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 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 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

or 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 actual argument supplied must be the array state supplied to the initialization functions 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 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.

9 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).

9.1 Program Text

Program Text (g05rdce.c)

9.2 Program Data

None.

9.3 Program Results

Program Results (g05rdce.r)

nag_rand_copula_normal (g05rdc) (PDF version)

g05 Chapter Contents

g05 Chapter Introduction

NAG C 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 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_rand_copula_normal (g05rdc)