rnorm.X generates multivariate normal random variates in
the space X.
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)number of datasets to be simulated
The mean of the dataset to be simulated
The variance covariance matrix
The expected total count
vectors in the sampling space
should the jacobian of the log or logratio transformation be included in the density calculations? defaults to FALSE (see details)
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
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.
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
# NOT RUN {
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)
# }
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