random_mvar_gaussian_copula
Given a Cholesky factorization of a correlation matrix, generates pseudorandom numbers from a Gaussian Copula distribution.
Synopsis
#include <imsls.h>
float *imsls_f_random_mvar_gaussian_copula(int n, float chol[], …, 0)
The type double function is imsls_d_ random_mvar_gaussian_copula.
Required Arguments
int n (Input)
Number of random numbers to generate.
float chol[] (Input)
Array of size n × n containing the upper‑triangular Cholesky factorization of the correlation matrix of order n.
Return Value
An array of length n containing the pseudorandom numbers from a multivariate Gaussian Copula distribution.
Synopsis with Optional Arguments
#include <imsls.h>
float *imsls_f_random_mvar_gaussian_copula (int n, float chol[],
IMSLS_RETURN_USER, r[],
0)
Optional Arguments
IMSLS_RETURN_USER, r[] (Output)
User-supplied array of length n containing the pseudorandom numbers from a multivariate Gaussian Copula distribution.
Description
Function imsls_f_random_mvar_gaussian_copula 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 × 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 Gaussian Copulas to Imprint and Extract Correlation Information
This example uses function imsls_f_random_mvar_gaussian_copula 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 function imsls_f_canonical_correlation 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 function imsls_f_random_mvar_gaussian_copula correctly embeds 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.
#include <imsls.h>
#include <imsl.h>
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#define NVAR 3
int main()
{
int lmax=15000, i, j, k, kmax, kk;
float chol[NVAR*NVAR], gcvart[NVAR], *gcdevt, corrout[NVAR*NVAR],
relerr, arg1=10.0, arg2=15.0, rs, rs00;
float corrin[] = {
1.0, -0.9486832, 0.8164965,
-0.9486832, 1.0, -0.6454972,
0.8164965, -0.6454972, 1.0
};
printf("Off-diagonal elements of Input Correlation Matrix:\n\n");
for (i = 1; i < NVAR; i++) {
for (j = 0; j < i; j++) {
printf(" CorrIn(%d,%d) = %10.6f\n",
i, j, corrin[i*NVAR + j]);
}
}
printf("\nOff-diagonal elements of Output Correlation Matrices\n");
printf("calculated from Gaussian Copula imprinted multivariate\n");
printf("sequence:\n");
/*
* Compute the Cholesky factorization of corrin
*
* Use IMSL function imsl_f_lin_sol_posdef to generate
* the NVAR by NVAR upper triangular matrix chol from
* the Cholesky decomposition R*RT of input correlation
* matrix corrin:
*/
imsl_f_lin_sol_posdef (NVAR, corrin, NULL,
IMSL_FACTOR_USER, chol,
IMSL_FACTOR_ONLY,
0);
kmax = lmax / 100;
for (kk = 1; kk <= 3; kk++) {
gcdevt = (float *) malloc(kmax * NVAR * sizeof(float));
printf("\n# of vectors in multivariate sequence: %7d\n\n",
kmax);
/* use Congruential RN generator, with multiplier 16807 */
imsls_random_option(1);
/* set RN generator seed to be 123457 */
imsls_random_seed_set(123457);
for (k = 0; k < kmax; k++) {
/*
* generate a NVAR-length random Gaussian Copula
* variate output vector gcvart which is uniformly
* distributed on the interval [0,1] and imprinted
* with correlation information from input Cholesky
* matrix chol:
*/
imsls_f_random_mvar_gaussian_copula(NVAR, chol,
IMSLS_RETURN_USER, gcvart,
0);
for (j = 0; j < 3; j++) {
/*
* invert Gaussian Copula probabilities to deviates
* using variable-specific inversions: j = 0: Chi
* Square; j = 1: F; j = 2: Normal(0,1); will end
* up with deviate sequences ready for mapping to
* canonical correlation matrix:
*/
if (j == 0) {
/* convert probs into ChiSquare(df=10) deviates */
gcdevt[k*NVAR + j] =
imsls_f_chi_squared_inverse_cdf(gcvart[j], arg1);
} else if (j == 1) {
/* convert probs into F(dfn=15,dfd=10) deviates */
gcdevt[k*NVAR + j] =
imsls_f_F_inverse_cdf(gcvart[j], arg2, arg1);
} else {
/*
* convert probs into Normal(mean=0,variance=1)
* deviates:
*/
gcdevt[k*NVAR + j] =
imsls_f_normal_inverse_cdf(gcvart[j]);
}
}
}
/*
* extract Canonical Correlation matrix from arbitrarily
* distributed deviate sequences gcdevt (k=1..kmax, j=1..NVAR)
* which have been imprinted with corrin (i=1..NVAR, j=1..NVAR)
* above:
*/
imsls_f_canonical_correlation(kmax, NVAR, gcdevt,
IMSLS_RETURN_USER, corrout,
0);
for (i = 1; i < NVAR; i++) {
for (j = 0; j <= i-1; j++) {
rs00 = corrin[i*NVAR + j];
rs = corrout[i*NVAR + j];
relerr = fabs((rs - rs00)/rs00);
printf(" CorrOut(%d,%d) = %10.6f; relerr = %10.6f\n",
i, j, corrout[i*NVAR + j], relerr);
}
}
free(gcdevt);
kmax *= 10;
}
}
Output
Off-diagonal elements of Input Correlation Matrix:
CorrIn(1,0) = -0.948683
CorrIn(2,0) = 0.816496
CorrIn(2,1) = -0.645497
Off-diagonal elements of Output Correlation Matrices
calculated from Gaussian Copula imprinted multivariate
sequence:
# of vectors in multivariate sequence: 150
CorrOut(1,0) = -0.940215; relerr = 0.008926
CorrOut(2,0) = 0.794511; relerr = 0.026927
CorrOut(2,1) = -0.616082; relerr = 0.045570
# of vectors in multivariate sequence: 1500
CorrOut(1,0) = -0.947444; relerr = 0.001306
CorrOut(2,0) = 0.808306; relerr = 0.010031
CorrOut(2,1) = -0.635650; relerr = 0.015255
# of vectors in multivariate sequence: 15000
CorrOut(1,0) = -0.948263; relerr = 0.000443
CorrOut(2,0) = 0.817261; relerr = 0.000936
CorrOut(2,1) = -0.646206; relerr = 0.001098