# An Affine Equivariant Multivariate Normal Score Transform for Compositional Data

- 501 Downloads
- 6 Citations

## Abstract

The geostatistical treatment of continuous variables often includes a transformation to normal scores. In the case of analysing a composition, it has been suggested that standard methods can be applied to (isometric) logratio transformed compositions. Several logratio transformations are available and invariance of the final results under the choice of logratio transform is desirable. However, a geostatistical procedure which includes marginal normal scores transformations of the individual logratio scores via quantile matching will not have this invariance property, nor will the resulting vectors of scores show a joint multivariate normal distribution. In this paper an affine-equivariant normal score transform is proposed. The method is based on a continuous deformation of the underlying logratio space to a Gaussian space. The properties and performance of this method are illustrated and compared with existing alternatives using a simulated setting and a case study from a banded iron formation ore mining operation from Western Australia. The proposed method is also suitable for the study of other multivariate non-compositional cases.

## Keywords

Additive logratio transform Flow Gaussian anamorphosis Ordinary differential equation## Notes

### Acknowledgments

The authors acknowledge financial support through the ECU CES Travel Scheme 2014 and the DAAD-UA Grant CodaBlockKriging.

## References

- Aitchison J (1986) The Statistical analysis of compositional data. Chapman & Hall Ltd., London (Reprinted in 2003 with additional material by The Blackburn Press, London)Google Scholar
- Aitchison J (1997) The one-hour course in compositional data analysis or compositional data analysis is simple. In: Pawlowsky-Glahn V (ed) Proceedings of IAMG’97—the III annual conference of the international association for mathematical geology, pp 3–35Google Scholar
- Barnett RM, Manchuk JG, Deutsch CV (2014) Projection pursuit multivariate transform. Math Geosci 46:337–360CrossRefGoogle Scholar
- Deutsch CV, Journel AG (1998) GSLIB: geostatistical software library and user’s guide. Oxford University Press, OxfordGoogle Scholar
- Egozcue JJ, Pawlowsky-Glahn V, Mateu-Figueras G, Barceló-Vidal C (2003) Isometric logratio transformations for compositional data analysis. Math Geol 35:279–300CrossRefGoogle Scholar
- Filzmoser P, Hron K (2008) Outlier detection for compositional data using robust methods. Math Geosci 40:233–248CrossRefGoogle Scholar
- Friedman JH, Tukey JW (1974) A projection pursuit algorithm for exploratory data analysis. IEEE T Comput c-23:881–890Google Scholar
- Korkmaz S, Goksuluk D, Zararsiz G (2014) MVN: an R package for assessing multivariate normality. R J 6:151–162Google Scholar
- Leuangthong O, Deutsch CV (2003) Stepwise conditional transformation for simulation of multiple variables. Math Geol 35:155–173CrossRefGoogle Scholar
- Rivoirard J (1984) Une méthode d’estimation du recuperable local multivariable. Note 894, CGMM, Mines-Paris Tech, ParisGoogle Scholar
- Szekely GJ, Rizzo ML (2013) Energy statistics: a class of statistics based on distances. J Stat Plan Infer. doi: 10.1016/j.jspi.2013.03.018 Google Scholar
- Tercan AE (1999) The importance of orthogonalization algorithm in modeling conditional distributions by orthogonal transformed indicator methods. Math Geol 31:155–173Google Scholar
- Tolosana-Delgado R (2006) Geostatistics for constrained variables: positive data, compositions and probabilities. Application to environmental hazard monitoring Dissertation, Universitat de GironaGoogle Scholar
- Ward C, Mueller U (2012) Multivariate estimation using logratios: a worked alternative. In: Abrahamsen P et al (eds) Geostatistics Oslo 2012, pp 333–343Google Scholar