[R] 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 __ [EMAIL PROTECTED] mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
Re: [R] Help : generating correlation matrix with a particular structure
Siew Leng TENG [EMAIL PROTECTED] 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 ~~ - ([EMAIL PROTECTED]) FAX: (+45) 35327907 __ [EMAIL PROTECTED] mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
Re: [R] Help : generating correlation matrix with a particular structure
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. __ [EMAIL PROTECTED] mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
Re: [R] Help : generating correlation matrix with a particular structure
Extending what Gabor and Peter have already said, the following
should provide a partial solution:
patternCor3 - function(A=diag(2), a=1:3){
# nk x nk covariance correlation matrices
# k = length(a); abs(a) = min(diag(A)
minV.A - min(diag(A))
if(any(adj.a - abs(a)minV.A)){
warning(abs(a) too large; can't exceed,
min(diag(A)) = , minV.A,
; forced into that range.)
a[adj.a] - sign(a[adj.a])*minV.A
}
Aa - kronecker(diag(3), A)
n - dim(A)[1]
i1 - n+1:n
i2 - n+i1
diag(Aa[1:n, i1]) - a[1]
diag(Aa[i1, 1:n]) - a[1]
diag(Aa[1:n, i2]) - a[2]
diag(Aa[i2, 1:n]) - a[2]
diag(Aa[i1, i2]) - a[3]
diag(Aa[i2, i1]) - a[3]
s.A - sqrt(diag(A))
r.Aa - (Aa/outer(rep(s.A,3), rep(s.A,3)))
eig.Aa - eigen(Aa)
list(Aa=Aa, corr.Aa=r.Aa, eigen.Aa=eig.Aa)
}
If this works, all(eigen.Aa$values=0). Thus, you can add a test
for this and have something close to what you want. You could add an
objective function that includes these eigenvalues with, say, minimum
adjustment of a and feed it to optim and let optim find a solution
that is closest in whatever sense you think is useful.
hope this helps. spencer graves
Gabor Grothendieck wrote:
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.
__
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--
Spencer Graves, PhD, Senior Development Engineer
O: (408)938-4420; mobile: (408)655-4567
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