The cpt-transform maps a composition in the D-part real-simplex
isometrically to a D-1 dimensional euclidian vector space, identified with a plane parallel
to the simplex but passing through the origin. However the
transformation is not injective and does not even reach the whole
plane. Thus resulting covariance matrices are always singular.
The data can then
be analysed in this transformed space by all classical multivariate
analysis tools not relying on a full rank of the covariance matrix. See
ipt and apt for alternatives. The
interpretation of the results is relatively easy since the relation of each
transformed component to the original parts is preserved.
The centered planar transform is given by
$$ cpt(x)_i := clo(x)_i - \frac1D $$
References
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, tools:::Rd_expr_doi("10.1016/j.cageo.2006.11.017").