acomparith {compositions} | R Documentation |
The Aitchison Simplex with its two operations perturbation as + and power transform as * is a vector space. This vector space is represented by these operations.
power.acomp(x,s) ## Methods for class "acomp" ## x*y ## x/y
x |
an acomp composition or dataset of compositions (or a number or a numeric vector) |
y |
a numeric vector of size 1 or nrow(x) |
s |
a numeric vector of size 1 or nrow(x) |
The power transform is the basic multiplication operation of the Aitchison simplex seen as a vector space. It is defined as:
(x*y)_i:= clo( (x_i^{y_i})_i )_i
The division operation is just the multiplication with 1/y.
An "acomp"
vector or matrix.
For *
the arguments x and y can be exchanged. Note that
this definition generalizes the power by a scalar, since y
or
s
may be given as a scalar, or as a vector with as many components as
the composition in acomp
x
. The result is then a matrix
where each row corresponds to the composition powered by one of the scalars
in the vector.
Aitchison, J. (1986) The Statistical Analysis of Compositional
Data Monographs on Statistics and Applied Probability. Chapman &
Hall Ltd., London (UK). 416p.
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
Pawlowsky-Glahn, V. and J.J. Egozcue (2001) Geometric approach to
statistical analysis on the simplex. SERRA 15(5), 384-398
http://ima.udg.es/Activitats/CoDaWork03
http://ima.udg.es/Activitats/CoDaWork05
acomp(1:5)* -1 + acomp(1:5) data(SimulatedAmounts) cdata <- acomp(sa.lognormals) plot( tmp <- (cdata-mean(cdata))/msd(cdata) ) class(tmp) mean(tmp) msd(tmp) var(tmp)