seen conflicting (or at least seemingly
conflicting to me, a relative
> statistics novice), views on whether one can use the correlation matrix
with these commands
> instead of the covariance matrix.  I thought that if the commands
standardized the covariance
> matrix, then it would not matter, but I end up with larger values when I
test the covariance
> matrix versus when I test rho.  So, my question is, if one does not know
sigma, can they use rho?
>  And, if so, which command (or is there another) is better to use?  I
gather that both use eigen
> decomposition?  Thank you so much  in advance for your help.
> Best,
> Matt
> 
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://www.stat.math.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide!
http://www.R-project.org/posting-guide.html
> 
> 
-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595