# Robust Principal Component Analysis for Compositional Tables

## TL;DR

This paper introduces a robust PCA method for compositional tables that uses a special coordinate system linked to clr coefficients, facilitating interpretation and analysis of relationships between factors in compositional data.

## Contribution

It proposes a new coordinate choice related to clr coefficients, enabling robust PCA for compositional tables with improved interpretability and handling of data singularities.

## Key findings

- Enables dimension reduction in compositional tables
- Facilitates interpretation of factor relationships
- Handles data singularity issues effectively

## Abstract

A data table which is arranged according to two factors can often be considered as a compositional table. An example is the number of unemployed people, split according to gender and age classes. Analyzed as compositions, the relevant information would consist of ratios between different cells of such a table. This is particularly useful when analyzing several compositional tables jointly, where the absolute numbers are in very different ranges, e.g. if unemployment data are considered from different countries. Within the framework of the logratio methodology, compositional tables can be decomposed into independent and interactive parts, and orthonormal coordinates can be assigned to these parts. However, these coordinates usually require some prior knowledge about the data, and they are not easy to handle for exploring the relationships between the given factors. Here we propose a special choice of coordinates with a direct relation to centered logratio (clr) coefficients, which are particularly useful for an interpretation in terms of the original cells of the tables. With these coordinates, robust principal component analysis (PCA) is performed for dimension reduction, allowing to investigate the relationships between the factors. The link between orthonormal coordinates and clr coefficients enables to apply robust PCA, which would otherwise suffer from the singularity of clr coefficients.