NAG CL Interface
g05rdc (copula_normal)
1
Purpose
g05rdc sets up a reference vector and generates an array of pseudorandom numbers from a Normal (Gaussian) copula with covariance matrix .
2
Specification
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,
, is defined by
where
is the number of dimensions,
is the multivariate Normal density function with mean zero and covariance matrix
and
is the inverse of the univariate Normal density function with mean zero and variance
.
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:
– 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
. 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:
or .
-
2:
– Nag_ModeRNG
Input
-
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 g05rdc.
- Set up reference vector and generate variates.
Constraint:
, or .
-
3:
– Integer
Input
-
On entry: , the number of random variates required.
Constraint:
.
-
4:
– Integer
Input
-
On entry: , the number of dimensions of the distribution.
Constraint:
.
-
5:
– const double
Input
-
Note: the dimension,
dim, of the array
c
must be at least
.
the
th element of the matrix
is stored in
- when ;
- when .
On entry: the covariance matrix of the distribution. Only the upper triangle need be set.
Constraint:
must be positive semidefinite to machine precision.
-
6:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
c.
Constraint:
.
-
7:
– double
Communication Array
-
On entry: if , the reference vector as set up by g05rdc in a previous call with or .
On exit: if or , the reference vector that can be used in subsequent calls to g05rdc with .
-
8:
– Integer
Input
-
On entry: the dimension of the array
r. If
, it must be the same as the value of
lr specified in the prior call to
g05rdc with
or
.
Constraint:
.
-
9:
– Integer
Communication Array
Note: the dimension,
, 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:
– double
Output
-
Note: the dimension,
dim, of the array
x
must be at least
-
when ;
-
when .
where
appears in this document, it refers to the array element
- when ;
- when .
On exit: the array of values from a multivariate Gaussian copula, with holding the th dimension for the th variate.
-
11:
– Integer
Input
-
On entry: the stride used in the array
x.
Constraints:
- if ,
;
- if , .
-
12:
– 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 had an illegal value.
- NE_INT
-
On entry,
lr is not large enough,
: minimum length required
.
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- 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 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
and
.
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.
The time taken by g05rdc is of order .
It is recommended that the diagonal elements of should not differ too widely in order of magnitude. This may be achieved by scaling the variables if necessary. The actual matrix decomposed is , where is a diagonal matrix with small positive diagonal elements. This ensures that, even when is singular, or nearly singular, the Cholesky factor corresponds to a positive definite covariance matrix that agrees with within machine precision.
10
Example
This example prints ten pseudorandom observations from a Normal copula with covariance matrix
generated by
g05rdc. All ten observations are generated by a single call to
g05rdc with
. The random number generator is initialized by
g05kfc.
10.1
Program Text
10.2
Program Data
None.
10.3
Program Results