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Weighted Pivot Coordinates for Compositional Data and Their Application to Geochemical Mapping

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Abstract

The log ratio methodology converts compositional data, such as concentrations of chemical elements in a rock, from their original Aitchison geometry to interpretable real orthonormal coordinates, thereby allowing meaningful statistical processing and visualization. However, it must be taken into account that the original concentrations can be flawed by detection limit or imprecision problems that can severely affect the resulting coordinates. This paper aims to construct such orthonormal log ratio coordinates, called weighted pivot coordinates, that capture the relevant relative information about an original component and treat the redundant information in a controlled manner. Theoretical developments are supported by a thorough simulation study. Weighted pivot coordinates are then applied to the geochemical mapping of catchment outlet sediments from the National Geochemical Survey of Australia illustrating their advantage over possible alternatives.

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Acknowledgements

The insightful and constructive reviewer reports from Dr. Raimon Tolosana-Delgado and two anonymous reviewers are gratefully acknowledged; they helped to greatly improve the paper. The authors thank and acknowledge also the participants of the first GeoMap Workshop (held in Olomouc, Czech Republic, 17–20 June 2014) which provoked discussions leading to the present contribution. The paper was supported by the Grant COST Action CRoNoS IC1408 and the infrastructural part was supported from NPU I (LO1304).

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Authors and Affiliations

  1. Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University Olomouc, 17. listopadu 12, 77146, Olomouc, Czech Republic

    Karel Hron & Eva Fišerová

  2. Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040, Vienna, Austria

    Peter Filzmoser

  3. Research School of Earth Sciences, The Australian National University, Canberra, ACT, 2601, Australia

    Patrice de Caritat

  4. Institute of Molecular and Translational Medicine, Faculty of Medicine and Dentistry, Palacký University Olomouc, Hněvotínská 5, 77900, Olomouc, Czech Republic

    Alžběta Gardlo

  5. Department of Clinical Biochemistry, University Hospital Olomouc, I. P. Pavlova 6, 77520, Olomouc, Czech Republic

    Alžběta Gardlo

Authors
  1. Karel Hron
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  2. Peter Filzmoser
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  3. Patrice de Caritat
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  4. Eva Fišerová
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  5. Alžběta Gardlo
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Correspondence to Karel Hron.

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Hron, K., Filzmoser, P., de Caritat, P. et al. Weighted Pivot Coordinates for Compositional Data and Their Application to Geochemical Mapping. Math Geosci 49, 797–814 (2017). https://doi.org/10.1007/s11004-017-9684-z

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  • DOI: https://doi.org/10.1007/s11004-017-9684-z

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