N-simplex random vectors generators

Contains 2 functions to generate random vectors in the \(n\)-simplex

\[\begin{align*} &\sum_{i=1}^n b_i = 1, \\ &b_i \geq 0\ \ \ \forall i=1,\cdots,n. \end{align*}\]

• runif_simplex : Uniformly distributed vectors in n-simplex,

• random_simplex : Dirichlet generator of vectors in n-simplex with permutations (for antithetic variance reduction applications).

Note, a Flat Dirichlet generator (i.e., all alpha Dirichlet coefficients set to \(1\)) is equivalent to an uniformly distributed n-simplex vector generators.

azapy.Util.randomgen.random_simplex(rng, simplex_dim, size=1, antithetic=False, dirichlet_alpha=None)

Returns random vectors in the n-simplex (based on Dirichlet generator) sum_{k=1}^n x_k = 1 for x_k >= 0

Parameters:

rngnumpy.random.mtrand.RandomState

Random number generator (e.g. as returned by numpy.random.RandomState(42)).

simplex_dimpositive int

Simplex dimension.

sizepositive int, optional

Sample size. The default is 1.

antitheticBoolean, optional

If it is set to True, then all the vector permutation will be added to the sample. The actual size of the sample will be size * factorial(simplex_dim). The default is False.

dirichlet_alphalist, optional

Dirichlet alpha coefficients. The length of the list must be equal to simplex_dim. If it is None, then uniform distributed random vectors are generated (equivalent to dirichlet_alpha = [1] * simple_dim). The default is None.

Returns:

numpy.array

A sample of size size of random vectors of length simplex_dim.

azapy.Util.randomgen.runif_simplex(rng, simplex_dim, size=1)

Returns uniformly distributed random vectors in the n-simplex sum_{k=1}^n x_k = 1 for x_k >= 0

Parameters:

rngnumpy.random.mtrand.RandomState

Random number generator (e.g. as returned by numpy.random.RandomState(42)).

simplex_dimpositive int

Simplex dimension.

sizepositive int, optional

Sample size. The default is 1.

Returns:

numpy.array

A sample of size size of random vectors of length simplex_dim.

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Examples of generating random vectors in a simplex

# Examples of how to call random simplex functions
import numpy as np
import azapy as az

# Build a random numbers generator 
rng = np.random.RandomState(42)

# Uniform random vectors (10) in a 5-simplex
simplex_dim = 5
size = 10
sq = az.runif_simplex(rng, simplex_dim, size)

# Dirichlet distributed random vectors (10) in a 5-simplex
simplex_dim = 5
alpha = [0.5] * simplex_dim
size = 10
sq = az.random_simplex(rng, simplex_dim, size=10, dirichlet_alpha=alpha)
print(sq)

# Same as above but with antithetic permutations 
# The effective size of the sample is size * factorial(simplex_dim)
# in this example 10 * 5! = 1200
simplex_dim = 5
alpha = [0.5] * simplex_dim
size = 10
sq = az.random_simplex(rng, simplex_dim, size=10, 
                       antithetic=True, dirichlet_alpha=alpha)
print(sq)

# Flat Dirichlet distributed random vectors (10) in a 5-simplex
# equivalent with the uniform distributed vectors in a simplex
simplex_dim = 5
alpha = [1] * simplex_dim
size = 10
sq = az.random_simplex(rng, simplex_dim, size=10, 
                       antithetic=True, dirichlet_alpha=alpha)
print(sq)

# Uniform distributed random vectors (10) in a 5-simplex
# equivalent with Flat Dirichlet
simplex_dim = 5
size = 10
sq = az.random_simplex(rng, simplex_dim, size=10, antithetic=True)
print(sq)

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