Extended continuous similarity indices: theory and application for QSAR descriptor selection

Extended (or n-ary) similarity indices have been recently proposed to extend the comparative analysis of binary strings. Going beyond the traditional notion of pairwise comparisons, these novel indices allow comparing any number of objects at the same time. This results in a remarkable efficiency gain with respect to other approaches, since now we can compare N molecules in O(N) instead of the common quadratic O(N²) timescale. This favorable scaling has motivated the application of these indices to diversity selection, clustering, phylogenetic analysis, chemical space visualization, and post-processing of molecular dynamics simulations. However, the current formulation of the n-ary indices is limited to vectors with binary or categorical inputs. Here, we present the further generalization of this formalism so it can be applied to numerical data, i.e. to vectors with continuous components. We discuss several ways to achieve this extension and present their analytical properties. As a practical example, we apply this formalism to the problem of feature selection in QSAR and prove that the extended continuous similarity indices provide a convenient way to discern between several sets of descriptors.