Log-contrast and Orthonormal Log-ratio Coordinates for Compositional Data with a Total

Compositional data require an appropriate statistical analysis because they provide the relative importance of the parts of a whole. Methods based on log-ratio coordinates give a consistent framework for analyzing this type of data. Any statistical model including variables created using the original parts should be formulated according to the geometry of the simplex. This geometry includes the log-contrast: a simple way to express a set of log-ratios in a linear form. Basic concepts and properties of log-ratios, log-contrasts, and orthonormal coordinates are revisited. In addition, we introduce an approach that includes both the log-ratio orthonormal coordinates and an auxiliary variable carrying absolute information. We illustrate the approach through the principal component analysis and discriminant analysis of real data sets.