Discriminant Analysis for Compositional Data Incorporating Cell-Wise Uncertainties

Abstract

Abstract: In the geosciences it is still uncommon to include measurement uncertainties into statistical methods such as discriminant analysis, but, especially for trace elements, measurement uncertainties are frequently of relevant size. Uncertainties can be reported by each measured variable, by each observation or by individual cells (i.e., each observation has an individual uncertainty for each variable). Most methods incorporating uncertainties use the uncertainties as weights for the variables or observations of the data set. The method proposed in this contribution uses variance additivity properties and generalised least squares to calculate better estimates of group variances and group means, which then influence the decision rules of linear and quadratic discrimination algorithms. This methodological framework allows incorporation of cell-wise uncertainties, and would be largely valid if the information about co-dependency between variable errors within each observation were reported. The method is also appropriate for incorporating uncertainties into compositional data sets—for example, those formed by concentrations, proportions, percentages or any other form of information about the relative abundance of a set of components forming a whole—even if such uncertainties are nearly never reported considering this compositional nature. The methods are illustrated by means of case studies with simulated data. Graphical abstract: [Figure not available: see fulltext.]