A New Approach to Deriving Jaccard Similarity and Jaccard Distance Properties with and without Considering Feature Weights

Jaccard similarity and Jaccard distance between feature sets find numerous applications in chemistry, bio-informatics, information retrieval and text mining. A typical task in these applications is to find most similar feature sets. While Jaccard similarity of two sets is defined as the ratio of the number of features shared by both sets to the number of features occurring in either set, their Jaccard distance is defined as 1 – Jaccard similarity of the two sets. Unlike Jaccard similarity, Jaccard distance is a metric. Hence, in particular, it preserves the triangle inequality property, which enables efficient discovery of most similar feature sets in terms of Jaccard similarity or, in other words, least distant sets in terms of Jaccard distance. Jaccard distance is a metric also when taking into account real positive weights of features. In the paper, we provide a new approach to derivation of properties of Jaccard similarity and Jaccard distance. In particular, we offer a new proof that Jaccard distance preserves the triangle inequality both in the case of not using weights of features as well as in the case of using them.