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 : Binary Vector 1
         B : Binary Vector 2 
         

• Output -

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

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

$\mathbf {M_{11}} :{\text{the number of attributes where A is 1 and B is 1}}$

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

Given two objects, A and B, each with n binary attributes, the Jaccard coefficient is a useful measure of the overlap that A and B share with their attributes. Each attribute of A and B can either be 0 or 1. The total number of each combination of attributes for both A and B are specified as follows:

   M11 represents the total number of attributes where A and B both have a value of 1.
   M01 represents the total number of attributes where the attribute of A is 0 and the attribute of B is 1.
   M10 represents the total number of attributes where the attribute of A is 1 and the attribute of B is 0.
   M00 represents the total number of attributes where A and B both have a value of 0.

Used in

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