NAG Library Function Document
nag_rand_matrix_multi_students_t (g05ryc)
1 Purpose
nag_rand_matrix_multi_students_t (g05ryc) sets up a reference vector and generates an array of pseudorandom numbers from a multivariate Student's distribution with degrees of freedom, mean vector and covariance matrix .
2 Specification
#include <nag.h> |
#include <nagg05.h> |
void |
nag_rand_matrix_multi_students_t (Nag_OrderType order,
Nag_ModeRNG mode,
Integer n,
Integer df,
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 degrees of freedom,
is the vector of means,
is the vector of positions and
is the covariance matrix.
The function returns the value
where
is generated by
nag_rand_normal (g05skc) from a Normal distribution with mean zero and covariance matrix
and
is generated by
nag_rand_chi_sq (g05sdc) from a
-distribution with
degrees of freedom.
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_students_t (g05ryc).
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_students_t (g05ryc).
- Set up reference vector and generate variates.
Constraint:
, or .
- 3:
n – IntegerInput
On entry: , the number of random variates required.
Constraint:
.
- 4:
df – IntegerInput
On entry: , the number of degrees of freedom of the distribution.
Constraint:
.
- 5:
m – IntegerInput
On entry: , the number of dimensions of the distribution.
Constraint:
.
- 6:
xmu[m] – const doubleInput
On entry: , the vector of means of the distribution.
- 7:
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: matrix which, along with
df, defines the covariance of the distribution. Only the upper triangle need be set.
Constraint:
c must be positive semidefinite to
machine precision.
- 8:
pdc – IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
c.
Constraint:
.
- 9:
r[lr] – doubleInput/Output
On entry: if , the reference vector as set up by nag_rand_matrix_multi_students_t (g05ryc) 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_students_t (g05ryc) with .
- 10:
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_students_t (g05ryc) with
or
.
Constraint:
.
- 11:
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.
- 12:
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 Student's vectors generated by the function, with holding the th dimension for the th variate.
- 13:
pdx – IntegerInput
-
On entry: the stride used in the array
x.
Constraints:
- if ,
;
- if , .
- 14:
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, .
Constraint: .
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_students_t (g05ryc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_rand_matrix_multi_students_t (g05ryc) 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_students_t (g05ryc) 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 Student's
-distribution with ten degrees of freedom, means vector
and
c matrix
generated by nag_rand_matrix_multi_students_t (g05ryc). All ten observations are generated by a single call to nag_rand_matrix_multi_students_t (g05ryc) with
. The random number generator is initialized by
nag_rand_init_repeatable (g05kfc).
10.1 Program Text
Program Text (g05ryce.c)
10.2 Program Data
None.
10.3 Program Results
Program Results (g05ryce.r)