NAG Library Function Document

nag_rngs_matrix_multi_students_t (g05lxc)

void	nag_rngs_matrix_multi_students_t (Nag_OrderType order, Integer mode, Integer df, Integer m, const double xmu[], const double c[], Integer pdc, Integer n, double x[], Integer pdx, Integer igen, Integer iseed[], double r[], Integer lr, NagError *fail)

3 Description

When the covariance matrix is nonsingular (i.e., strictly positive definite), the distribution has probability density function

f (x) = \frac{Γ (\frac{(ν + m)}{2})}{{(π v)}^{m / 2} Γ (ν / 2) {|C|}^{\frac{1}{2}}} {[1 + \frac{{(x - a)}^{T} C^{- 1} (x - a)}{ν}]}^{\frac{- (ν + m)}{2}}

where

m

is the number of dimensions,

ν

is the degrees of freedom,

a

is the vector of means,

x

is the vector of positions and

\frac{ν}{ν - 2} C

is the covariance matrix.

The function returns the value

x = a + \sqrt{\frac{v}{s}} z

where

z

is generated by nag_rgsn_matrix_multi_normal (g05lyc) from a Normal distribution with mean zero and covariance matrix

C

and

s

is generated by nag_rngs_chi_sq (g05lcc) from a

χ^{2}

-distribution with

ν

degrees of freedom.

One of the initialization functions nag_rngs_init_repeatable (g05kbc) (for a repeatable sequence if computed sequentially) or nag_rngs_init_nonrepeatable (g05kcc) (for a non-repeatable sequence) must be called prior to the first call to nag_rngs_matrix_multi_students_t (g05lxc).

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

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: mode – IntegerInput

On entry: selects the operation to be performed.

$mode = 0$: Initialize and generate random numbers.
$mode = 1$: Initialize only (i.e., set up reference vector).
$mode = 2$: Generate random numbers using previously set up reference vector.

Constraint:

mode = 0

1

2

3: df – IntegerInput

On entry:

ν

, the number of degrees of freedom of the distribution.

Constraint:

df \geq 3

4: m – IntegerInput

On entry:

m

, the number of dimensions of the distribution.

Constraint:

m > 0

5: xmu[m] – const doubleInput

On entry:

a

, the vector of means of the distribution.

6: c[ $\dim$ ] – const doubleInput

Note: the dimension, dim, of the array c must be at least

pdc \times m

The

(i, j)

th element of the matrix

C

is stored in

$c [(j - 1) \times pdc + i - 1]$ when $order = Nag_ColMajor$ ;
$c [(i - 1) \times pdc + j - 1]$ when $order = Nag_RowMajor$ .

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.

7: pdc – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array c.

Constraint:

pdc \geq m

8: n – IntegerInput

On entry:

n

, the number of random variates required.

Constraint:

n \geq 1

9: x[ $\dim$ ] – doubleOutput

Note: the dimension, dim, of the array x must be at least

$\max (1, pdx \times m)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdx)$ when $order = Nag_RowMajor$ .

Where

X (i, j)

appears in this document, it refers to the array element

$x [(j - 1) \times pdx + i - 1]$ when $order = Nag_ColMajor$ ;
$x [(i - 1) \times pdx + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the array of pseudorandom multivariate Student's

t

vectors generated by the function, with

X (i, j)

holding the

j

th dimension for the

i

th variate.

10: pdx – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array x.

Constraints:

if $order = Nag_ColMajor$ , $pdx \geq n$ ;
if $order = Nag_RowMajor$ , $pdx \geq m$ .

11: igen – IntegerInput

On entry: must contain the identification number for the generator to be used to return a pseudorandom number and should remain unchanged following initialization by a prior call to nag_rngs_init_repeatable (g05kbc) or nag_rngs_init_nonrepeatable (g05kcc).

12: iseed[ $4$ ] – IntegerCommunication Array

On entry: contains values which define the current state of the selected generator.

On exit: contains updated values defining the new state of the selected generator.

13: r[lr] – doubleInput/Output

On entry: if

mode = 2

, the reference vector as set up by nag_rngs_matrix_multi_students_t (g05lxc) in a previous call with

mode = 0

1

On exit: if

mode = 0

1

, the reference vector that can be used in subsequent calls to nag_rngs_matrix_multi_students_t (g05lxc) with

mode = 2

14: lr – IntegerInput

On entry: the dimension of the array r. If

mode = 2

, it must be the same as the value of lr specified in the prior call to nag_rngs_matrix_multi_students_t (g05lxc) with

mode = 0

1

Constraint:

lr > m \times (m + 1) + 1

15: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $df = 〈value〉$ .
Constraint: $df > 2$ .; On entry, invalid value for igen. Ensure igen has not changed since random number generator was initialized.; On entry, $m = 〈value〉$ .
Constraint: $m > 0$ .; On entry, $m = 〈value〉$ .
Constraint: $m \geq 1$ .; On entry, $mode = 〈value〉$ .
Constraint: $mode \geq 0$ and $mode \leq 2$ .; On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .; On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .; On entry, $pdc = 〈value〉$ .
Constraint: $pdc > 0$ .; On entry, $pdx = 〈value〉$ .
Constraint: $pdx > 0$ .
NE_INT_2: m is not the same as when r was set up in a previous call. Previous value $= 〈value〉$ and $m = 〈value〉$ .; On entry, $lr = 〈value〉$ , $(m \times m + m + 1) = 〈value〉$ .
Constraint: $lr \geq m \times m + m + 1$ .; On entry, $pdc = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdc \geq m$ .; On entry, $pdx = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdx \geq m$ .; On entry, $pdx = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdx \geq n$ .
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_POS_DEF: The covariance matrix c is not positive semidefinite to machine precision.

7 Accuracy

The maximum absolute error in

L L^{T}

, and hence in the covariance matrix of the resulting vectors, is less than

(m ε + (m + 3) ε / 2)

times the maximum element of

C

, where

ε

is the machine precision. Under normal circumstances, the above will be small compared to sampling error.

8 Further Comments

The time taken by nag_rngs_matrix_multi_students_t (g05lxc) is of order

n m^{3}

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 = L L^{T}

, 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 multivariate Student's

t

-distribution ten degrees of freedom, means vector

[\begin{matrix} 1.0 \\ 2.0 \\ - 3.0 \\ 0.0 \end{matrix}]

and c matrix

[\begin{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 \end{matrix}],

generated by nag_rngs_matrix_multi_students_t (g05lxc). All ten observations are generated by a single call to nag_rngs_matrix_multi_students_t (g05lxc) with