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NAG Toolbox: nag_rand_dist_dirichlet (g05se)

Purpose

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

Syntax

[state, x, ifail] = g05se(n, a, state, 'm', m)
[state, x, ifail] = nag_rand_dist_dirichlet(n, a, state, 'm', m)

Description

The distribution has PDF (probability density function)
f(x) = m 1/(B(α)) ∏ xi α_i − 1   and i = 1 B(α) = ( ∏ i = 1m Γ (αi) )/(
Γ(m ) ∑ αii = 1
)
f(x) = 1 B(α) i=1 m x i αi - 1 and B(α) = i=1 m Γ (αi) Γ ( i=1 m αi )
where x = {x1,x2,,xm} x = {x1,x2,,xm}  is a vector of dimension mm, such that xi > 0xi>0 for all ii and i = 1m xi = 1 i=1 m xi=1.
nag_rand_dist_dirichlet (g05se) generates a draw from a Dirichlet distribution by first drawing mm independent samples, yigamma(αi,1)yigamma(αi,1), i.e., independent draws from a gamma distribution with parameters αi > 0αi>0 and one, and then setting xi = yi / j = 1m yjxi=yi/ j=1 m yj.
One of the initialization functions nag_rand_init_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_dist_dirichlet (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

Parameters

Compulsory Input Parameters

1:     n – int64int32nag_int scalar
nn, the number of pseudorandom numbers to be generated.
Constraint: n0n0.
2:     a(m) – double array
m, the dimension of the array, must satisfy the constraint m > 0m>0.
The parameter vector for the distribution.
Constraint: a(i) > 0.0ai>0.0, for i = 1,2,,mi=1,2,,m.
3:     state( : :) – int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains information on the selected base generator and its current state.

Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The dimension of the array a.
mm, the number of dimensions of the distribution.
Constraint: m > 0m>0.

Input Parameters Omitted from the MATLAB Interface

ldx

Output Parameters

1:     state( : :) – int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains updated information on the state of the generator.
2:     x(ldx,m) – double array
ldxnldxn.
The nn pseudorandom numbers from the specified Dirichlet distribution, with x(i,j)xij holding the jjth dimension for the iith variate.
3:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
On entry, n < 0n<0.
  ifail = 2ifail=2
On entry, m < 1m<1.
  ifail = 3ifail=3
On entry, at least one a(i)0.0ai0.0.
  ifail = 4ifail=4
On entry,state vector was not initialized or has been corrupted.
  ifail = 6ifail=6
On entry, ldx < nldx<n.

Accuracy

Not applicable.

Further Comments

None.

Example

function nag_rand_dist_dirichlet_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
a = [2; 2; 2; 2];
% Initialize the generator to a repeatable sequence
[state, ifail] = nag_rand_init_repeat(genid, subid, seed);
[state, x, ifail] = nag_rand_dist_dirichlet(n, a, state)
 

state =

                   17
                 1234
                    1
                    0
                 4694
                17841
                 5384
                14764
                17917
                13895
                19930
                    8
                    0
                 1234
                    1
                    1
                 1234


x =

    0.3600    0.3138    0.0837    0.2426
    0.2874    0.5121    0.1497    0.0509
    0.2286    0.2190    0.3959    0.1566
    0.1744    0.3961    0.2764    0.1530
    0.1522    0.2845    0.2074    0.3559


ifail =

                    0


function g05se_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
a = [2; 2; 2; 2];
% Initialize the generator to a repeatable sequence
[state, ifail] = g05kf(genid, subid, seed);
[state, x, ifail] = g05se(n, a, state)
 

state =

                   17
                 1234
                    1
                    0
                 4694
                17841
                 5384
                14764
                17917
                13895
                19930
                    8
                    0
                 1234
                    1
                    1
                 1234


x =

    0.3600    0.3138    0.0837    0.2426
    0.2874    0.5121    0.1497    0.0509
    0.2286    0.2190    0.3959    0.1566
    0.1744    0.3961    0.2764    0.1530
    0.1522    0.2845    0.2074    0.3559


ifail =

                    0



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