# Gaussian Asymptotic Limits for the *α*-transformation in the Analysis of Compositional Data

## Abstract

Compositional data consists of vectors of proportions whose components sum to 1. Such vectors lie in the standard simplex, which is a manifold with boundary. One issue that has been rather controversial within the field of compositional data analysis is the choice of metric on the simplex. One popular possibility has been to use the metric implied by log-transforming the data, as proposed by Aitchison (*Biometrika***70**, 57–65, 1983, 1986) and another popular approach has been to use the standard Euclidean metric inherited from the ambient space. Tsagris et al. (2011) proposed a one-parameter family of power transformations, the *α*-transformations, which include both the metric implied by Aitchison’s transformation and the Euclidean metric as particular cases. Our underlying philosophy is that, with many datasets, it may make sense to use the data to help us determine a suitable metric. A related possibility is to apply the *α*-transformations to a parametric family of distributions, and then estimate *α* along with the other parameters. However, as we shall see, when one follows this last approach with the Dirichlet family, some care is needed in a certain limiting case which arises (*α* → 0), as we found out when fitting this model to real and simulated data. Specifically, when the maximum likelihood estimator of *α* is close to 0, the other parameters tend to be large. The main purpose of the paper is to study this limiting case both theoretically and numerically and to provide insight into these numerical findings.

## Keywords and phrases

Dirichlet distribution Log-ratio transformation Manifold Metric Power transformation.## AMS (2000) subject classification

Primary 62E20 Secondary 62H12## Preview

Unable to display preview. Download preview PDF.

## Notes

### Acknowledgements

This work was partially supported by EPSRC grant EP/K022547/1, for which we are grateful. Partial results from this research were obtained when the second author was a PhD student at the University of Nottingham.

## References

- Aitchison, J. (1983). Principal components analysis of compositional data.
*Biometrika***70**, 57–65.MathSciNetCrossRefzbMATHGoogle Scholar - Aitchison, J. (1986). The Statistical Analysis of Compositional Data. Monographs on Statistics and Applied Probability. Chapman & Hall Ltd, London. Reprinted in 2003 with additional material by The Blackburn Press.Google Scholar
- Baxter, M.J. (1995). Standardization and transformation in principal component analysis, with applications to archaeometry.
*Appl. Stat.***44**, 513–527.CrossRefGoogle Scholar - Baxter, M.J. (2001). Statistical modelling of artefact compositional data.
*Archaeometry***43**, 131–147.CrossRefGoogle Scholar - Baxter, M.J., Beardah, C.C., Cool, H.E.M. and Jackson, C.M. (2005). Compositional data analysis of some alkaline glasses.
*Math. Geol.***37**, 183–196.CrossRefGoogle Scholar - Baxter, M.J. and Freestone, I.C. (2006). Log-ratio compositional data analysis in archaeometry.
*Archaeometry***48**, 511–531.CrossRefGoogle Scholar - Bhattacharya, A. and Bhattacharya, R.N. (2012).
*Nonparametric inference on manifolds with applications to shape spaces*. Cambridge University Press, Cambridge.CrossRefzbMATHGoogle Scholar - Dryden, I.L., Koloydenko, A. and Zhou, D. (2009). Non-euclidean statistics for covariance matrices, with applications to diffusion tensor imaging.
*Ann. Appl. Statist.***3**, 1102–1123.MathSciNetCrossRefzbMATHGoogle Scholar - Dryden, I.L., Le, H., Preston, S.P. and Wood, A.T.A. (2014). Mean shapes, projections and intrinsic limiting distributions. [Discussion contribution].
*Journal of Statistical Planning and Inference***145**, 25–32.MathSciNetCrossRefzbMATHGoogle Scholar - Dryden, I.L. and Mardia, K.V. (1998).
*Statistical Shape Analysis*. Wiley, New York.zbMATHGoogle Scholar - Dryden, I.L. and Mardia, K.V. (2016).
*Statistical Shape Analysis with Applications in r*, 2nd edn. Wiley, New York.CrossRefzbMATHGoogle Scholar - Fisher, R.A. (1953). Dispersion on a sphere.
*Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. The Royal Society***217**, 295–305.MathSciNetCrossRefzbMATHGoogle Scholar - Fisher, N.I., Lewis, T. and Embleton, B.J.J. (1987).
*Statistical Analysis of Spherical Data*. Cambridge University Press, Cambridge.CrossRefzbMATHGoogle Scholar - Hartigan, J.A. (1975).
*Clustering Algorithms*. Wiley, New York.zbMATHGoogle Scholar - Hotz, T. and Huckemann, S. (2015). Intrinsic means on the circle: uniqueness, Locus and Asymptotics.
*Ann. Inst. Stat. Math.***67**, 177–193.MathSciNetCrossRefzbMATHGoogle Scholar - Kendall, D.G., Barden, D., Carne, T.K. and Le, H. (1999).
*Shape and Shape Theory*. Wiley, New York.CrossRefzbMATHGoogle Scholar - Mardia, K.V. (1972).
*Statistics of Directional Data*. Academic Press, London.zbMATHGoogle Scholar - Mardia, K.V. and Jupp, P.E. (2000).
*Directional Statistics*. John Wiley & Sons, Chichester.zbMATHGoogle Scholar - Scealy, J.L. and Welsh, A.H. (2014). Colours and cocktails: compositional data analysis. 2013 Lancaster lecture.
*Aust. N. Z. J. Stat.***56**, 145–169.MathSciNetCrossRefzbMATHGoogle Scholar - Small, C.G. (1996).
*The Statistical Theory of Shape*. Springer, New York.CrossRefzbMATHGoogle Scholar - Tsagris, M.T., Preston, S. and Wood, A.T.A. (2011). A data-based power transformation for compositional data. In: Proceedings of the 4th Compositional Data Analysis Workshop, Girona, Spain.Google Scholar
- Tsagris, M., Preston, S. and Wood, A.T.A. (2016). Improved classification for compositional data using the
*α*-transformation.*J. Classif.***33**, 243–261.MathSciNetCrossRefzbMATHGoogle Scholar - Tsagris, M. and Stewart, C. (2018). A folded model for compositional data analysis. arXiv:1802.07330.