  A data table which is arranged according to two factors can often be
considered as a compositional table. An example is the number of unemployed
people, split according to gender and age classes. Analyzed as compositions,
the relevant information would consist of ratios between different cells of
such a table. This is particularly useful when analyzing several compositional
tables jointly, where the absolute numbers are in very different ranges, e.g.
if unemployment data are considered from different countries. Within the
framework of the logratio methodology, compositional tables can be decomposed
into independent and interactive parts, and orthonormal coordinates can be
assigned to these parts. However, these coordinates usually require some prior
knowledge about the data, and they are not easy to handle for exploring the
relationships between the given factors.
  Here we propose a special choice of coordinates with a direct relation to
centered logratio (clr) coefficients, which are particularly useful for an
interpretation in terms of the original cells of the tables. With these
coordinates, robust principal component analysis (PCA) is performed for
dimension reduction, allowing to investigate the relationships between the
factors. The link between orthonormal coordinates and clr coefficients enables
to apply robust PCA, which would otherwise suffer from the singularity of clr
coefficients.
