# rDirichlet: Dirichlet distribution In compositions: Compositional Data Analysis

## Description

The Dirichlet distribution on the simplex.

## Usage

 ```1 2 3 4 5``` ```dDirichlet(x, alpha, log=FALSE, measure="Lebesgue") rDirichlet.acomp(n, alpha) rDirichlet.rcomp(n, alpha) ```

## Arguments

 `n` number of datasets to be simulated `alpha` parameters of the Dirichlet distribution `x` a data set (acomp, rcomp, data.frame, matrix; even one-row) of point(s) on the simplex `log` a boolean, controlling if the density or the log-density is returned `measure` one of: "Lebesgue" or "Aitchison" (partial match applies), or else a function returning the reference LOG-density (see details below)

## Details

The Dirichlet distribution is the result of closing a vector of equally-scaled Gamma-distributed variables. It the conjugate prior distribution for a vector of probabilities of a multinomial distribution. Thus, it generalizes the beta distribution for more than two parts.

For the density, the implementation allows to obtain the conventional density (with respect to the Lebesgue measure, default behaviour or giving `measure="Lebuesgue"`), the compositional density (with respect to the Aitchison measure, giving `measure="Aitchison"`), or else w.r.to any other reference density (giving at `measure` a function returning the log-density of the reference measure for any point of the simplex)

## Value

For `rDirichlet.*` a generated random dataset of class `"acomp"` or `"rcomp"`, drawn from a Dirichlet distribution with the given parameter `alpha`. The names of `alpha` are used to name the parts.

For `dDirichlet`, the (conventional) Dirichlet density

## Author(s)

K.Gerald v.d. Boogaart http://www.stat.boogaart.de, Raimon Tolosana-Delgado

## References

Aitchison, J. (1986) The Statistical Analysis of Compositional Data Monographs on Statistics and Applied Probability. Chapman & Hall Ltd., London (UK). 416p.

Mateu-Figueras, G.; Pawlowsky-Glahn, V. (2005). The Dirichlet distribution with respect to the Aitchison measure on the simplex, a first approach. In: Mateu-Figueras, G. and Barcel\'o-Vidal, C. (Eds.) Proceedings of the 2nd International Workshop on Compositional Data Analysis, Universitat de Girona, ISBN 84-8458-222-1, http://ima.udg.es/Activitats/CoDaWork05/

`rnorm.acomp`

## Examples

 ```1 2 3 4``` ```tmp <- rDirichlet.acomp(10,alpha=c(A=2,B=0.2,C=0.2)) plot(tmp) dDirichlet(tmp, alpha=c(A=2,B=0.2,C=0.2)) dDirichlet(tmp[1,]*0, alpha=c(A=2,B=0.2,C=0.2)) ```

compositions documentation built on Jan. 5, 2022, 5:09 p.m.