iSIM-Sigma: Efficient Standard Deviation Calculation for Molecular Similarity

Abstract

The average and variance of the molecular similarities in a set are of high value and useful for cheminformatics tasks such as chemical space exploration and subset selection. However, the calculation of the variance of the complete similarity matrix has a quadratic complexity, O(N²). As the sizes of molecular libraries constantly increase, this pairwise approach is unfeasible. In this work, we present an approach to calculate the exact standard deviation of molecular similarities in a set (with N molecules and M features) for the Russell–Rao (RR) and Sokal–Michener (SM) similarity indexes in O(NM²) complexity. Furthermore, we present a highly accurate linear complexity approximation, O(N), based on sampling representative molecules from the set. The proposed approximation can be extended to other similarity indices, including the popular Jaccard–Tanimoto (JT). With only the sampling of 50 molecules, the proposed method can estimate the standard deviation of similarities in a set with an RMSE lower than 0.01 for sets of up to 50,000 molecules. In comparison, random sampling does not warrant a good approximation with the same number of selected molecules as shown in our results.