DUGi

The shifted-scaled Dirichlet Distribution in the Simplex

http://dugi.udg.edu/item/http:@@@@hdl.handle.net@@2072@@273623

Perturbation and powering are two operations in the simplex that define a vector-space structure.Perturbation and powering in the simplex play the same role as the sum and product byscalars in real space. A standard Dirichlet random composition can be shifted by perturbation, andscaled powering by a real scalar. The obtained random composition has a shifted-scaled Dirichletdistribution. The procedure is analogous to standardization of real random variables. The deriveddistribution is a generalization of the Dirichlet one, and it is studied from a probabilistic pointof view. In the simplex, considered as an Euclidean space, the Aitchison measure is the natural(Lebesgue type) measure, which is compatible with its operations and metrics. Therefore, a naturalway of describing the generalized (shifted-scaled) Dirichlet probability distributions is using probabilitydensities with respect to the Aitchison measure. This density representation is comparedwith the traditional probability density with respect to the Lebesgue measure. In particular, thecenter and variability for both representations are compared

Universitat de Girona. Departament d’Informàtica i Matemàtica Aplicada