NAG Library Function Document
nag_rand_matrix_multi_normal (g05rzc)
1 Purpose
nag_rand_matrix_multi_normal (g05rzc) sets up a reference vector and generates an array of pseudorandom numbers from a multivariate Normal distribution with mean vector and covariance matrix .
2 Specification
#include <nag.h> |
#include <nagg05.h> |
void |
nag_rand_matrix_multi_normal (Nag_OrderType order,
Nag_ModeRNG mode,
Integer n,
Integer m,
const double xmu[],
const double c[],
Integer pdc,
double r[],
Integer lr,
Integer state[],
double x[],
Integer pdx,
NagError *fail) |
|
3 Description
When the covariance matrix is nonsingular (i.e., strictly positive definite), the distribution has probability density function
where
is the number of dimensions,
is the covariance matrix,
is the vector of means and
is the vector of positions.
Covariance matrices are symmetric and positive semidefinite. Given such a matrix , there exists a lower triangular matrix such that . is not unique, if is singular.
nag_rand_matrix_multi_normal (g05rzc) decomposes to find such an . It then stores , and in the reference vector which is used to generate a vector of independent standard Normal pseudorandom numbers. It then returns the vector , which has the required multivariate Normal distribution.
It should be noted that this function will work with a singular covariance matrix
, provided
is positive semidefinite, despite the fact that the above formula for the probability density function is not valid in that case.
Wilkinson (1965) should be consulted if further information is required.
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_matrix_multi_normal (g05rzc).
4 References
Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley
Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford
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
. See
Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
mode – Nag_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_matrix_multi_normal (g05rzc).
- Set up reference vector and generate variates.
Constraint:
, or .
- 3:
n – IntegerInput
On entry: , the number of random variates required.
Constraint:
.
- 4:
m – IntegerInput
On entry: , the number of dimensions of the distribution.
Constraint:
.
- 5:
xmu[m] – const doubleInput
On entry: , the vector of means of the distribution.
- 6:
c[] – const doubleInput
-
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.
- 7:
pdc – IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
c.
Constraint:
.
- 8:
r[lr] – doubleInput/Output
On entry: if , the reference vector as set up by nag_rand_matrix_multi_normal (g05rzc) in a previous call with or .
On exit: if or , the reference vector that can be used in subsequent calls to nag_rand_matrix_multi_normal (g05rzc) with .
- 9:
lr – IntegerInput
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 nag_rand_matrix_multi_normal (g05rzc) with
or
.
Constraint:
.
- 10:
state[] – IntegerCommunication 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.
- 11:
x[] – doubleOutput
-
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 pseudorandom multivariate Normal vectors generated by the function, with holding the th dimension for the th variate.
- 12:
pdx – IntegerInput
-
On entry: the stride used in the array
x.
Constraints:
- if ,
;
- if , .
- 13:
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 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.
- NE_INVALID_STATE
-
On entry,
state vector has been corrupted or not initialized.
- 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
Not applicable.
8 Parallelism and Performance
nag_rand_matrix_multi_normal (g05rzc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_rand_matrix_multi_normal (g05rzc) 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.
The time taken by nag_rand_matrix_multi_normal (g05rzc) 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 multivariate Normal distribution with means vector
and covariance matrix
generated by nag_rand_matrix_multi_normal (g05rzc). All ten observations are generated by a single call to nag_rand_matrix_multi_normal (g05rzc) with
. The random number generator is initialized by
nag_rand_init_repeatable (g05kfc).
10.1 Program Text
Program Text (g05rzce.c)
10.2 Program Data
None.
10.3 Program Results
Program Results (g05rzce.r)