Tf–idf

Monday, 25 June 2012

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Branches of Mathematics

Pure mathematics

Applied mathematics

Motivation [link]

Suppose we have a set of English text documents and wish to determine which document is most relevant to the query "the brown cow". A simple way to start out is by eliminating documents that do not contain all three words "the", "brown", and "cow", but this still leaves many documents. To further distinguish them, we might count the number of times each term occurs in each document and sum them all together; the number of times a term occurs in a document is called its ''term frequency''.

However, because the term "the" is so common, this will tend to incorrectly emphasize documents which happen to use the word "the" more frequently, without giving enough weight to the more meaningful terms "brown" and "cow". The term "the" is not a good keyword to distinguish relevant and non-relevant documents and terms, unlike the less common words "brown" and "cow". Hence an ''inverse document frequency'' factor is incorporated which diminishes the weight of terms that occur very frequently in the collection and increases the weight of terms that occur rarely.

Mathematical details [link]

The term count in the given document is simply the number of times a given term appears in that document. This count is usually normalized to prevent a bias towards longer documents (which may have a higher term count regardless of the actual importance of that term in the document) to give a measure of the importance of the term

t

within the particular document

d

. Thus we have the term frequency

\mathrm{tf}(t,d)

. (Many variants have been suggested; see e.g. Manning, Raghavan and Schütze, p. 118.)

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.

: $\mathrm{idf}(t, D) = \log \frac{|D|}{|\{d \in D: t \in d\}|}$

with

|D|

: cardinality of D, or the total number of documents in the corpus

|\{d \in D: t \in d\}|

: number of documents where the term

t

appears (i.e.,

\mathrm{tf}(t,d) \neq 0

). 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

1 + |\{d \in D: t \in d\}|

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

Then the tf*idf is calculated as

: $\mathrm{tf*idf}(t,d,D) = \mathrm{tf}(t,d) \times \mathrm{idf}(t, D)$

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 1, the value of idf (and tf-idf) is greater than 0. As a term appears in more documents then ratio inside the log approaches 1 and making idf and tf-idf approaching 0. If a 1 is added to the denominator, a term that appears in all documents will have negative idf, and a term that occurs in all but one document will have an idf equal to zero.

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

Example [link]

Consider a document containing 100 words wherein the word ''cow'' appears 3 times. Following the previously defined formulas, the term frequency (TF) for ''cow'' is then (3 / 100) = 0.03. Now, assume we have 10 million documents and ''cow'' appears in one thousand of these. Then, the inverse document frequency is calculated as log(10 000 000 / 1 000) = 4. The tf*idf score is the product of these quantities: 0.03 × 4 = 0.12.

References [link]

External links [link]

Gensim is a Python+NumPy framework for Vector Space modelling. It contains incremental (memory-efficient) algorithms for Tf–idf, Latent Semantic Indexing and Latent Dirichlet Allocation.

Term Weighting Approaches in Automatic Text Retrieval

Robust Hyperlinking: An application of tf–idf for stable document addressability.

A demo of using tf–idf with PHP and Euclidean distance for Classification

Anatomy of a search engine

tf–idf support in scikit-learn

Text to Matrix Generator (TMG) MATLAB toolbox that can be used for various tasks in text mining (TM) specifically i) indexing, ii) retrieval, iii) dimensionality reduction, iv) clustering, v) classification. The indexing step offers the user the ability to apply local and global weighting methods, including tf–idf.

Pyevolve: A tutorial series explaining the tf-idf calculation.

Category:Information retrieval Category:Artificial intelligence applications Category:Statistical natural language processing Category:Ranking functions

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