Cosine similarity is a measure of similarity between two vectors by measuring the cosine of the angle between them. The result of the Cosine function is equal to 1 when the angle is 0, and it is less than 1 when the angle is of any other value. Calculating the cosine of the angle between two vectors thus determines whether two vectors are pointing in roughly the same direction.
This is often used to compare documents in text mining. In addition, it is used to measure cohesion within clusters in the field of Data Mining.
Cosine of two vectors can be easily derived by using the Euclidean Dot Product formula:
Given two vectors of attributes, A and B, the cosine similarity, θ, is represented using a dot product and magnitude as
The resulting similarity ranges from −1 meaning exactly opposite, to 1 meaning exactly the same, with 0 usually indicating independence, and in-between values indicating intermediate similarity or dissimilarity.
For text matching, the attribute vectors A and B are usually the term frequency vectors of the documents. The cosine similarity can be seen as a method of normalizing document length during comparison.
In the case of information retrieval, the cosine similarity of two documents will range from 0 to 1, since the term frequencies (tf-idf weights) cannot be negative. The angle between two term frequency vectors cannot be greater than 90°.
This cosine similarity metric may be extended such that it yields the Jaccard coefficient in the case of binary attributes. This is the Tanimoto coefficient, T(A, B), represented as
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