Coherent Ratios for Compositional Data Analysis with Zeros

Compositional data (CoDa) are scale-invariant by nature, and their analysis traditionally relies on log-ratio transformations. A cornerstone of the Aitchison school is the principle of subcompositional coherence: analyses should be consistent when focusing on any subset of parts, a property that follows from the isomorphism property of the logarithm. However, zeros in the data render log-ratio transformations undefined, posing a fundamental challenge. This paper introduces a family of transformations based on a ratio-based homeomorphism, which maps extended non-negative numbers to the unit interval. By applying it to suitable ratios of components, we construct both i) local representations that require a reference component (or a set of components) to be non-zero, and ii) global representations that handle zeros in all components simultaneously. We show that the local representations preserve subcompositional coherence for subcompositions containing the reference part, a property deemed essential by the log-ratio community, while the global representations trade this coherence for the ability to accommodate zeros everywhere. Thus, no single transformation satisfies all desirable properties, but the local approach offers a principled compromise: it retains subcompositional coherence and enables a quasi-global treatment of zeros. The practical utility of the approach is illustrated on a glass dataset for predicting the refractive index, where the proposed local representation outperforms standard log-ratio methods with zero imputation, matches industry benchmarks, and yields interpretable coefficients consistent with domain knowledge. The proposed framework provides a flexible, imputation-free toolbox for analyzing compositional data with zeros, allowing analysts to choose between coherence and full coverage depending on the application, and enabling the use of standard multivariate techniques on bounded, interpretable coordinates.