Modeling and Analysis of Compositional Data

Chapter 3The Aitchison geometry

3.1 General comments

In real space, we are used to add vectors, to multiply them by a constant or scalar value, to look for properties such as orthogonality, or to compute the distance between two points. All this, and much more, is possible because the real space is a linear vector space with a metric structure. We are familiar with its geometric structure, the Euclidean geometry, and we are used to represent our observations within this geometry. But this geometry is not a proper geometry for compositional data.

To illustrate this assertion, consider the compositions , , , and . Intuitively, we would say that the difference between and is not the same as the difference between and . The Euclidean distance between them is certainly the same, as there is ...