NAG Library Function Document

nag_rand_copula_normal (g05rdc)

+− 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

1 Purpose

nag_rand_copula_normal (g05rdc) sets up a reference vector and generates an array of pseudorandom numbers from a Normal (Gaussian) copula with covariance matrix

C

2 Specification

#include <nag.h>

#include <nagg05.h>

void	nag_rand_copula_normal (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)

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}

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

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

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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: mode – Nag_ModeRNGInput

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 nag_rand_copula_normal (g05rdc).
$mode = Nag_InitializeAndGenerate$: Set up reference vector and generate variates.

Constraint:

mode = Nag_InitializeReference

Nag_GenerateFromReference

Nag_InitializeAndGenerate

3: n – IntegerInput

On entry:

n

, the number of random variates required.

Constraint:

n \geq 0

4: m – IntegerInput

On entry:

m

, the number of dimensions of the distribution.

Constraint:

m > 0

5: c[ $\dim$ ] – const doubleInput

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 – IntegerInput

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] – doubleCommunication Array

On entry: if

mode = Nag_GenerateFromReference

, the reference vector as set up by nag_rand_copula_normal (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 nag_rand_copula_normal (g05rdc) with

mode = Nag_GenerateFromReference

8: lr – IntegerInput

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 nag_rand_copula_normal (g05rdc) with

mode = Nag_InitializeReference

Nag_InitializeAndGenerate

Constraint:

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

9: state[ $\dim$ ] – IntegerCommunication 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[ $\dim$ ] – doubleOutput

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 – IntegerInput

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

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.
NE_INVALID_STATE: On entry, state vector has been corrupted or not initialized.
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 nag_rand_matrix_multi_normal (g05rzc) for an indication of the accuracy of the underlying multivariate Normal distribution.

8 Parallelism and Performance

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

nag_rand_copula_normal (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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken by nag_rand_copula_normal (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 nag_rand_copula_normal (g05rdc). All ten observations are generated by a single call to nag_rand_copula_normal (g05rdc) with

mode = Nag_InitializeAndGenerate

. The random number generator is initialized by nag_rand_init_repeatable (g05kfc).

10.1 Program Text

Program Text (g05rdce.c)

10.2 Program Data

None.

10.3 Program Results

Program Results (g05rdce.r)

nag_rand_copula_normal (g05rdc) (PDF version)

g05 Chapter Contents

g05 Chapter Introduction

NAG Library Manual

NAG Library Function Documentnag_rand_copula_normal (g05rdc)

+− 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 Library Function Document

nag_rand_copula_normal (g05rdc)