Chapter 18: Random Number Generation

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



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