Estimation of colour volumes as concave hypervolumes using α‐shapes

Abstract

Abstract Organisms often display multiple colour patches and for many analyses, it may be useful to take into account all these patches at the same time, and reconstruct the colour volume of the organisms. Stoddard and Prum (2008, The American Naturalist, 171, 755) proposed to use convex hulls to reconstruct the colour volume of a species. Convex hull volume has since then often been used as an index of colourfulness, and the intersection of multiple convex hulls is used to study the colour similarity between two objects. In this article, I outline the limitations of convex hulls in this context. In particular, multiple studies have reported that the convex hull overestimates the actual colour volume. I argued for the use of a more general tool, developed as a more flexible extension of the convex hulls: α‐shapes. Depending on the parameter α, α‐shapes can reconstruct concave (i.e. non‐convex) volumes with voids or pockets that are better suited for the estimation of colour volumes. To determine the optimal value of the parameter α, I point out two expected properties of multidimensional trait spaces, which translate into two conditions providing a lower and upper bound on α, and I propose technical tools to identify the α value satisfying these two conditions. Using colour data from the whole bird community from the biological station of the Nouragues, French Guiana, I show that using α‐shapes rather than convex hulls results in possibly major differences in the estimation of the colour volume. I discuss possible future developments of this new framework in both colour science and other areas of ecology dealing with multidimensional trait spaces, such as community ecology where α‐shape volumes could serve as a replacement for the functional richness FRic, or morphometrics.