Given a Cholesky factorization of a correlation matrix, generates pseudorandom numbers from a Gaussian Copula distribution.

RNMVGC generates pseudorandom numbers from a multivariate Gaussian Copula distribution which are uniformly distributed on the interval (0,1) representing the probabilities associated with standard normal N(0,1) deviates imprinted with correlation information from input upper‑triangular Cholesky matrix CHOL. Cholesky matrix CHOL is defined as the “square root” of a user‑defined correlation matrix, that is CHOL is an upper triangular matrix such that the transpose of CHOL times CHOL is the correlation matrix. First, a length n array of independent random normal deviates with mean 0 and variance 1 is generated, and then this deviate array is post‑multiplied by Cholesky matrix CHOL. Finally, the Cholesky‑imprinted random N(0,1) deviates are mapped to output probabilities using the N(0,1) cumulative distribution function (CDF).

Random deviates from arbitrary marginal distributions which are imprinted with the correlation information contained in Cholesky matrix CHOL can then be generated by inverting the output probabilities using user‑specified inverse CDF functions.

This example uses subroutine RNMVGC to generate a multivariate sequence gcdevt whose marginal distributions are user‑defined and imprinted with a user‑specified input correlation matrix corrin and then uses subroutine CANCOR to extract an output canonical correlation matrix corrout from this multivariate random sequence.

This example illustrates two useful copula related procedures. The first procedure generates a random multivariate sequence with arbitrary user‑defined marginal deviates whose dependence is specified by a user‑defined correlation matrix. The second procedure is the inverse of the first: an arbitrary multivariate deviate input sequence is first mapped to a corresponding sequence of empirically derived variates, i.e. cumulative distribution function values representing the probability that each random variable has a value less than or equal to the input deviate. The variates are then inverted, using the inverse standard normal CDF function, to N(0,1) deviates; and finally, a canonical covariance matrix is extracted from the multivariate N(0,1) sequence using the standard sum of products.

This example demonstrates that subroutine RNMVGC correctly imbeds the user‑defined correlation information into an arbitrary marginal distribution sequence by extracting the canonical correlation from these sequences and showing that they differ from the original correlation matrix by a small relative error, which generally decreases as the number of multivariate sequence vectors increases.