Preprints
https://doi.org/10.31223/X5TM02
https://doi.org/10.31223/X5TM02
 
14 Nov 2022
14 Nov 2022
Status: this preprint is open for discussion.

Comparing detrital age spectra, and other geological distributions, using the Wasserstein distance

Alex Lipp1 and Pieter Vermeesch2 Alex Lipp and Pieter Vermeesch
  • 1Merton College, University of Oxford, Oxford, UK
  • 2Department of Earth Sciences, University College London, London, UK

Abstract. Distributional data such as detrital age populations or grain size distributions are common in the geological sciences. As analytical techniques become more sophisticated, increasingly large amounts of distributional data are being gathered. These advances require quantitative and objective methods, such as multidimensional scaling (MDS), to analyse large numbers of samples. Crucial to such methods is choosing a sensible measure of dissimilarity between samples. At present, the Kolmogorov-Smirnov (KS) statistic is the most widely used of these dissimilarity measures. However, the KS statistic has some limitations. It is very sensitive to differences between the modes of two distributions, and relatively insensitive to differences between their tails. Here we introduce the Wasserstein-2 distance (W2) as an alternative to address this issue. Whereas the KS-distance is defined as the maximum vertical distance between two empirical cumulative distribution functions, the W2-distance is a function of the horizontal distances (i.e., age differences) between individual observations. Using a combination of synthetic examples and a published zircon U-Pb dataset, we show that the W2 distance produces similar MDS results to the KS-distance in most cases, but significantly different results in some cases. Where the results differ, the W2 results are geologically more sensible. For the case study, we find that the MDS map that is produced using W2 can be readily interpreted in terms of the shape and average age of the age spectra. The W2-distance has been added to the R package IsoplotR, for immediate use in detrital geochronology and other applications. The W2 distance can be generalised to multiple dimensions, which opens opportunities beyond distributional data.

Alex Lipp and Pieter Vermeesch

Status: open (until 26 Dec 2022)

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Alex Lipp and Pieter Vermeesch

Alex Lipp and Pieter Vermeesch

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Short summary
The Wasserstein distance is shown to be an appropriate dissimilarity metric for comparing distributional data such as detrital mineral ages. Using synthetic and real data we compare the Wasserstein distance to the commonly used Kolmogorov-Smirnov distance. The results are, in general, similar, but where they differ the Wasserstein distance is found to have more geologically sensible results. Code required to calculate the Wasserstein distance between distributions is provided in python and R.