g05se generates a vector of pseudorandom numbers taken from a Dirichlet distribution.

Syntax

C#
public static void g05se( int n, int m, double[] a, G05..::..G05State g05state, double[,] x, out int ifail )

Visual Basic
Public Shared Sub g05se ( _ n As Integer, _ m As Integer, _ a As Double(), _ g05state As G05..::..G05State, _ x As Double(,), _ <OutAttribute> ByRef ifail As Integer _ )

Visual C++
public: static void g05se( int n, int m, array<double>^ a, G05..::..G05State^ g05state, array<double,2>^ x, [OutAttribute] int% ifail )

F#
static member g05se : n : int * m : int * a : float[] * g05state : G05..::..G05State * x : float[,] * ifail : int byref -> unit

Parameters

n: Type: System..::..Int32
On entry: $n$ , the number of pseudorandom numbers to be generated.

Constraint: $n \geq 0$ .

m: Type: System..::..Int32
On entry: $m$ , the number of dimensions of the distribution.

Constraint: $m > 0$ .

a: Type: array<System..::..Double>[]()[][]
An array of size [m]
On entry: the parameter vector for the distribution.

Constraint: $a [i - 1] > 0.0$ , for $i = 1, 2, \dots, m$ .

g05state: Type: NagLibrary..::..G05..::..G05State
An Object of type G05.G05State.

x: Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, _tdx]
Note: dim1 must satisfy the constraint: $dim1 \geq n$
On exit: the $n$ pseudorandom numbers from the specified Dirichlet distribution, with $x [i - 1, j - 1]$ holding the $j$ th dimension for the $i$ th variate.

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

The distribution has PDF (probability density function)

\begin{array}{l} f (x) & = & \frac{1}{B (α)} \prod_{i = 1}^{m} x_{i}^{α_{i} - 1} and \\ B (α) & = & \frac{\prod_{i = 1}^{m} Γ (α_{i})}{Γ (\sum_{i = 1}^{m} α_{i})} \end{array}

where

x = \{x_{1}, x_{2}, \dots, x_{m}\}

is a vector of dimension

m

, such that

x_{i} > 0

for all

i

and

\sum_{i = 1}^{m} x_{i} = 1

g05se generates a draw from a Dirichlet distribution by first drawing

m

independent samples,

y_{i} \sim gamma (α_{i}, 1)

, i.e., independent draws from a gamma distribution with parameters

α_{i} > 0

and one, and then setting

x_{i} = y_{i} / \sum_{j = 1}^{m} y_{j}

One of the initialization methods (G05KFF not in this release) (for a repeatable sequence if computed sequentially) or (G05KGF not in this release) (for a non-repeatable sequence) must be called prior to the first call to g05se.

References

Dagpunar J (1988) Principles of Random Variate Generation Oxford University Press

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

Error Indicators and Warnings

Errors or warnings detected by the method:

Some error messages may refer to parameters that are dropped from this interface (LDX) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.

$ifail = 1$: On entry, $n < 0$ .

$ifail = 2$: On entry, $m < 1$ .

$ifail = 3$: On entry, at least one $a [i - 1] \leq 0.0$ .

$ifail = 4$

On entry,

state vector was not initialized or has been corrupted.

$ifail = 6$: On entry, $ldx < n$ .

$ifail = -9000$: An error occured, see message report.
$ifail = -6000$: Invalid Parameters $〈value〉$
$ifail = -4000$: Invalid dimension for array $〈value〉$
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

Not applicable.

Parallelism and Performance

None.

Further Comments

None.

Example

Example program (C#): g05see.cs

Example program data: g05see.d

Example program results: g05see.r