Modeling and Analysis of Compositional Data

Chapter 1Introduction

Compositional data describe parts of some whole. They are commonly presented as vectors of proportions, percentages, concentrations, or frequencies. As proportions are expressed as real numbers, one is tempted to interpret, or even analyze, them as real multivariate data. This practice can lead to paradoxes and/or misinterpretations, some of them well known even a century ago, but mostly forgotten and neglected over the years. Some simple examples illustrate the anomalous behavior of proportions when analyzed without taking into account the special characteristics of compositional data.

Example 1.1 (Intervals covering negative proportions)

Daily measurements of an air pollutant are reported as . The given interval of concentration covers a nonsensical range of concentrations that includes negative values. It is probably generated by an average of concentrations which contain some values much higher than . For instance, the following is a set of rounded random percentages: . Their mean is , while their standard deviation is . Thus a typical -interval for the ...