Extending compositional data analysis from a graph signal processing perspective

TL;DR

This paper introduces a novel framework that combines compositional data analysis with graph signal processing, allowing for selective consideration of variable relationships, which enhances interpretability in fields like bioinformatics and geochemistry.

Contribution

It extends Aitchison geometry to incorporate graph-based relationships, enabling analysis of specific variable pairs while maintaining key properties like scale invariance.

Findings

Framework retains scale invariance and compositional coherence.

Application examples show improved interpretability over standard methods.

Extensions to include absolute information are straightforward.

Abstract

Traditional methods for the analysis of compositional data consider the log-ratios between all different pairs of variables with equal weight, typically in the form of aggregated contributions. This is not meaningful in contexts where it is known that a relationship only exists between very specific variables (e.g.~for metabolomic pathways), while for other pairs a relationship does not exist. Modeling absence or presence of relationships is done in graph theory, where the vertices represent the variables, and the connections refer to relations. This paper links compositional data analysis with graph signal processing, and it extends the Aitchison geometry to a setting where only selected log-ratios can be considered. The presented framework retains the desirable properties of scale invariance and compositional coherence. An additional extension to include absolute information is readily made. Examples from bioinformatics and geochemistry underline the usefulness of thisapproach in comparison to standard methods for compositional data analysis.