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Universitat de Girona. Departament dâ€™InformÃ tica i MatemÃ tica Aplicada  
Monti, Gianna Serafina
Mateu i Figueras, GlÃ²ria PawlowskyGlahn, Vera Egozcue, Juan JosÃ© 

2011 May 12  
Perturbation and powering are two operations in the simplex that define a vectorspace 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 shiftedscaled 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 (shiftedscaled) 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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http://hdl.handle.net/10256/13637  
eng  
Universitat de Girona. Departament dâ€™InformÃ tica i MatemÃ tica Aplicada  
CoDaWork 2011. The 4th International Workshop on Compositional Data Analysis  
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The shiftedscaled Dirichlet Distribution in the Simplex  
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