Means and Covariance Functions for Geostatistical Compositional Data: an Axiomatic Approach
This work focuses on the characterization of the central tendency of a sample of compositional data. It provides new results about theoretical properties of means and covariance functions for compositional data, with an axiomatic perspective. Original results that shed new light on geostatistical modeling of compositional data are presented. As a first result, it is shown that the weighted arithmetic mean is the only central tendency characteristic satisfying a small set of axioms, namely continuity, reflexivity, and marginal stability. Moreover, this set of axioms also implies that the weights must be identical for all parts of the composition. This result has deep consequences for spatial multivariate covariance modeling of compositional data. In a geostatistical setting, it is shown as a second result that the proportional model of covariance functions (i.e., the product of a covariance matrix and a single correlation function) is the only model that provides identical kriging weights for all components of the compositional data. As a consequence of these two results, the proportional model of covariance function is the only covariance model compatible with reflexivity and marginal stability.
KeywordsAitchison geometry Central tendency Functional equation Geostatistics Multivariate covariance function model
We are truly indebted to one Advisory Editor and to the Editor-in-Chief for their very constructive comments, which helped to improve this paper.
- Aczél J (1966) Lectures on functional equations and their applications. Academic Press, New YorkGoogle Scholar
- Aitchison J (1982) The statistical analysis of compositional data. J R Stat Soc Ser B (Stat Methodol) 44(2):139–177Google Scholar
- Cressie N (1993) Statistics for spatial data, Revised edn. Wiley, LondonGoogle Scholar
- Egozcue JJ, Pawlowksy-Glahn V (2011) Basic concepts and procedures. In: Pawlowsky-Glahn V, Buccianti A (eds) Compositional data analysis: theory and applications. Wiley, LondonGoogle Scholar
- Griva I, Nash SG, Sofer A (2009) Linear and nonlinear optimization, 2nd edn. SIAMGoogle Scholar
- Kolmogorov A (1930) Sur la notion de la moyenne. Atti R Accad Naz Lincei Mem Cl Sci Fis Mat Natur Sez 12:323–343Google Scholar
- Mateu-Figueras G, Pawlowksy-Glahn V, Egozcue JJ (2011) The principle of working on coordinates. In: Pawlowsky-Glahn V, Buccianti A (eds) Compositional data analysis: theory and applications. Wiley, LondonGoogle Scholar
- Pawlowksy-Glahn V, Olea RA (2004) Geostatistical analysis of compositional data. Oxford University Press, OxfordGoogle Scholar
- Tolosana-Delgado R, van den Boogaart K, Pawlowsky-Glahn V (2011) Geostatistics for compositions. In: Pawlowsky-Glahn V, Buccianti A (eds) Compositional data analysis: theory and applications. Wiley, LondonGoogle Scholar
- Wackernagel H (2013) Multivariate geostatistics: an introduction with applications. Springer, BerlinGoogle Scholar