|
Cosine similarity is a measure of similarity between two vectors of an inner product space that measures the cosine of the angle between them. The cosine of 0° is 1, and it is less than 1 for any other angle. It is thus a judgment of orientation and not magnitude: two vectors with the same orientation have a cosine similarity of 1, two vectors at 90° have a similarity of 0, and two vectors diametrically opposed have a similarity of -1, independent of their magnitude. Cosine similarity is particularly used in positive space, where the outcome is neatly bounded in (). Note that these bounds apply for any number of dimensions, and cosine similarity is most commonly used in high-dimensional positive spaces. For example, in information retrieval and text mining, each term is notionally assigned a different dimension and a document is characterised by a vector where the value of each dimension corresponds to the number of times that term appears in the document. Cosine similarity then gives a useful measure of how similar two documents are likely to be in terms of their subject matter.〔Singhal, Amit (2001). "Modern Information Retrieval: A Brief Overview". Bulletin of the IEEE Computer Society Technical Committee on Data Engineering 24 (4): 35–43.〕 The technique is also used to measure cohesion within clusters in the field of data mining.〔P.-N. Tan, M. Steinbach & V. Kumar, "Introduction to Data Mining", , Addison-Wesley (2005), ISBN 0-321-32136-7, chapter 8; page 500.〕 ''Cosine distance'' is a term often used for the complement in positive space, that is: . It is important to note, however, that this is not a proper distance metric as it does not have the triangle inequality property and it violates the coincidence axiom; to repair the triangle inequality property while maintaining the same ordering, it is necessary to convert to angular distance (see below.) One of the reasons for the popularity of cosine similarity is that it is very efficient to evaluate, especially for sparse vectors, as only the non-zero dimensions need to be considered. ==Definition== The cosine of two vectors can be derived by using the Euclidean dot product formula: : Given two vectors of attributes, ''A'' and ''B'', the cosine similarity, ''cos(θ)'', is represented using a dot product and magnitude as : , where and are components of vector and respectively. The resulting similarity ranges from −1 meaning exactly opposite, to 1 meaning exactly the same, with 0 indicating orthogonality (decorrelation), 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°. If the attribute vectors are normalized by subtracting the vector means (e.g., ), the measure is called centered cosine similarity and is equivalent to the Pearson Correlation Coefficient. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「cosine similarity」の詳細全文を読む スポンサード リンク
|