Means and covariance functions for geostatistical compositional data: an axiomatic approach

TL;DR

This paper characterizes the central tendency and covariance functions for compositional data using an axiomatic approach, revealing unique properties and models that satisfy specific axioms in geostatistics.

Contribution

It establishes that the weighted arithmetic mean and proportional covariance models are uniquely determined by axioms like continuity, reflexivity, and marginal stability.

Findings

Weighted arithmetic mean is the only central tendency satisfying key axioms.

Proportional covariance models are the only models with identical kriging weights for all components.

Proportional covariance functions are compatible with key axioms for geostatistical compositional data.

Abstract

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 the 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 on the 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.