[Rd] matrix not positive definite (while it should be)

[Petr Savicky]

On Thu, May 05, 2011 at 02:31:59PM -0400, Arthur Charpentier wrote:
> I do have some trouble with matrices. I want to build up a covariance matrix
> with a hierarchical structure). For instance, in dimension n=10, I have two
> subgroups (called REGION).
> 
> NR=2; n=10
> CORRELATION=matrix(c(0.4,-0.25,
>                      -0.25,0.3),NR,NR)
> REGION=sample(1:NR,size=n,replace=TRUE)
> R1=REGION%*%t(rep(1,n))
> R2=rep(1,n)%*%t(REGION)
> SIGMA=matrix(NA,n,n)
> 
> for(i in 1:NR){
> for(j in 1:NR){
> SIGMA[(R1==i)&(R2==j)]=CORRELATION[i,j]
> }}
> 
> If I run quickly some simulations, I build up the following matrix
> 
> > CORRELATION
>       [,1]  [,2]
> [1,]  0.40 -0.25
> [2,] -0.25  0.30
> > REGION
>  [1] 2 2 1 1 2 1 2 1 1 2
> > SIGMA
>        [,1]  [,2]  [,3]  [,4]  [,5]  [,6]  [,7]  [,8]  [,9] [,10]
>  [1,]  0.30  0.30 -0.25 -0.25  0.30 -0.25  0.30 -0.25 -0.25  0.30
>  [2,]  0.30  0.30 -0.25 -0.25  0.30 -0.25  0.30 -0.25 -0.25  0.30
>  [3,] -0.25 -0.25  0.40  0.40 -0.25  0.40 -0.25  0.40  0.40 -0.25
>  [4,] -0.25 -0.25  0.40  0.40 -0.25  0.40 -0.25  0.40  0.40 -0.25
>  [5,]  0.30  0.30 -0.25 -0.25  0.30 -0.25  0.30 -0.25 -0.25  0.30
>  [6,] -0.25 -0.25  0.40  0.40 -0.25  0.40 -0.25  0.40  0.40 -0.25
>  [7,]  0.30  0.30 -0.25 -0.25  0.30 -0.25  0.30 -0.25 -0.25  0.30
>  [8,] -0.25 -0.25  0.40  0.40 -0.25  0.40 -0.25  0.40  0.40 -0.25
>  [9,] -0.25 -0.25  0.40  0.40 -0.25  0.40 -0.25  0.40  0.40 -0.25
> [10,]  0.30  0.30 -0.25 -0.25  0.30 -0.25  0.30 -0.25 -0.25  0.30

Hi.

If X is a random vector from the 2 dimensional normal distribution
with the covariance matrix

        [,1]  [,2]
  [1,]  0.40 -0.25
  [2,] -0.25  0.30

then the vector X[REGION], which consists of replicated components
of X, has the expanded covariance matrix n times n, which you ask
for. Since the mean and the covariance matrix determine the distribution
uniquely, this is also a description of the required distribution.

The distribution is concentrated in a 2 dimensional subspace, since
the covariance matrix has rank 2.

Hope this helps.

Petr Savicky.