Compositional data

In statistics, compositional data are quantitative descriptions of the parts of some whole, conveying relative information. Mathematically, compositional data is represented by points on a simplex. Measurements involving probabilities, proportions, percentages, and ppm can all be thought of as compositional data.

All analyses of real data deals with noisy measurements. The simpler case is in empirical studies such as questionnaire surveys, where each respondent's total allocation is fixed and one must deal solely with the effects of closure, odds and odds ratios serve as invariant statistics. In such cases, associations are determined solely by the magnitude of the odds ratio relative to 1. But in compositional data containing noise, the noise is present in the total sum and is distributed asymmetrically among the individual components. Since noise and signal are inseparable, the data can only be dichotomized into zeros and noise-contaminated components. Therefore, without explicit noise filtering, even statistics based on odds remain mathematically ill-defined. Indeed, odds and their ratios can only be calculated across samples in the absolute absence of any noise other than that induced by the compositionality itself (except for normalization).