The simplex

2016-10-25 — 2016-10-25

Wherein a method for producing a uniform point on the n‑simplex is presented, in which n independent Uniform(0,1) draws are ordered and their successive differences are taken as the simplex coordinates.

classification
metrics
probability
statistics

The space of convex combinations of things. Hacks for it.

1 Simulating uniformly from the simplex

This one is apparently “folk wisdom”.

But say I wish to simulate a vector drawn uniformly from the \(n\)-simplex.

  • Simulate \(n\) random uniform variables on the unit interval, \((u_1, u_2, \dots, u_n)\)
  • Sort them in decreasing order, \((u'_1, u'_2, \dots, u'_n)\)
  • The random vector is \((u'_1-0, u'_2-u'_1, u'_3-u'_2, \dots, u'_n-u'_{n-1})\)

2 Simulating Dirichlet distributions

See Dirichlet variables, which are distributions over the simplex. C&C gumbel max tricks.

3 References

Friedman. 2008. “An Elementary Illustrated Introduction to Simplicial Sets.” arXiv:0809.4221 [Math].