NAG Library Routine Document
G05RDF
1 Purpose
G05RDF sets up a reference vector and generates an array of pseudorandom numbers from a Normal (Gaussian) copula with covariance matrix $C$.
2 Specification
SUBROUTINE G05RDF ( 
MODE, N, M, C, LDC, R, LR, STATE, X, LDX, IFAIL) 
INTEGER 
MODE, N, M, LDC, LR, STATE(*), LDX, IFAIL 
REAL (KIND=nag_wp) 
C(LDC,M), R(LR), X(LDX,M) 

3 Description
The Gaussian copula,
$G$, is defined by
where
$m$ is the number of dimensions,
${\Phi}_{C}$ is the multivariate Normal density function with mean zero and covariance matrix
$C$ and
${\varphi}_{{C}_{\mathit{ii}}}^{1}$ is the inverse of the univariate Normal density function with mean zero and variance
${C}_{\mathit{ii}}$.
G05RZF is used to generate a vector from a multivariate Normal distribution and
G01EAF is used to convert each element of that vector into a uniformly distributed value between zero and one.
One of the initialization routines
G05KFF (for a repeatable sequence if computed sequentially) or
G05KGF (for a nonrepeatable sequence) must be called prior to the first call to G05RDF.
4 References
Nelsen R B (1998) An Introduction to Copulas. Lecture Notes in Statistics 139 Springer
Sklar A (1973) Random variables: joint distribution functions and copulas Kybernetika 9 499–460
5 Parameters
 1: MODE – INTEGERInput
On entry: a code for selecting the operation to be performed by the routine.
 ${\mathbf{MODE}}=0$
 Set up reference vector only.
 ${\mathbf{MODE}}=1$
 Generate variates using reference vector set up in a prior call to G05RDF.
 ${\mathbf{MODE}}=2$
 Set up reference vector and generate variates.
Constraint:
${\mathbf{MODE}}=0$, $1$ or $2$.
 2: N – INTEGERInput
On entry: $n$, the number of random variates required.
Constraint:
${\mathbf{N}}\ge 0$.
 3: M – INTEGERInput
On entry: $m$, the number of dimensions of the distribution.
Constraint:
${\mathbf{M}}>0$.
 4: C(LDC,M) – REAL (KIND=nag_wp) arrayInput
On entry: the covariance matrix of the distribution. Only the upper triangle need be set.
Constraint:
$C$ must be positive semidefinite to machine precision.
 5: LDC – INTEGERInput
On entry: the first dimension of the array
C as declared in the (sub)program from which G05RDF is called.
Constraint:
${\mathbf{LDC}}\ge {\mathbf{M}}$.
 6: R(LR) – REAL (KIND=nag_wp) arrayCommunication Array
On entry: if ${\mathbf{MODE}}=1$, the reference vector as set up by G05RDF in a previous call with ${\mathbf{MODE}}=0$ or $2$.
On exit: if ${\mathbf{MODE}}=0$ or $2$, the reference vector that can be used in subsequent calls to G05RDF with ${\mathbf{MODE}}=1$.
 7: LR – INTEGERInput
On entry: the dimension of the array
R as declared in the (sub)program from which G05RDF is called. If
${\mathbf{MODE}}=1$, it must be the same as the value of
LR specified in the prior call to G05RDF with
${\mathbf{MODE}}=0$ or
$2$.
Constraint:
${\mathbf{LR}}\ge {\mathbf{M}}\times \left({\mathbf{M}}+1\right)+1$.
 8: STATE($*$) – INTEGER arrayCommunication Array

Note: the actual argument supplied must be the array
STATE supplied to the initialization routines
G05KFF or
G05KGF.
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.
 9: X(LDX,M) – REAL (KIND=nag_wp) arrayOutput
On exit: the array of values from a multivariate Gaussian copula, with ${\mathbf{X}}\left(i,j\right)$ holding the $j$th dimension for the $i$th variate.
 10: LDX – INTEGERInput
On entry: the first dimension of the array
X as declared in the (sub)program from which G05RDF is called.
Constraint:
${\mathbf{LDX}}\ge {\mathbf{N}}$.
 11: IFAIL – INTEGERInput/Output
On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$
On entry, ${\mathbf{MODE}}\ne 0$, $1$ or $2$.
 ${\mathbf{IFAIL}}=2$
On entry, ${\mathbf{N}}<0$.
 ${\mathbf{IFAIL}}=3$
On entry, ${\mathbf{M}}<1$.
 ${\mathbf{IFAIL}}=4$
The covariance matrix $C$ is not positive semidefinite to machine precision.
 ${\mathbf{IFAIL}}=5$
On entry, ${\mathbf{LDC}}<{\mathbf{M}}$.
 ${\mathbf{IFAIL}}=6$
The reference vector
R has been corrupted or
M has changed since
R was set up in a previous call to G05RDF with
${\mathbf{MODE}}=0$ or
$2$.
 ${\mathbf{IFAIL}}=7$
On entry, ${\mathbf{LR}}\le {\mathbf{M}}\times \left({\mathbf{M}}+1\right)$.
 ${\mathbf{IFAIL}}=8$
On entry,  STATE vector was not initialized or has been corrupted. 
 ${\mathbf{IFAIL}}=10$
On entry, ${\mathbf{LDX}}<{\mathbf{N}}$.
7 Accuracy
See
Section 7 in G05RZF for an indication of the accuracy of the underlying multivariate Normal distribution.
The time taken by G05RDF is of order $n{m}^{3}$.
It is recommended that the diagonal elements of $C$ should not differ too widely in order of magnitude. This may be achieved by scaling the variables if necessary. The actual matrix decomposed is $C+E=L{L}^{\mathrm{T}}$, where $E$ is a diagonal matrix with small positive diagonal elements. This ensures that, even when $C$ is singular, or nearly singular, the Cholesky factor $L$ corresponds to a positive definite covariance matrix that agrees with $C$ within machine precision.
9 Example
This example prints ten pseudorandom observations from a Normal copula with covariance matrix
generated by G05RDF. All ten observations are generated by a single call to G05RDF with
${\mathbf{MODE}}=2$. The random number generator is initialized by
G05KFF.
9.1 Program Text
Program Text (g05rdfe.f90)
9.2 Program Data
Program Data (g05rdfe.d)
9.3 Program Results
Program Results (g05rdfe.r)