Modeling and Analysis of Compositional Data

Chapter 4Coordinate representation

4.1 Introduction

To introduce transformations based on ratios, Aitchison (1986) used the fact that, for compositional data, size is irrelevant—as interest lies in relative proportions of the components measured. The essential transformations introduced by him are the additive logratio transformation () and the centered logratio transformation (). He applied classical statistical analysis to the transformed observations, using the transformation for modeling, and the transformation for those techniques based on a metric. Although not explicit, the underlying reason was that the transformation does not preserve distances, whereas the transformation preserves distances but leads to a singular covariance matrix. In mathematical terms, we say that the