RNMVGC
Given a Cholesky factorization of a correlation matrix, generates pseudorandom numbers from a Gaussian Copula distribution.
Required Arguments
CHOL — Array of size n by n containing the upper‑triangular Cholesky factorization of the correlation matrix of order n where n = size(CHOL,1). (Input)
R — Array of length n containing the pseudorandom numbers from a multivariate Gaussian Copula distribution. (Output)
FORTRAN 90 Interface
Generic: RNMVGC (CHOL, R)
Specific: The specific interface names are S_RNMVGC and D_RNMVGC.
Description
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.
Example: Using Copulas to Imprint and Extract Correlation Information
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.
 
use rnmvgc_int
use cancor_int
use anorin_int
use chiin_int
use fin_int
use amach_int
use rnopt_int
use rnset_int
use umach_int
use chfac_int
implicit none
 
integer, parameter :: lmax=15000, nvar=3
real corrin(nvar,nvar), tol, chol(nvar,nvar), gcvart(nvar), &
gcdevt(lmax,nvar), corrout(nvar,nvar), relerr
integer irank, k, kmax, kk, i, j, nout
 
data corrin /&
1.0, -0.9486832, 0.8164965, &
-0.9486832, 1.0, -0.6454972, &
0.8164965, -0.6454972, 1.0/
 
call umach (2, nout)
 
write(nout,*)
write(nout,*) "Off-diagonal Elements of Input " // &
"Correlation Matrix: "
write(nout,*)
 
do i = 2, nvar
do j = 1, i-1
write(nout,'(" CorrIn(",i2,",",i2,") = ", f10.6)') &
i, j, corrin(i,j)
end do
end do
 
write(nout,*)
write(nout,*) "Off-diagonal Elements of Output Correlation " // &
"Matrices calculated from"
write(nout,*) "Gaussian Copula imprinted multivariate sequence:"
 
! Compute the Cholesky factorization of CORRIN.
tol=amach(4)
tol=100.0*tol
call chfac (corrin, irank, chol, tol=tol)
 
kmax = lmax/100
do kk = 1, 3
write (*, '(/" # vectors in multivariate sequence: ", &
i7/)') kmax
 
call rnopt(1)
call rnset (123457)
 
do k = 1, kmax
 
! Generate an array of Gaussian Copula random numbers.
call rnmvgc (chol, gcvart)
do j = 1, nvar
! Invert Gaussian Copula probabilities to deviates.
if (j .eq. 1) then
! ChiSquare(df=10) deviates:
gcdevt(k, j) = chiin(gcvart(j), 10.e0)
else if (j .eq. 2) then
! F(dfn=15,dfd=10) deviates:
gcdevt(k, j) = fin(gcvart(j), 15.e0, 10.e0)
else
! Normal(mean=0,variance=1) deviates:
gcdevt(k, j) = anorin(gcvart(j))
end if
end do
end do
! Extract Canonical Correlation matrix.
call cancor (gcdevt(:kmax,:), corrout)
 
do i = 2, nvar
do j = 1, i-1
relerr = abs(1.0 - (corrout(i,j) / corrin(i,j)))
write(nout,'(" CorrOut(",i2,",",i2,") = ", '// &
'f10.6, "; relerr = ", f10.6)') &
i, j, corrout(i,j), relerr
end do
end do
kmax = kmax*10
end do
end
Output
 
Off-diagonal Elements of Input Correlation Matrix:
CorrIn( 2, 1) = -0.948683
CorrIn( 3, 1) = 0.816496
CorrIn( 3, 2) = -0.645497
Off-diagonal Elements of Output Correlation Matrices calculated from
Gaussian Copula imprinted multivariate sequence:
 
# vectors in multivariate sequence: 150
 
CorrOut( 2, 1) = -0.940215; relerr = 0.008926
CorrOut( 3, 1) = 0.794511; relerr = 0.026927
CorrOut( 3, 2) = -0.616082; relerr = 0.045569
 
# vectors in multivariate sequence: 1500
 
CorrOut( 2, 1) = -0.947443; relerr = 0.001308
CorrOut( 3, 1) = 0.808307; relerr = 0.010031
CorrOut( 3, 2) = -0.635650; relerr = 0.015256
 
# vectors in multivariate sequence: 15000
 
CorrOut( 2, 1) = -0.948267; relerr = 0.000439
CorrOut( 3, 1) = 0.817261; relerr = 0.000936
CorrOut( 3, 2) = -0.646208; relerr = 0.001101
Published date: 03/19/2020
Last modified date: 03/19/2020