To make David's approach a little more concrete:
You can always have correlations all equal to 1 --
the variables are all the same, except for the names
you've given them. You can have two variables
with correlation -1, but you can't get a third variable
that has -1 correlation to both of the first two.
Patrick Burns
[EMAIL PROTECTED]
+44 (0)20 8525 0696
http://www.burns-stat.com
(home of S Poetry and "A Guide for the Unwilling S User")
[EMAIL PROTECTED] wrote:
Well, if you think about the geometry, all correlations equal usually
won't work. Think of the SDs as the sides of a simplex and the
correlations as the cosines of the angles between the sides (pick one
variable as the 'origin'.) Only certain values will give a valid
covariance or correlation matrix.
HTH,
David L. Reiner, PhD
Head Quant
Rho Trading Securities, LLC
-----Original Message-----
From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED]
On Behalf Of Mizanur Khondoker
Sent: Thursday, June 26, 2008 11:11 AM
To: r-help@r-project.org
Subject: [SPAM] - [R] constructing arbitrary (positive definite)
covariance matrix - Found word(s) list error in the Text body
Dear list,
I am trying to use the 'mvrnorm' function from the MASS package for
simulating multivariate Gaussian data with given covariance matrix.
The diagonal elements of my covariance matrix should be the same,
i.e., all variables have the same marginal variance. Also all
correlations between all pair of variables should be identical, but
could be any value in [-1,1]. The problem I am having is that the
matrix I create is not always positive definite (and hence mvrnorm
fails).
Is there any simple way of constructing covariance matrix of the above
structure (equal variance, same pairwise correlation from [-1, 1])
that will always be positive definite?
I have noticed that covraince matrices created using the following COV
function are positive definite for -0.5 < r <1. However, for r <
-0.5, the matrix is not positive definite.
Does anyone have any idea why this is the case? For my simualtion, I
need to generate multivariate data for the whole range of r, [-1, 1]
for a give value of sd.
Any help/ suggestion would be greatly appreciated.
Examples
########
COV<-function (p = 3, sd = 1, r= 0.5){
cov <- diag(sd^2, ncol=p, nrow=p)
for (i in 1:p) {
for (j in 1:p) {
if (i != j) {
cov[i, j] <- r * sd*sd
}
}
}
cov
}
library(MASS)
### Simualte multivarite gaussin data (works OK)
Sigma<-COV(p = 3, sd = 2, r= 0.5)
mu<-1:3
mvrnorm(5, mu=mu, Sigma=Sigma)
[,1] [,2] [,3]
[1,] 1.2979984 1.843248 4.460891
[2,] 2.1061054 1.457201 3.774833
[3,] 2.1578538 2.761939 4.589977
[4,] 0.8775056 4.240710 2.203712
[5,] 0.2698180 2.075759 2.869573
### Simualte multivarite gaussin data ( gives Error)
Sigma<-COV(p = 3, sd = 2, r= -0.6)
mu<-1:3
mvrnorm(5, mu=mu, Sigma=Sigma)
Error in mvrnorm(5, mu = mu, Sigma = Sigma) :
'Sigma' is not positive definite
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