# Orthonormal pairwise logratio selection (OPALS) algorithm for compositional data analysis in high dimensions

## TL;DR

The paper introduces the OPALS algorithm to efficiently analyze high-dimensional compositional data using pairwise logratios.

## Contribution

The novel contribution is an algorithm that computes orthonormal pairwise logratios efficiently using Latin squares theory.

## Key findings

- OPALS reduces computational burden for high-dimensional compositional data analysis.
- The algorithm is demonstrated on molecular biology data examples.
- Relationships between orthonormal pairwise logratios and pivot coordinates are explored in regression and classification.

## Abstract

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.

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.