Robust principal component analysis for compositional tables

Abstract

A data table arranged according to two factors can often be considered a compositional table. An example is the number of unemployed people, split according to gender and age classes. Analyzed as compositions, the relevant information consists 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 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 (rPCA) is performed for dimension reduction, allowing to investigate relationships between the factors. The link between orthonormal coordinates and clr coefficients enables to apply rPCA, which would otherwise suffer from the singularity of clr coefficients.

Keywords:

• Compositional data
• compositional table
• robust principal component analysis
• independence table
• interaction table
• pivot coordinates

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by COST Action CRoNoS [grant number IC1408] and Mathematical Models of the Internal Grant Agency of the Palacký University in Olomouc [grant number IGA_PrF_2019_006].