Tf*idf - Mathematical Details

Mathematical Details

Tf–idf is the product of two statistics, term frequency and inverse document frequency. Various ways for determining the exact values of both statistics exist. In the case of the term frequency tf(t,d), the simplest choice is to use the raw frequency of a term in a document, i.e. the number of times that term t occurs in document d. If we denote the raw frequency of t by f(t,d), then the simple tf scheme is tf(t,d) = f(t,d). Other possibilities include

• boolean "frequencies": tf(t,d) = 1 if t occurs in d and 0 otherwise;
• logarithmically scaled frequency: tf(t,d) = log (f(t,d) + 1);
• augmented frequency, to prevent a bias towards longer documents, e.g. raw frequency divided by the maximum raw frequency of any term in the document:

The inverse document frequency is a measure of whether the term is common or rare across all documents. It is obtained by dividing the total number of documents by the number of documents containing the term, and then taking the logarithm of that quotient.

with

• : cardinality of D, or the total number of documents in the corpus
• : number of documents where the term appears (i.e., ). If the term is not in the corpus, this will lead to a division-by-zero. It is therefore common to adjust the formula to .

Mathematically the base of the log function does not matter and constitutes a constant multiplicative factor towards the overall result.

Then tf–idf is calculated as

A high weight in tf–idf is reached by a high term frequency (in the given document) and a low document frequency of the term in the whole collection of documents; the weights hence tend to filter out common terms. Since the ratio inside the idf's log function is always greater than or equal to 1, the value of idf (and tf-idf) is greater than or equal to 0. As a term appears in more documents, the ratio inside the logarithm approaches 1, bringing the idf and tf-idf closer to 0.

Various (mathematical) forms of the tf–idf term weight can be derived from a probabilistic retrieval model that mimicks human relevance decision making.