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

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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. Orig- inal 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 con- sequences 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.
Keywords
Aitchison geometry, Central tendency, Functional equation, Geostatistics, Multivariate covariance function model

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Please use this url to cite or link to this publication:

Chicago
Allard, Denis, and Thierry Marchant. 2018. “Means and Covariance Functions for Geostatistical Compositional Data : an Axiomatic Approach.” Mathematical Geosciences .
APA
Allard, D., & Marchant, T. (2018). Means and covariance functions for geostatistical compositional data : an axiomatic approach. MATHEMATICAL GEOSCIENCES .
Vancouver
1.
Allard D, Marchant T. Means and covariance functions for geostatistical compositional data : an axiomatic approach. MATHEMATICAL GEOSCIENCES . Springer Nature; 2018;
MLA
Allard, Denis, and Thierry Marchant. “Means and Covariance Functions for Geostatistical Compositional Data : an Axiomatic Approach.” MATHEMATICAL GEOSCIENCES (2018): n. pag. Print.
@article{8546246,
  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. Orig- inal 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 con- sequences 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.},
  author       = {Allard, Denis and Marchant, Thierry},
  issn         = {1874-8961},
  journal      = {MATHEMATICAL GEOSCIENCES },
  keyword      = {Aitchison geometry,Central tendency,Functional equation,Geostatistics,Multivariate covariance function model},
  language     = {eng},
  publisher    = {Springer Nature},
  title        = {Means and covariance functions for geostatistical compositional data : an axiomatic approach},
  url          = {http://dx.doi.org/10.1007/s11004-017-9713-y},
  year         = {2018},
}

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