Problem when creating matrix of values based on covariance matrix

-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org] On Behalf
Of Michael Dewey
Sent: Sunday, August 12, 2012 6:54 AM
To: Boel Brynedal; R-help Mailing List
Subject: Re: [R] Problem when creating matrix of values based on covariance matrix

At 15:17 11/08/2012, Boel Brynedal wrote:

Hi,

I want to simulate a data set with similar covariance structure as my
observed data, and have calculated a covariance matrix (dimensions
8368*8368). So far I've tried two approaches to simulating data:
rmvnorm from the mvtnorm package, and by using the Cholesky
decomposition
(http://www.cerebralmastication.com/2010/09/cholesk-post-on-correlated-random-

normal-generation/).

The problem is that the resulting covariance structure in my simulated
data is very different from the original supplied covariance vector.

It is, of course, not guaranteed to be the same as you are only
sampling from the distribution. In your example below you draw a
sample of size 1000 from a 8368 variable distribution so I suspect it
is almost sure to be different although I am surprised how different.
What happens if you increase the sample size?

Lets just look at some of the values:

cov8[1:4,1:4] # covariance of simulated data

            X1          X2         X3         X4
X1 34515296.00    99956.69   369538.1  1749086.6
X2    99956.69 34515296.00  2145289.9  -624961.1
X3   369538.08  2145289.93 34515296.0  -163716.5
X4  1749086.62  -624961.09  -163716.5 34515296.0

CEUcovar[1:4,1:4]

             [,1]         [,2]          [,3]         [,4]
[1,] 0.1873402987  0.001837229  0.0009009272  0.010324521
[2,] 0.0018372286  0.188665853  0.0124216535 -0.001755035
[3,] 0.0009009272  0.012421654  0.1867835412 -0.000142395
[4,] 0.0103245214 -0.001755035 -0.0001423950  0.192883488

So the distribution of the observed covariance is very narrow compared
to the simulated data.

None of the eigenvalues of the observed covariance matrix are
negative, and it appears to be a positive definite matrix. Here is
what I did to create the simulated data:

Chol <- chol(CEUcovar)
Z <- matrix(rnorm(20351 * 8368), 8368)
X <- t(Chol) %*% Z
sample8 <- data.frame(as.matrix(t(X)))

dim(sample8)

[1] 20351  8368
cov8=cov(sample8,method='spearman')

[earlier I've also tried sample8 <- rmvnorm(1000,
mean=rep(0,ncol(CEUcovar)), sigma=CEUcovar, method="eigen") with as
'bad' results, much larger covariance values in the simulated data ]

Any ideas of WHY the simulated data have such a different covariance?
Any experience with similar issues? Would be happy to supply the
covariance matrix if anyone wants to give it a try.
Any suggestions? Anything apparent that I left our or neglected?

Any advice would be highly appreciated.
Best,
Bo

Michael Dewey
info at aghmed.fsnet.co.uk
http://www.aghmed.fsnet.co.uk/home.html