Cosine similarity

Refers to measuring the angular distance (cosine) between two vectors. Cosine of two vectors can be easily derived by using the Euclidean Dot Product formula:

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

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

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

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.