thanks for the tip
actually, I know that the covariance matrix has rank 2, but it should still
be definite positive (not strictly positive, but positive)
my problem is that Cholesky needs a positive matrix...
my concern is that I have
> min(eigen(SIGMA)$values)
[1] -2.109071e-17
while theoretically it should be 0 (if it was [1] 2.109071e-17 I guess it
would work, the problem is the minus sign)
Arthur
2011/5/6 Petr Savicky <savi...@cs.cas.cz>
> 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.
>
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