  In compositional data analysis an observation is a vector containing
non-negative values, only the relative sizes of which are considered to be of
interest. Without loss of generality, a compositional vector can be taken to be
a vector of proportions that sum to one. Data of this type arise in many areas
including geology, archaeology, biology, economics and political science. In
this paper we investigate methods for classification of compositional data. Our
approach centres on the idea of using the $\alpha$-transformation to transform
the data and then to classify the transformed data via regularised discriminant
analysis and the k-nearest neighbours algorithm. Using the
$\alpha$-transformation generalises two rival approaches in compositional data
analysis, one (when $\alpha=1$) that treats the data as though they were
Euclidean, ignoring the compositional constraint, and another (when $\alpha=0$)
that employs Aitchison's centred log-ratio transformation. A numerical study
with several real datasets shows that whether using $\alpha=1$ or $\alpha=0$
gives better classification performance depends on the dataset, and moreover
that using an intermediate value of $\alpha$ can sometimes give better
performance than using either 1 or 0.
