[10] Simplex Uniformity#

pysht.simplex.uniformity tests whether compositional observations are uniform with respect to volume on a probability simplex. The null is the Dirichlet model with every concentration parameter equal to one.

The model selector controls the alternative:

model

Alternative model

Wilks degrees of freedom

"symmetric"

one common positive concentration

1

"general"

one positive concentration per component

number of components

Inputs must lie in the strict simplex interior. Boundary zeros are not perturbed. The optimizer controls tolerance and max_iter are explicit and fail loudly when a finite maximum cannot be established.

See the simplex-uniformity validation ledger for the likelihood, independent optimization oracle, null calibration, and MLE boundary cases.

Functions#

Goodness-of-fit tests for observations on a probability simplex.

pysht.simplex.uniformity(x, *, model='symmetric', tolerance=1.0e-10, max_iter=200)[source]#

Test uniformity on a probability simplex with a Dirichlet LRT.

Parameters:
x

An (n, k) matrix whose rows lie in the strict interior of the probability simplex. At least two rows and two components are needed. Rows whose sums differ from one by no more than a documented floating-point tolerance are normalized before fitting.

model

"symmetric" fits one common Dirichlet concentration parameter; "general" fits one positive parameter per component.

tolerance

Positive convergence tolerance for the score equations.

max_iter

Positive maximum number of bracketing/root or Newton iterations.

Returns:
HypothesisTestResult

The likelihood-ratio statistic calibrated by Wilks’ chi-square approximation with one or k degrees of freedom.

Parameters:

  • x (ArrayLike)

  • model (str)

  • tolerance (float)

  • max_iter (int)

Return type:

HypothesisTestResult

Notes

The null is Dirichlet alpha=(1, ..., 1), the uniform distribution with respect to simplex volume. Boundary components are rejected instead of being silently perturbed. The general-model fit rejects identical rows, for which its concentration MLE is unbounded. The symmetric fit is unbounded only when every row is the simplex barycenter.