Amounts analysed in log-scale
A class to analyse positive amounts in a logistic framework.
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vector or dataset of positive numbers
vector containing the indices xor names of the columns to be used
a numeric vectors giving the total amounts of each dataset.
should the user be warned in case of NA,NaN or 0 coding different types of missing values?
a number, vector or matrix of positive numbers giving the detection limit of all values, all columns or each value, respectively
the code for 'Below Detection Limit' in X
the code for 'Structural Zero' in X
the code for 'Missing At Random' in X
the code for 'Missing Not At Random' in X
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.
Like compositions, amounts have some important properties. Amounts are always positive. An amount of exactly zero essentially means that we have a substance of another quality. Different amounts - spanning different orders of magnitude - are often given in different units (ppm, ppb, 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, vector space operations are introduced 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 perform classical multivariate analysis on ilt-transformed coordinates (i.e., logs) 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
rcomp to analyse the same data
as compositions disregarding the total amounts, focusing on relative
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
ilr are mirrored
in the aplus class by the single bijective isometric transform
a vector of class
"aplus" representing a vector of amounts
or a matrix of class
multiple vectors of amounts, each vector in one row.
The policy of treatment of zeroes, missing values and values below detecion limit is explained in depth in compositions.missing.
Raimon Tolosana-Delgado, K.Gerald v.d. Boogaart http://www.stat.boogaart.de
van den Boogaart, K.G. and R. Tolosana-Delgado (2008) "compositions": a unified R package to analyze Compositional Data, Computers & Geosciences, 34 (4), pages 320-338, doi:10.1016/j.cageo.2006.11.017.
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