Vector Space Model

Vector space model or term vector model is an algebraic model for representing text documents (and any objects, in general) as vectors of identifiers, such as, for example, index terms. It is used in information filtering, information retrieval, indexing and relevancy rankings. Its first use was in the SMART Information Retrieval System.

Documents and queries are represented as vectors.

Each dimension corresponds to a separate term. If a term occurs in the document, its value in the vector is non-zero. Several different ways of computing these values, also known as (term) weights, have been developed. One of the best known schemes is tf-idf weighting (see the example below).

The definition of term depends on the application. Typically terms are single words, keywords, or longer phrases. If words are chosen to be the terms, the dimensionality of the vector is the number of words in the vocabulary (the number of distinct words occurring in the corpus).

Vector operations can be used to compare documents with queries.

Relevance rankings of documents in a keyword search can be calculated, using the assumptions of document similarities theory, by comparing the deviation of angles between each document vector and the original query vector where the query is represented as the same kind of vector as the documents.

In practice, it is easier to calculate the cosine of the angle between the vectors, instead of the angle itself:

Where ${\displaystyle \mathbf {d_{2}} \cdot \mathbf {q} }$ is the intersection (i.e. the dot product) of the document (d₂ in the figure to the right) and the query (q in the figure) vectors, ${\displaystyle \left\|\mathbf {d_{2}} \right\|}$ is the norm of vector d₂, and ${\displaystyle \left\|\mathbf {q} \right\|}$ is the norm of vector q. The norm of a vector is calculated as such:

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