aplus {compositions}R Documentation

Amounts analysed in log-scale

Description

A class to analyse positive amounts in a logistic framework.

Usage

          aplus(X,parts=1:NCOL(oneOrDataset(X)),total=NA)
          

Arguments

X vector or dataset of positive numbers
parts vector containing the indices xor names of the columns to be used
total a numeric vectors giving the total amounts of each dataset.

Details

Many multivariate datasets essentially describe amounts of D different parts in a whole. When the whole is large in relation to the considered parts, such that they do not exclude each other, or when the total amount of each componenten is indeed determined by the phenomenon under investigation and not by sampling artifacts (such as dilution or sample preparation), then the parts can be treated as amounts rather than as a composition (cf. acomp, rcomp).
Like compositions, amounts have some important properties. Amounts are always positive. An amount of exactly zero essentially means that we have a substance of an other quality. Different amounts - spanning different orders of magnitude - are often given in different units (ppm, ppb, %) and conversion factors need not to be fixed (e.g. for ppm, g/l, vol.%, mass %, molar fraction). Often, these amounts are also taken as indicators of other non-measured components (e.g. K as indicator for potassium feldspar), which might be proportional to the measured amount. However, in contrast to compositions, amounts themselves do matter. Amounts are typically heavily skewed and in many practical cases a log-transform makes their distribution roughly symmetric, even normal.
In full analogy to Aitchison's compositions, we introduce vector space operations for amounts: the perturbation perturbe.aplus as a vector space addition (corresponding to change of units), the power transformation power.aplus as scalar multiplication describing the law of mass action, and a distance dist which is independent of the chosen units. The induced vector space is mapped isometrically to a classical R^D by a simple log-transformation called ilt, resembling classical log transform approaches.
The general approach in analysing aplus objects is thus to performe classical multivariate analysis on ilt-transformed coordinates and to backtransform or display the results in such a way that they can be interpreted in terms of the original amounts.
The class aplus is complemented by the rplus, allowing to analyse amounts directly as real numbers, and by the classes acomp and rcomp to analyse the same data as compositions disregarding the total amounts, focusing on relative amounts only.
The classes rcomp, acomp, aplus, and rplus are designed as similar as possible in order to allow direct comparison between results achieved by the different approaches. Especially the acomp simplex transforms clr, alr, ilr are mirrored in the aplus class by the single bijective isometric transform ilt

Value

a vector of class "aplus" representing a vector of amounts or a matrix of class "aplus" representing multiple vectors of amounts, each vector in one row.

Author(s)

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

References

See Also

ilt,acomp, rplus, princomp.aplus, plot.aplus, boxplot.aplus, barplot.aplus, mean.aplus, var.aplus, variation.aplus, cov.aplus, msd

Examples

data(SimulatedAmounts)
plot(aplus(sa.lognormals))

[Package compositions version 0.9-11 Index]