nag_rand_copula_normal (g05rdc) (PDF version)
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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 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.
Set up reference vector only.
Generate variates using reference vector set up in a prior call to nag_rand_copula_normal (g05rdc).
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.
  • 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

On entry, argument value had an illegal value.
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.
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.
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.
On entry, state vector has been corrupted or not initialized.
On entry, the covariance matrix C is not positive semidefinite to machine precision.
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


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