Multivariate Statistical Analysis of Compositional Data within the Simplex

Abstract

Many multivariate statistical tools assume the data is drawn from a continuous and unbounded sample space, such as the d-dimensional set of real numbers R. In these spaces, variables can take on values ranging from −∞ to +∞, including zero. In contrast, many of the data that appear in quantitative microanalysis experiments are drawn from more restricted spaces. The counts in an XEDS detector channel, for example, are constrained to be non-negative. It does not make sense to speak of a 10 eV wide X-ray channel with negative counts, and when commonly-applied algorithms such as Principal Components Analysis (PCA) return results with negative values, many analysts are confused. Compositional data are even more constrained than count data. Consider the microanalysis of a Ni-Al-Fe alloy in an SEM using XEDS. The measured composition at each pixel on the sample is expressed in terms of the mass fraction of each component of the mixture, such as Ni0.25Al0.5Fe0.25. The mass fraction data for each component is constrained to vary from 0 to 1, an extremely narrow range. To complicate matters further, the three component values are jointly constrained to sum to one, i.e. xNi + xAl + xFe = 1. Mathematically, the components are said to fall within a simplex, a severely restricted subspace of R.