Jaccard Distance

Jaccard Distance is a measure used to quantify the dissimilarity between two sets. It is derived from the Jaccard index (also known as the Jaccard similarity coefficient), which measures the similarity between finite sample sets. The Jaccard index is calculated as the size of the intersection divided by the size of the union of the sample sets.

The Jaccard Distance, which quantifies how dissimilar two sets are, is calculated as the complement of the Jaccard index. It is defined as:

$\text{Jaccard Distance} = 1 - \text{Jaccard Index}$

Or, in terms of set notation:

$\text{Jaccard Distance} (A, B) = 1 - \frac{|A \cap B|}{|A \cup B|}$

where:

• $A$ and $B$ are two sets,
• $|A \cap B|$ is the size of the intersection of the sets $A$ and $B$, and
• $|A \cup B|$ is the size of the union of the sets $A$ and $B$.

The Jaccard Distance ranges from 0 to 1, where 0 indicates that the sets are identical, and 1 indicates that the sets have no elements in common. This measure is widely used in various fields such as computational biology, information retrieval, and machine learning, particularly in clustering and similarity measurement tasks.