naginterfaces.library.rand.dist_​dirichlet

naginterfaces.library.rand.dist_dirichlet(n, a, statecomm)

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

For full information please refer to the NAG Library document for g05se

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/g05/g05sef.html

Parameters

nint

$n$, the number of pseudorandom numbers to be generated.

afloat, array-like, shape $\left(m\right)$

The parameter vector for the distribution.

statecommdict, RNG communication object, modified in place

RNG communication structure.

This argument must have been initialized by a prior call to init_repeat() or init_nonrepeat().

Returns

xfloat, ndarray, shape $(n,m)$

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.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $2$)

On entry, $m=⟨value⟩$.

Constraint: $m>0$.

(errno $3$)

On entry, at least one $a\left[i\right]\le 0$.

(errno $4$)

On entry, $statecomm$[‘state’] vector has been corrupted or not initialized.

Notes

The distribution has PDF (probability density function)

$$\begin{array}{c}\begin{array}{ccc}f\left(x\right)& =& \frac{1}{B\left(\alpha \right)}{\prod }_1^m{x}_i^{{\alpha }_i-1}\text{and}\\ B\left(\alpha \right)& =& \frac{{\prod }_1^m\Gamma \left({\alpha }_i\right)}{\Gamma \left({\sum }_1^m{\alpha }_i\right)}\end{array}\end{array}$$

where $x=\{x_1,x_2,\dots ,x_m\}$ is a vector of dimension $m$, such that $x_i>0$ for all $i$ and ${\sum }_1^m{x}_i=1$.

dist_dirichlet generates a draw from a Dirichlet distribution by first drawing $m$ independent samples, $y_i\sim gamma({\alpha }_i,1)$, i.e., independent draws from a gamma distribution with parameters ${\alpha }_i>0$ and one, and then setting $x_i=y_i/{\sum }_1^m{y}_j$.

One of the initialization functions init_repeat() (for a repeatable sequence if computed sequentially) or init_nonrepeat() (for a non-repeatable sequence) must be called prior to the first call to dist_dirichlet.

References

Dagpunar, J, 1988, Principles of Random Variate Generation, Oxford University Press

Hastings, N A J and Peacock, J B, 1975, Statistical Distributions, Butterworth