R help - Dec 2004 - Help : generating correlation matrix with a particular structure

Hi,

I would like to generate a correlation matrix with a
particular structure. For example, a 3n x 3n matrix :
A_(nxn)   aI_(nxn)  bI_(nxn)
aI_(nxn)  A_(nxn)   cI_(nxn)
aI_(nxn)  cI_(nxn)  A_(nxn)

where
- A_(nxn) is a *specified* symmetric, positive
definite nxn matrix.
- I_(nxn) is an identity matrix of order n
- a, b, c are (any) real numbers

Many attempts have been unsuccessful because a
resulting matrix with any a, b, c may not be a
positive definite one, and hence cannot qualify as a
correlation matrix. Trying to first generate a
covariance matrix however, does not guarantee a
corresponding correlation matrix with the above
structure.

My larger purpose is to use this correlation matrix to
generate multivariate normal observations from the
corresponding covariance matrix (derived via cholesky
decomposition of the cor matrix).

Greatly appreciate any comments, if this is possible
or how this can be done.

Many grateful thanks and good day,
Melinda

R-version used :
---------------
Windows version
R-1.8.1
Running on Windows XP

Siew Leng TENG <siewlengteng at yahoo.com> writes:
> Hi,
> 
> I would like to generate a correlation matrix with a
> particular structure. For example, a 3n x 3n matrix :
> A_(nxn)   aI_(nxn)  bI_(nxn)
> aI_(nxn)  A_(nxn)   cI_(nxn)
> aI_(nxn)  cI_(nxn)  A_(nxn)
> 
> where
> - A_(nxn) is a *specified* symmetric, positive
> definite nxn matrix.
> - I_(nxn) is an identity matrix of order n
> - a, b, c are (any) real numbers
> 
> Many attempts have been unsuccessful because a
> resulting matrix with any a, b, c may not be a
> positive definite one, and hence cannot qualify as a
> correlation matrix. Trying to first generate a
> covariance matrix however, does not guarantee a
> corresponding correlation matrix with the above
> structure.
Er, a correlation matrix *is* a covariance matrix with 1 down the
diagonal... 

You need to sort out the parametrization issues. What you're trying to
achieve is quite hard. Consider the simpler case of two blocks and
n=2; what you're asking for is a covariance matrix of the form

1 r a 0
r 1 0 a
a 0 1 r
0 a r 1

so if this is the correlation matrix of (X1,Y1,X2,Y2) you want

X1 and Y1 correlated 
X2 and Y2 correlated
X1 and X2 correlated
Y1 and Y2 correlated

but

X1 and Y2 uncorrelated
Y1 and X2 uncorrelated


One approach is to work out the conditional variance of (X2,Y2) given
(X1,Y1) and check for positive semidefiniteness. You do the math...

(Some preliminary experiments suggest that the criterion could be
abs(a)+abs(r) <= 1, but don't take my word for it)
> R-version used :
> ---------------
> Windows version
> R-1.8.1
> Running on Windows XP
You might want to upgrade, but it might not do anything for you in
this respect.

-- 
   O__  ---- Peter Dalgaard             Blegdamsvej 3  
  c/ /'_ --- Dept. of Biostatistics     2200 Cph. N   
 (*) \(*) -- University of Copenhagen   Denmark      Ph: (+45) 35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)             FAX: (+45) 35327907

Siew Leng TENG <siewlengteng <at> yahoo.com> writes:

: 
: Hi,
: 
: I would like to generate a correlation matrix with a
: particular structure. For example, a 3n x 3n matrix :
: A_(nxn)   aI_(nxn)  bI_(nxn)
: aI_(nxn)  A_(nxn)   cI_(nxn)
: aI_(nxn)  cI_(nxn)  A_(nxn)
: 
: where
: - A_(nxn) is a *specified* symmetric, positive
: definite nxn matrix.
: - I_(nxn) is an identity matrix of order n
: - a, b, c are (any) real numbers
: 
: Many attempts have been unsuccessful because a
: resulting matrix with any a, b, c may not be a
: positive definite one, and hence cannot qualify as a
: correlation matrix. Trying to first generate a
: covariance matrix however, does not guarantee a
: corresponding correlation matrix with the above
: structure.
: 
: My larger purpose is to use this correlation matrix to
: generate multivariate normal observations from the
: corresponding covariance matrix (derived via cholesky
: decomposition of the cor matrix).

This can be formulated a semidefinite programming problem.
I don't think R has any packages that do that but a google
search for "semidefinite programming" will find more info and 
some free non-R software which you could consider interfacing 
to R.