rAitchison {compositions}  R Documentation 
Aitchison Distribution
Description
The Aitchison distribution is a class of distributions the simplex, containing the normal and the Dirichlet as subfamilies.
Usage
dAitchison(x,
theta=alpha+sigma %*% clr(mu),
beta=1/2*gsi.svdinverse(sigma),
alpha=mean(theta),
mu=clrInv(c(sigma%*%(thetaalpha))),
sigma=1/2*gsi.svdinverse(beta),
grid=30,
realdensity=FALSE,
expKappa=AitchisonDistributionIntegrals(theta,beta,
grid=grid,mode=1)$expKappa)
rAitchison(n,
theta=alpha+sigma %*% clr(mu),
beta=1/2*gsi.svdinverse(sigma),
alpha=mean(theta),
mu=clrInv(c(sigma%*%(thetaalpha))),
sigma=1/2*gsi.svdinverse(beta), withfit=FALSE)
AitchisonDistributionIntegrals(
theta=alpha+sigma %*% clr(mu),
beta=1/2*gsi.svdinverse(sigma),
alpha=mean(theta),
mu=clrInv(c(sigma%*%(thetaalpha))),
sigma=1/2*gsi.svdinverse(beta),
grid=30,
mode=3)
Arguments
x 
acompcompositions the density should be computed for. 
n 
integer: number of datasets to be simulated 
theta 
numeric vector: Location parameter vector 
beta 
matrix: Spread parameter matrix (clr or ilr) 
alpha 
positiv scalar: departure from normality parameter (positive scalar) 
mu 
acompcomposition, normal reference mean parameter composition 
sigma 
matrix: normal reference variance matrix (clr or ilr) 
grid 
integer: number of discretisation points along each side of the simplex 
realdensity 
logical: if true the density is given with respect to the Haar measure of the real simplex, if false the density is given with respect to the Aitchison measure of the simplex. 
mode 
integer: desired output:
1: Compute nothing, only transform parameters, 
expKappa 
The closing divisor of the density 
withfit 
Should a prespliting of the Aitchison density be used for simulation? 
Details
The Aitchison Distribution is a joint generalisation of the Dirichlet
Distribution and the additive lognormal distribution (or normal on the
simplex). It can be parametrized by Ait(theta,beta) or by
Ait(alpha,mu,Sigma). Actually, beta and Sigma can be easily
transformed into each other, such that only one of them is
necessary. Parameter theta is a vector in R^D
, alpha is its sum, mu is a
composition in S^D
, and beta and sigma are symmetric matrices, which
can either be expressed in ilr or clr space.
The parameters are transformed as
\beta=1/2 \Sigma^{1}
\theta=clr(\mu)\Sigma+\alpha (1,\ldots,1)^t
The distribution exists, if either,
\alpha\geq 0
and Sigma is positive definite (or beta
negative definite) in
ilrcoordinates, or if each theta is strictly positive and Sigma has
at least one positive eigenvalue (or beta has at least one negative
eigenvalue). The simulation procedure currently only works with the
first case.
AitchisonDistributionIntegral is a convenience function to compute the
parameter transformation and several functions of these
parameters. This is done by numerical integration over a
multinomial simplex lattice of D parts with grid
many elements
(see xsimplex
).
The density of the Aitchison distribution is given by:
f(x,\theta,\beta)=exp((\theta1)^t \log(x)+ilr(x)^t \beta ilr(x))/exp(\kappa_{Ait(\theta,\beta)})
with respect to the classical Haar measure on the simplex, and as
f(x,\theta,\beta)=exp(\theta^t \log(x)+ilr(x)^t \beta
ilr(x))/exp(\kappa_{Ait(\theta,\beta)})
with respect to the Aitchison
measure of the simplex. The closure constant expKappa is computed
numerically, in AitchisonDistributionIntegrals
.
The random composition generation is done by rejection sampling based on an optimally fitted additive logistic normal distribution. Thus, it only works if the correponding Sigma in ilr would be positive definite.
Value
dAitchison 
Returns the density of the Aitchison distribution evaluated at x as a numeric vector. 
rAitchison 
Returns a sample of size n of simulated compostions as an acomp object. 
AitchisondistributionIntegrals 
Returns a list with

Note
The simulation procedure currently only works with a positive definite Sigma. You need a relatively high grid constant for precise values in the numerical integration.
Author(s)
K.Gerald v.d. Boogaart, R. TolosanaDelgado http://www.stat.boogaart.de
References
Aitchison, J. (1986) The Statistical Analysis of Compositional
Data Monographs on Statistics and Applied Probability. Chapman &
Hall Ltd., London (UK). 416p.
See Also
runif.acomp
, rnorm.acomp
,
rDirichlet.acomp
Examples
(erg<AitchisonDistributionIntegrals(c(1,3,2),ilrvar2clr(diag(c(1,2))),grid=20))
(myvar<with(erg, 1/2*ilrvar2clr(solve(clrvar2ilr(beta)))))
(mymean<with(erg,myvar%*%theta))
with(erg,myvarclrVar)
with(erg,mymeanclrMean)