Jaccard similarity

This is a technical method discussed in Social Media Analysis 10-802 in Spring 2010.

What problem does it address

Quantifying similarity between two vectors. Refers to measuring the angular distance (cosine) between two vectors. In text domains, a document is generally treated as a bag of words where each unique word in the vocabulary is a dimension of the vector. Thus similarity between two documents can be assessed by finding the cosine similarity between the vectors corresponding to these two documents. Each element of vector A and vector B is generally taken to be tf-idf weight.

Algorithm

• Input -

         A : Vector 1
         B : Vector 2 
         

• Output - cosine : cosine of angle between the vectors

$\mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta$

Given two vectors of attributes, A and B, the cosine similarity, θ, is represented using a dot product and magnitude as

${\text{similarity}}=\cos(\theta )={A\cdot B \over \|A\|\|B\|}={\frac {\sum _{i=1}^{n}{A_{i}\times B_{i}}}{{\sqrt {\sum _{i=1}^{n}{(A_{i})^{2}}}}\times {\sqrt {\sum _{i=1}^{n}{(B_{i})^{2}}}}}}$

Used in

Widely used for calculating the similarity of documents using the bag-of-words and vector space models