A review of compositional data analysis and recent advances

Abstract

Compositional data are positive multivariate data with unity sum constraint that have emerged over the last years in numerous scientific fields. Ever since, numerous models and approaches have been proposed for analyzing such data in the last 40 years. We list some of their properties and difficulties and review many techniques proposed over this period. In particular, we focus on transformations, distributions, regression models, discriminant analysis and clustering techniques, dimensionality reduction techniques, variable selection algorithms and finally we list some books and R packages developed for compositional data analysis.

Keywords:

• Compositional data
• transformations
• distributions
• regression models
• R packages

MATHEMATICS SUBJECT CLASSIFICATION:

• 62H99

Acknowledgments

The author would like to acknowledge Michail Tsagris for their fruitful conversations and thank the reviewers for their constructive comments that significantly improved the paper.

Notes

1 See Tsagris and Stewart (2020) for many examples of applications involving compositional data.

2 On the contrary, the review of Greenacre (2021) is narrow as it deals solely on log-ratio transformation based techniques neglecting a large strand of the literature.

3 This is equal to the trace of the covariance matrix of the data after the ILR transformed data.

4 The relationship between the CLR and ALR can be found in Aitchison (2003).

5 This is the orthonormal D × D Helmert matrix (Lancaster 1965) after deletion of the first row.

6 To the best of our knowledge, this transformation has not been studied by any researcher in the context of compositional data.