Modeling and Analysis of Compositional Data

Chapter 2Compositional data and their sample space

2.1 Basic concepts

Let us start with the definition of the central concept in this book, namely, the type of data the methods presented here have been developed for.

Definition 2.1 (-part composition)

A (row) vector, , is a -part composition when all its components are strictly positive real numbers and carry only relative information.

The meaning of relative information refers to that where the only information is contained in the ratios between the components of the composition and the numerical value of each component by itself is irrelevant. Indeed, that compositional information is relative is implicitly stated in the units, as they are usually parts of a whole. The most common examples have a constant sum and are known in the literature as closed data (Chayes, 1971). Frequently, , which means that measurements have been made in, or transformed to, parts per unit (proportions), or , for measurements represented in percentages. ...