Multidimensional Scaling of Varietal Data in Sedimentary Provenance Analysis. Issue 3 (14th March 2023) - Vermeesch, P.; Lipp, A. G.; Hatzenbühler, D.; Caracciolo, L.; Chew, D.
Abstract: Varietal studies of sedimentary provenance use the properties of individual minerals or mineral groups. These are recorded as lists of numerical tables that can be difficult to interpret. Multidimensional Scaling (MDS) is a popular multivariate ordination technique for analyzing other types of provenance data based on, for example, detrital geochronology or petrography. Applying MDS to varietal data would allow them to be treated on an equal footing with those other provenance proxies. MDS requires a method to quantify the dissimilarity between two samples. This paper introduces three ways to do so. The first method ("treatment‐by‐row") turns lists of (compositional) data tables into lists of vectors, using principal component analysis. These lists of vectors can then be treated as "distributional" data and subjected to MDS analysis using dissimilarity measures such as the Kolmogorov‐Smirnov statistic. The second method ("treatment‐by‐column") turns lists of compositional data tables into multiple lists of vectors, each representing a single component of the varietal data. These multiple distributional data sets are subsequently subjected to Procrustes analysis or 3‐way MDS. The third method uses the Wasserstein‐2 distance to jointly compare the rows and columns of varietal data. This arguably makes the best use of the data but acts more like a "black box" than the other two methods. Applying the three methods to a detrital titanite data set from Colombia yields similar results. After converting varietal data to dissimilarity matrices, they can be combined with other types of provenance data, again using Procrustes analysis or 3‐way MDS. Plain Language Summary: The source of modern or ancient sediment can be determined by examining either the overall characteristics of the sediment or the chemical composition of individual sediment particles. With the help of recent analytical advancements, geologists can now analyze the complete chemical makeup of single grains of sand or silt. These types of data sets, known as "varietal" data sets, have the ability to uncover differences between sediments that are not visible through traditional methods. However, varietal data are incompatible with the statistical methods that geologists typically use to determine the origin of sediment. This paper addresses this issue by presenting three methods for quantifying the differences between varietal data sets, which is a crucial step in any further statistical analysis. Testing these methods on a varietal data set from Colombia shows similar outcomes. By using the techniques described in this paper, varietal data can now be combined with other conventional methods for determining sediment origin. Key Points: Varietal data are defined as lists of compositional tables Given an appropriate dissimilarity measure, varietal data can be subjected to multidimensional scaling This paper introduces three ways to quantify the pairwise dissimilarity of varietal data