Jaccard

The Jaccard index, also known as the Jaccard similarity coefficient, is a statistic used to measure the similarity between two finite sets. For sets A and B, it is defined as J(A,B) = |A ∩ B| / |A ∪ B|. The value lies between 0 and 1, with J(A,B) = 1 when the sets are identical and J(A,B) = 0 when they have no elements in common. When both sets are empty, the index is conventionally defined as 1. The Jaccard distance, D(A,B) = 1 − J(A,B), serves as a dissimilarity metric.

Named after the Swiss botanist Paul Jaccard, the measure was introduced in the early 20th century to compare species distributions. Since then it has become widely used across disciplines, including ecology, genomics, text mining, and machine learning, for comparing presence/absence data or binary feature vectors. It emphasizes shared positives and is insensitive to shared negatives, which makes it suited for sparse data.

Extensions of the Jaccard index include the weighted or generalized Jaccard, which accommodates nonnegative real-valued data. For vectors a and b with nonnegative components, the weighted Jaccard is defined as sum_i min(a_i, b_i) / sum_i max(a_i, b_i). This variant is applied to multisets and to applications requiring abundance or frequency information. In practice, the Jaccard statistic is implemented in many software libraries for evaluating similarity between samples, sets, or binary attributes.