The shifted-scaled Dirichlet Distribution in the Simplex

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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 by scalars in real space. A standard Dirichlet random composition can be shifted by perturbation, and scaled powering by a real scalar. The obtained random composition has a shifted-scaled Dirichlet distribution. The procedure is analogous to standardization of real random variables. The derived distribution is a generalization of the Dirichlet one, and it is studied from a probabilistic point of 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 natural way of describing the generalized (shifted-scaled) Dirichlet probability distributions is using probability densities with respect to the Aitchison measure. This density representation is compared with the traditional probability density with respect to the Lebesgue measure. In particular, the center and variability for both representations are compared ​
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