Compositional Closure—Its Origin Lies Not in Mathematics but Rather in Nature Itself

Abstract

Compositional closure, spurious negative correlations in data sets of a fixed sum (e.g., fractions and percent), is often encountered in geostatistical analyses, particularly in mineralogy, petrology, and geochemistry. Techniques to minimize the effects of closure (e.g., log-ratio transformations) can provide consistent geostatistical results. However, such approaches do not remove these effects because closure does not result from mathematical operations but is an inherent property of the physical systems under study. The natural world causes physical closure; mathematics simply describes that closure and cannot alter it by manipulations. Here, we examine the distinct types of geologic systems and samples to determine in which situations closure (physical and mathematical) does or does not ensue and the reasons therefor. We parse compositional systems based on (1) types of components under study, immutable (e.g., elements) or reactive (minerals), and (2) whether the system is open or closed to component transfer. Further, open systems can be (1) displacive in which addition of a component physically crowds out others, or (2) accommodative in which addition or subtraction of components does not affect the others. Only displacive systems are subject to compositional closure. Accommodative systems, even with components expressed as percent or fractions, are not closed physically or, therefore, mathematically.