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Error Propagation in Isometric Log-ratio Coordinates for Compositional Data: Theoretical and Practical

Mehmet Can Mert1, Peter Filzmoser1, Karel Hron2

  • 1Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria.

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Summary
This summary is machine-generated.

Understanding measurement errors in geochemical data is crucial. This study investigates how errors affect log-ratio coordinates, offering guidance for accurate soil sample analysis.

Keywords:
Aitchison geometryCompositional differential calculusDetection limitOrthonormal coordinatesTaylor approximation

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Area of Science:

  • Geochemistry
  • Statistical analysis
  • Data science

Background:

  • Geochemical soil sample analysis often involves compositional data.
  • Log-ratio coordinates (e.g., centered log-ratio, isometric log-ratio) are essential for analyzing the relative structure of this data.
  • The impact of measurement errors and detection limits on these coordinates is not well understood.

Purpose of the Study:

  • To theoretically investigate the effect of measurement errors on log-ratio coordinate transformations in geochemistry.
  • To assess the influence of error propagation on the geometric means used in these transformations.
  • To provide practical recommendations for handling errors in compositional data analysis.

Main Methods:

  • Theoretical investigation using the theory of error propagation.
  • Simulation studies to complement theoretical limitations.
  • Analysis of how errors affect centered log-ratio and isometric log-ratio coordinates.

Main Results:

  • Measurement errors can significantly distort the presentation of geochemical data in log-ratio coordinates.
  • The extent of distortion depends on the magnitude of the error and the specific log-ratio method used.
  • Simulations provided insights into error propagation effects where theoretical approaches were limited.

Conclusions:

  • Practitioners need to be aware of potential distortions caused by measurement errors in compositional geochemical data.
  • Recommendations are provided regarding acceptable error levels and expected data distortion for different analytical purposes.
  • Careful consideration of error propagation is vital for reliable interpretation of soil geochemistry studies.