SUMMARY: In the analysis of compositional data, the most fundamental information is conveyed by the pairwise logratios between components. While logratio coordinate representations, such as balances and pivot coordinates, are widely used to aggregate such information into higher-level relationships, there are instances where a fine-grained representation using all pairwise logratios can be advantageous. Performing this within an orthonormal (or orthogonal) logratio coordinate framework becomes particularly challenging for high-dimensional compositions, since a composition with D parts results in D ( D - 1 ) / 2 pairwise logratios (excluding reciprocals). This work presents an efficient algorithm (OPALS) based on Latin squares theory to obtain all orthonormal pairwise logratios from just D - 1 logratio coordinate systems. Thus, the computational burden associated with using such representation for data analysis and modelling in high dimensions is notably alleviated, or even made feasible. Moreover, the relationship between estimates from orthonormal pairwise logratios and ordinary pivot coordinates is discussed in the context of regression and classification analysis. AVAILABILITY AND IMPLEMENTATION: The OPALS algorithm is described in detail in this article and can be implemented directly from the provided methodology. The performance and properties of the method are illustrated through two examples using contemporary molecular biology data.

Department of Algebra and Geometry Faculty of Science Palacký University Olomouc Olomouc 77146 Czech Republic

Department of Computer Science Applied Mathematics and Statistics University of Girona Girona 17003 Spain

Department of mathematical analysis and applications of mathematics Faculty of Science Palacký University Olomouc Olomouc 77146 Czech Republic

Institute of Active Lifestyle Faculty of Physical Culture Palacký University Olomouc Olomouc 77111 Czech Republic

School of Business FHNW University of Applied Sciences and Arts Northwestern Switzerland Olten 4600 Switzerland

Université Paris Saclay INRAE Jouy en Josa MGP 78350 France

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