Parthenope University of Naples
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Mapping the Reliability of the Additive Log-Ratio Transformation
No full textData for which ratios of parts are more important than absolute values
have a compositional nature and should be analyzed in the (D − 1)-dimensional
Simplex, but in order to use standard multivariate analysis techniques they are
often mapped bijectively from the Simplex into the ordinary euclidean space.
The additive log-ratio (alr) is one popular ad hoc transformation, that has
already been shown to amplify the relative errors in data, in some cases by an
unbounded factor. In this paper an explicit mapping of the reliability of the alr
transformed data is derived, given the desired accuracy and the relative error of
the sensors acquiring them. To this purpose, first the relative condition number
for the alr transformation has been defined and plotted in the 3-Simplex, then a
generalized characterization of well conditioned compositions (with a condition
number less than or equal to an arbitrary threshold) has been derived and finally
isoconditioning surfaces have been defined, analytically derived and plotted in
the 3-Simple
