These packages are archived in zip format. All code is available in
the public domain under the GPL unless otherwise noted.
Generating Uniform Random Vectors Given Correlation Matrix
MatlabForUnifVecGenUsingNORTA.zip
This Matlab code generates uniform random vectors while "closely"
matching a given correlation matrix. The generation procedure used is
the NORTA method (Normal to anything), which is essentially a Gaussian copula.
It is known that NORTA cannot exactly match all correlation matrices. This code
augments the usual NORTA procedure to deal with this problem. The augmentation ensures that the
correlation matrix of the generated vectors is exactly the desired one in many
cases, but if that is not possible, then the correlation matrix will be "close" to the
desired one. The augmentation can be done in many ways. The code provides a menu of four different heuristics, three of which solve a semi-definite problem (and will need
the LMI toolbox installed within Matlab) while the last one is a
heuristic that provides a good feasible solution without
solving a semi-definite program, and is thus recommended.
For full details of why this problem occurs, and the various heuristics for dealing with them,
please see Soumyadip Ghosh's thesis. Some aspects of the problem are also discussed in the references below.
- S. Ghosh and S. G. Henderson. 2002. Chessboard
distributions and random vectors with specified marginals and
covariance matrix Operations Research 50, 820-834.
- S. Ghosh and S. G. Henderson. 2003. Behaviour of
the NORTA method for correlated random vector generation as the
dimension increases. ACM TOMACS 13, 276-294.
- S. Ghosh and S. G. Henderson. 2005. Corrigendum: Behaviour of
the NORTA method for correlated random vector generation as the
dimension increases. ACM TOMACS 16, 93-94.
The same code can be used to generate samples for general random
marginals (exponentials etc.) if one aims to match rank
correlations instead of product moment correlations, and uses
continuous marginals. The desired rank correlation matrix is used as
the input uniform vector correlation matrix, and the generated uniform
vector sample is passed through a final step where each component is
transformed by the appropriate inverse of the desired marginal
distribution. Restricting to continuous marginal distributions ensures
that the final transformation step preserves the rank correlation
values. For discrete marginals, the final rank correlations will be close,
but not exact.
Generating Matrices Uniformly from Space of P.S.D. Matrices
MatlabforSamplingCorrelMatrix.zip
This Matlab code illustrates and implements how matrices can be
generated uniformly from the space of all n-dimensional
matrices that are positive semi-definite and have the a pre-selected
set of diagonal values. The simplest case is where all diagonals equal
1, i.e., the set of all n-dimensional matrices that are
commonly referred to as correlation matrices. An exact definition of this space and the details of this algorithm can be found in
- S. Ghosh and S. G. Henderson. 2003. Behaviour of
the NORTA method for correlated random vector generation as the
dimension increases. ACM TOMACS 13, 276-294.
- S. Ghosh and S. G. Henderson. 2005. Corrigendum: Behaviour of
the NORTA method for correlated random vector generation as the
dimension increases. ACM TOMACS 16, 93-94.