rnorm: Normal distributions on special spaces

In compositions: Compositional Data Analysis

Normal distributions on special spaces

Description

rnorm.X generates multivariate normal random variates in the space X.

Usage

rnorm.acomp(n,mean,var)
rnorm.rcomp(n,mean,var)
rnorm.aplus(n,mean,var)
rnorm.rplus(n,mean,var)
rnorm.rmult(n,mean,var)
rnorm.ccomp(n,mean,var,lambda)
dnorm.acomp(x,mean,var,withJacobian=FALSE)
dnorm.aplus(x,mean,var,withJacobian=FALSE)
dnorm.rmult(x,mean,var)
          

Arguments

n	number of datasets to be simulated
mean	The mean of the dataset to be simulated
var	The variance covariance matrix
lambda	The expected total count
x	vectors in the sampling space
withJacobian	should the jacobian of the log or logratio transformation be included in the density calculations? defaults to FALSE (see details)

Details

The normal distributions in the various spaces dramatically differ. The normal distribution in the rmult space is the commonly known multivariate joint normal distribution. For rplus this distribution has to be somehow truncated at 0. This is here done by setting negative values to 0, i.e. this simulation function produces a sort of multivariate tobit model.
The normal distribution of rcomp is seen as a normal distribution within the simplex as a geometrical portion of the real vector space. The variance is thus forced to be singular and restricted to the affine subspace generated by the simplex. The necessary truncation of negative values is currently done by setting them explicitly to zero and reclosing afterwards, again in the fashion of a tobit model.
The "acomp" and "aplus" are themselves metric vector spaces and thus a normal distribution is defined in them just as in the real space. The resulting distribution almost correspond to multivariate lognormal in the case of "aplus" and Aitchison normal distribution in the simplex in the case of "acomp". These models are equivalent in probability to the multivariate lognormal distribution and the addditive logistic normal distribution respectively, albeit without including the jacobian of the log or the logratio transformation. If you are interested in the density of the additive logistic normal model, give the extra argument withJacobian=TRUE. If you are interested in the multivariate lognormal density cou can either do the same, or better call dlnorm.rplus.
Densities are only provided for the models constructed for rmult, aplus and acomp because they do exist w with repect to the Lebesgue measure of each of these spaces. In the other cases it is not possible to compute a measure, since the truncation at zero values produce distributions that are not absolutely continuous with respect to the real, conventional Lebesgue measure.
For count compositions ccomp a rnorm.acomp is realized and used as a parameter to a Poisson distribution (see rpois.ccomp). So, this is in reality no normal model, but a double stochastic counting process.

Value

a random dataset of the given class generated by a normal distribution with the given mean and variance in the given space. For the density functions d*, the value of the probability density at the values of x provided

Author(s)

K.Gerald v.d. Boogaart 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.

Pawlowsky-Glahn, V. and J.J. Egozcue (2001) Geometric approach to statistical analysis on the simplex. SERRA 15(5), 384-398

Aitchison, J, C. Barcel'o-Vidal, J.J. Egozcue, V. Pawlowsky-Glahn (2002) A consise guide to the algebraic geometric structure of the simplex, the sample space for compositional data analysis, Terra Nostra, Schriften der Alfred Wegener-Stiftung, 03/2003

See Also

runif.acomp, rlnorm.rplus, rDirichlet.acomp

Examples

MyVar <- matrix(c(
0.2,0.1,0.0,
0.1,0.2,0.0,
0.0,0.0,0.2),byrow=TRUE,nrow=3)
MyMean <- c(1,1,2)

plot(rnorm.acomp(100,MyMean,MyVar))
plot(rnorm.rcomp(100,MyMean,MyVar))
plot(rnorm.aplus(100,MyMean,MyVar))
plot(rnorm.rplus(100,MyMean,MyVar))
plot(rnorm.rmult(100,MyMean,MyVar))
x <- rnorm.aplus(5,MyMean,MyVar)
dnorm.acomp(x,MyMean,MyVar)
dnorm.aplus(x,MyMean,MyVar)
dnorm.rmult(x,MyMean,MyVar)

compositions documentation built on June 22, 2024, 12:15 p.m.