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In probability theory and statistics, the Zipf–Mandelbrot law is a discrete probability distribution. Also known as the Pareto-Zipf law, it is a power-law distribution on ranked data, named after the linguist George Kingsley Zipf who suggested a simpler distribution called Zipf's law, and the mathematician Benoît Mandelbrot, who subsequently generalized it.
The probability mass function is given by:
where HN,q,s is given by:
which may be thought of as a generalization of a harmonic number. In the limit as N approaches infinity, this becomes the Hurwitz zeta function ζ(q,s). For finite N and q = 0 the Zipf–Mandelbrot law becomes Zipf's law. For infinite N and q = 0 it becomes a Zeta distribution.
Applications
The distribution of words ranked by their frequency in a random text corpus is generally a power-law distribution, known as Zipf's law.
If one plots the frequency rank of words contained in a large corpus of text data versus the number of occurrences or actual frequencies, one obtains a power-law distribution, with exponent close to one (but see Gelbukh and Sidorov 2001).
References and links
- B. Mandelbrot (1965). "Information Theory and Psycholinguistics". in B.B. Wolman and E. Nagel. Scientific psychology, Basic Books. Reprinted as
- B. Mandelbrot (1968). "Information Theory and Psycholinguistics". in R.C. Oldfield and J.C. Marchall. Language, Penguin Books.
- Z. K. Silagadze: Citations and the Zipf-Mandelbrot's law
- NIST: Zipf's law
- W. Li's References on Zipf's law
- Gelbukh and Sidorov 2001: Zipf and Heaps Laws’ Coefficients Depend on Language
Wikipedia content modification information:
- This page was last modified on 6 October 2008, at 04:07.
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