nag_rand_copula_normal (g05rdc) (PDF version)
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g05 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_rand_copula_normal (g05rdc)

+ Contents

    1  Purpose
    7  Accuracy

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 u1 , u2 ,, um ; C = ΦC ϕ C11 -1 u1 , ϕ C22 -1 u2 ,, ϕ Cmm -1 um
where m is the number of dimensions, ΦC  is the multivariate Normal density function with mean zero and covariance matrix C and ϕ Cii -1  is the inverse of the univariate Normal density function with mean zero and variance Cii .
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:     orderNag_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:     modeNag_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 or Nag_InitializeAndGenerate.
3:     nIntegerInput
On entry: n, the number of random variates required.
Constraint: n0.
4:     mIntegerInput
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×m.
The i,jth element of the matrix C is stored in
  • c[j-1×pdc+i-1] when order=Nag_ColMajor;
  • c[i-1×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:     pdcIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraint: pdcm.
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 or Nag_InitializeAndGenerate.
On exit: if mode=Nag_InitializeReference or Nag_InitializeAndGenerate, the reference vector that can be used in subsequent calls to nag_rand_copula_normal (g05rdc) with mode=Nag_GenerateFromReference.
8:     lrIntegerInput
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 or Nag_InitializeAndGenerate.
Constraint: lrm×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
  • max1,pdx×m when order=Nag_ColMajor;
  • max1,n×pdx when order=Nag_RowMajor.
Where Xi,j appears in this document, it refers to the array element
  • x[j-1×pdx+i-1] when order=Nag_ColMajor;
  • x[i-1×pdx+j-1] when order=Nag_RowMajor.
On exit: the array of values from a multivariate Gaussian copula, with Xi,j holding the jth dimension for the ith variate.
11:   pdxIntegerInput
On entry: the stride used in the array x.
Constraints:
  • if order=Nag_ColMajor, pdxn;
  • if order=Nag_RowMajor, pdxm.
12:   failNagError *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: n0.
NE_INT_2
On entry, pdc=value and m=value.
Constraint: pdcm.
On entry, pdx=value and m=value.
Constraint: pdxm.
On entry, pdx=value and n=value.
Constraint: pdxn.
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 nm3.
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=LLT, 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
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 ,
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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012