void	g05rdc (Nag_OrderType order, Nag_ModeRNG mode, Integer n, Integer m, const double c[], Integer pdc, double r[], Integer lr, Integer state[], double x[], Integer pdx, NagError *fail)

The function may be called by the names: g05rdc or nag_rand_copula_normal.

3 Description

The Gaussian copula,

G

, is defined by

G (u_{1}, u_{2}, \dots, u_{m}; C) = Φ_{C} (ϕ_{C_{11}}^{−1} (u_{1}), ϕ_{C_{22}}^{−1} (u_{2}), \dots, ϕ_{C_{mm}}^{−1} (u_{m}))

where

m

is the number of dimensions,

Φ_{C}

is the multivariate Normal density function with mean zero and covariance matrix

C

and

ϕ_{C_{ii}}^{−1}

is the inverse of the univariate Normal density function with mean zero and variance

C_{ii}

g05rzc is used to generate a vector from a multivariate Normal distribution and g01eac is used to convert each element of that vector into a uniformly distributed value between zero and one.

One of the initialization functions g05kfc (for a repeatable sequence if computed sequentially) or g05kgc (for a non-repeatable sequence) must be called prior to the first call to g05rdc.

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 Arguments

1: $order$ – Nag_OrderType Input

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $mode$ – Nag_ModeRNG Input

On entry: a code for selecting the operation to be performed by the function.

$mode = Nag_InitializeReference$: Set up reference vector only.
$mode = Nag_GenerateFromReference$: Generate variates using reference vector set up in a prior call to g05rdc.
$mode = Nag_InitializeAndGenerate$: Set up reference vector and generate variates.

Constraint:

mode = Nag_InitializeReference

Nag_GenerateFromReference

Nag_InitializeAndGenerate

3: $n$ – Integer Input

On entry:

n

, the number of random variates required.

Constraint:

n \geq 0

4: $m$ – Integer Input

On entry:

m

, the number of dimensions of the distribution.

Constraint:

m > 0

5: $c [\dim]$ – const double Input

Note: the dimension, dim, of the array c must be at least

pdc \times m

the

(i, j)

th element of the matrix

C

is stored in

$c [(j - 1) \times pdc + i - 1]$ when $order = Nag_ColMajor$ ;
$c [(i - 1) \times pdc + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the covariance matrix of the distribution. Only the upper triangle need be set.

Constraint:

C

must be positive semidefinite to machine precision.

6: $pdc$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array c.

Constraint:

pdc \geq m

7: $r [lr]$ – double Communication Array

On entry: if

mode = Nag_GenerateFromReference

, the reference vector as set up by g05rdc in a previous call with

mode = Nag_InitializeReference

Nag_InitializeAndGenerate

On exit: if

mode = Nag_InitializeReference

Nag_InitializeAndGenerate

, the reference vector that can be used in subsequent calls to g05rdc with

mode = Nag_GenerateFromReference

8: $lr$ – Integer Input

On entry: the dimension of the array r. If

mode = Nag_GenerateFromReference

, it must be the same as the value of lr specified in the prior call to g05rdc with

mode = Nag_InitializeReference

Nag_InitializeAndGenerate

Constraint:

lr \geq m \times (m + 1) + 1

9: $state [\dim]$ – Integer Communication Array

Note: the dimension,

\dim

, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).

On entry: contains information on the selected base generator and its current state.

On exit: contains updated information on the state of the generator.

10: $x [\dim]$ – double Output

Note: the dimension, dim, of the array x must be at least

$\max (1, pdx \times m)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdx)$ when $order = Nag_RowMajor$ .

where

X (i, j)

appears in this document, it refers to the array element

$x [(j - 1) \times pdx + i - 1]$ when $order = Nag_ColMajor$ ;
$x [(i - 1) \times pdx + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the array of values from a multivariate Gaussian copula, with

X (i, j)

holding the

j

th dimension for the

i

th variate.

11: $pdx$ – Integer Input

On entry: the stride used in the array x.

Constraints:

if $order = Nag_ColMajor$ , $pdx \geq n$ ;
if $order = Nag_RowMajor$ , $pdx \geq m$ .

12: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, lr is not large enough, $lr = ⟨ value ⟩$ : minimum length required $= ⟨ value ⟩$ .

On entry, $m = ⟨ value ⟩$ .
Constraint: $m > 0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pdc = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $pdc \geq m$ .

On entry, $pdx = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $pdx \geq m$ .

On entry, $pdx = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdx \geq n$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_INVALID_STATE: On entry, state vector has been corrupted or not initialized.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_POS_DEF: On entry, the covariance matrix $C$ is not positive semidefinite to machine precision.
NE_PREV_CALL: m is not the same as when r was set up in a previous call.
Previous value of $m = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .

7 Accuracy

See Section 7 in g05rzc for an indication of the accuracy of the underlying multivariate Normal distribution.

8 Parallelism and Performance

g05rdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g05rdc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken by g05rdc 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^{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.

10 Example

This example prints ten pseudorandom observations from a Normal copula with covariance matrix

[\begin{matrix} 1.69 & 0.39 & - 1.86 & 0.07 \\ 0.39 & 98.01 & - 7.07 & - 0.71 \\ - 1.86 & - 7.07 & 11.56 & 0.03 \\ 0.07 & - 0.71 & 0.03 & 0.01 \end{matrix}],

generated by g05rdc. All ten observations are generated by a single call to g05rdc with

mode = Nag_InitializeAndGenerate

. The random number generator is initialized by g05kfc.

10.1 Program Text

Program Text (g05rdce.c)

10.2 Program Data

None.

10.3 Program Results

Program Results (g05rdce.r)

NAG Library Manual, Mark 28.3

Interfaces: FL CL CPP AD

NAG CL Interface Introduction

G05 (Rand) Chapter Contents

G05 (Rand) Chapter Introduction

g05rd: FL CL CPP AD

NAG CL Interfaceg05rdc (copula_​normal)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
g05rdc (copula_normal)