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

Abstract

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(N 2 ) 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.Recently, we have introduced several methodological frameworks to extend the usage of similarity measures beyond the common cases mentioned above. Most importantly, we have demonstrated that the mathematical expansion of the core concepts of similarity measures can provide a way to quantify the similarity of an arbitrary number of objects at the same time. We first showed this on binary (molecular) fingerprints: the resulting similarity measures were termed extended (or n-ary) similarity measures [15]. They employ the core concept of similarity and dissimilarity counters, which have replaced the a, b, c and d terms that are commonly applied in the well-known, pairwise definitions of the similarity measures to describe the number of bit positions where two fingerprints have co-occurring one (a) or zero (d) bits, or a one bit that is exclusive to either of the fingerprints (b and c). In our framework, the 1-similarity, 0-similarity, and dissimilarity counters express the number of bit positions where the number of co-occurring one (or zero) bits is above, or below, a predefined coincidence threshold, respectively. For pairwise comparisons, these generalizations naturally revert to the well-known definitions of the classical, pairwise similarity measures.We have shown that the new methodology is not only computationally efficient, scaling as O(n) with the number of compared objects n, but it can be successfully applied for tasks such as diversity selection, clustering, as well as the visualization of large sections of chemical space [16][17][18][19]. A further generalization involved the extension of this framework to allow for more than two possible characters (t = 2) in an object (vector), opening the possibility to apply the extended similarity measures in bioinformatics, for the comparison of nucleotide (t = 4) or prot...

Molecular Dynamics Simulations and Diversity Selection by Extended Continuous Similarity Indices

“…In fact, we regard the recently introduced extended continuous similarity indices as a set of new similarity measures altogether, since they include completely original concepts to allow for the similarity calculation of an arbitrary number of continuous vectors. 20 Nonetheless, we decided to keep the names of the existing similarity metrics that served as their basis ( e.g ., Russell-Rao, Jaccard-Tanimoto, etc . ), so that they can be more easily traced back to the widely known, “traditional” measures.…”

“…This also means that our similarity indices are capable of processing these vectors in other fields as well. 20 The real-valued vectors in this situation demand two key aspects: first, we need to find a suitable way to normalize the coordinate values, since the extended continuous indices are defined over the [0, 1] interval. There are many (in principle, infinite) ways to perform a normalization, but the nature of this problem leads to a very natural decision.…”

“…Hence, this will be the procedure that we will follow here, with the sum of each column of the normalized matrix taken as the central elements of the formalism. 20 …”

“…The algorithm is inspired by the diversity pickers commonly applied in cheminformatics to sample large chemical spaces, usually based on the use of binary molecular fingerprints. 18 The various versions of the extended similarity indices 18 − 20 have shown great promise in the problems of diversity selection 21 and exploration of large and various data sets 22 , 23 including complex biological ensembles. 24 The keys to this success are the ability of the extended indices to quantify similarities between any number of objects and the fact that they can do so with linear scaling.…”

Molecular dynamics simulations and diversity selection by extended continuous similarity indices

“…The algorithm is inspired by the diversity pickers commonly applied in cheminformatics to sample large chemical spaces, usually based on the use of binary molecular fingerprints. 18 The various versions of the extended similarity indices [18][19][20] have shown great promise in the problems of diversity selection 21 and exploration of large and various datasets 22,23 including complex biological ensembles. 24 The keys to this success are the ability of the extended indices to quantify similarities between any number of objects, and the fact that they can do so with linear scaling.…”