where $m$ is the number of dimensions, ${\Phi}_{C}$ is the multivariate Normal density function with mean zero and covariance matrix $C$ and ${\varphi}_{{C}_{\mathit{ii}}}^{\mathrm{-1}}$ 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 non-repeatable sequence) must be called prior to the first call to g05rdf.
4References
Nelsen R B (1998) An Introduction to Copulas. Lecture Notes in Statistics 139 Springer
Sklar A (1973) Random variables: joint distribution functions and copulas Kybernetika9 499–460
5Arguments
1: $\mathbf{mode}$ – 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: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of random variates required.
Constraint:
${\mathbf{n}}\ge 0$.
3: $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of dimensions of the distribution.
Constraint:
${\mathbf{m}}>0$.
4: $\mathbf{c}({\mathbf{ldc}},{\mathbf{m}})$ – 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: $\mathbf{ldc}$ – 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: $\mathbf{r}\left({\mathbf{lr}}\right)$ – 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: $\mathbf{lr}$ – 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 $2$.
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: $\mathbf{x}({\mathbf{ldx}},*)$ – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array x
must be at least
${\mathbf{m}}$.
On exit: the array of values from a multivariate Gaussian copula, with ${\mathbf{x}}(i,j)$ holding the $j$th dimension for the $i$th variate.
10: $\mathbf{ldx}$ – 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: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, 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, ${\mathbf{mode}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{mode}}=0$, $1$ or $2$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}>0$.
${\mathbf{ifail}}=4$
On entry, the covariance matrix $C$ is not positive semidefinite to machine precision.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{ldc}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldc}}\ge {\mathbf{m}}$.
${\mathbf{ifail}}=6$
m is not the same as when r was set up in a previous call.
Previous value of ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=7$
On entry, lr is not large enough, ${\mathbf{lr}}=\u27e8\mathit{\text{value}}\u27e9$: minimum length required $\text{}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=8$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=10$
On entry, ${\mathbf{ldx}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
See Section 7 in g05rzf for an indication of the accuracy of the underlying multivariate Normal distribution.
8Parallelism and Performance
g05rdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g05rdf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
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.
10Example
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.