Tf–idf
In information retrieval, tf–idf is a measure of importance of a word to a document in a collection or corpus, adjusted for the fact that some words appear more frequently in general. Like the bag-of-words model, it models a document as a multiset of words, without word order. It is a refinement over the simple bag-of-words model, by allowing the weight of words to depend on the rest of the corpus.
It was often used as a weighting factor in searches of information retrieval, text mining, and user modeling. A survey conducted in 2015 showed that 83% of text-based recommender systems in digital libraries used tf–idf. Variations of the tf–idf weighting scheme were often used by search engines as a central tool in scoring and ranking a document's relevance given a user query. In applied search engine optimization practice, tf–idf is also described as a method for analysing term importance within web pages and supporting semantic SEO techniques. One of the simplest ranking functions is computed by summing the tf–idf for each query term; many more sophisticated ranking functions are variants of this simple model.
Motivations
conceived a statistical interpretation of term-specificity called Inverse Document Frequency, which became a cornerstone of term weighting:For example, the df and idf for some words in Shakespeare's 37 plays might be represented as follows:
| Word | df | idf |
| Romeo | 1 | 1.57 |
| salad | 2 | 1.27 |
| Falstaff | 4 | 0.966 |
| forest | 12 | 0.489 |
| battle | 21 | 0.246 |
| wit | 34 | 0.037 |
| fool | 36 | 0.012 |
| good | 37 | 0 |
| sweet | 37 | 0 |
We see that "Romeo", "Falstaff", and "salad" appears in very few plays, so seeing these words, one could get a good idea as to which play it might be. In contrast, "good" and "sweet" appears in every play and are completely uninformative as to which play it is.
Definition
- The tf–idf is the product of two statistics, term frequency and inverse document frequency. There are various ways for determining the exact values of both statistics.
- A formula that aims to define the importance of a keyword or phrase within a document or a web page.
Term frequency
where is the raw count of a term in a document, i.e., the number of times that term occurs in document. Note the denominator is simply the total number of terms in document . There are various other ways to define term frequency:
- the raw count itself:
- Boolean "frequencies": if occurs in and 0 otherwise;
- logarithmically scaled frequency: ;
- augmented frequency, to prevent a bias towards longer documents, e.g. raw frequency divided by the raw frequency of the most frequently occurring term in the document:
Inverse document frequency
| weighting scheme | idf weight |
| unary | 1 |
| inverse document frequency | |
| inverse document frequency smooth | |
| inverse document frequency max | |
| probabilistic inverse document frequency |
The inverse document frequency is a measure of how much information the word provides, i.e., how common or rare it is across all documents. It is the logarithmically scaled inverse fraction of the documents that contain the word :
with
- : is the set of all documents in the corpus
- : total number of documents in the corpus
- : number of documents where the term appears. If the term is not in the corpus, this will lead to a division-by-zero. It is therefore common to adjust the numerator to and the denominator to.
Term frequency–inverse document frequency
A high weight in tf–idf is reached by a high term frequency 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 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.
Justification of idf
Idf was introduced as "term specificity" by Karen Spärck Jones in a 1972 paper. Although it has worked well as a heuristic, its theoretical foundations have been troublesome for at least three decades afterward, with many researchers trying to find information theoretic justifications for it.Spärck Jones's own explanation did not propose much theory, aside from a connection to Zipf's law. Attempts have been made to put idf on a probabilistic footing, by estimating the probability that a given document contains a term as the relative document frequency,
so that we can define idf as
Namely, the inverse document frequency is the logarithm of "inverse" relative document frequency.
This probabilistic interpretation in turn takes the same form as that of self-information. However, applying such information-theoretic notions to problems in information retrieval leads to problems when trying to define the appropriate event spaces for the required probability distributions: not only documents need to be taken into account, but also queries and terms.
Link with information theory
Both term frequency and inverse document frequency can be formulated in terms of information theory; it helps to understand why their product has a meaning in terms of joint informational content of a document. A characteristic assumption about the distribution is that:This assumption and its implications, according to Aizawa: "represent the heuristic that tf–idf employs."
The conditional entropy of a "randomly chosen" document in the corpus, conditional to the fact it contains a specific term is:
In terms of notation, and are "random variables" corresponding to respectively draw a document or a term. The mutual information can be expressed as
The last step is to expand, the unconditional probability to draw a term, with respect to the choice of a document, to obtain:
This expression shows that summing the Tf–idf of all possible terms and documents recovers the mutual information between documents and term taking into account all the specificities of their joint distribution. Each Tf–idf hence carries the "bit of information" attached to a term x document pair.
Link with statistical theory
Tf–idf is closely related to the negative logarithmically transformed p-value from a one-tailed formulation of Fisher's exact test when the underlying corpus documents satisfy certain idealized assumptions.Example of tf–idf
Suppose that we have term count tables of a corpus consisting of only two documents:| Term | Term Count |
| this | 1 |
| is | 1 |
| a | 2 |
| sample | 1 |
The calculation of tf–idf for the term "this" is performed as follows:
In its raw frequency form, tf is just the frequency of the "this" for each document. In each document, the word "this" appears once; but as the document 2 has more words, its relative frequency is smaller.
An idf is constant per corpus, and accounts for the ratio of documents that include the word "this". In this case, we have a corpus of two documents and all of them include the word "this".
So tf–idf is zero for the word "this", which implies that the word is not very informative as it appears in all documents.
The word "example" is more interesting - it occurs three times, but only in the second document:
Finally,
.