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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.
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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 & Sidorov, 2001).
In ecological field studies, the relative abundance distribution (i.e. the graph of the number of species observed as a function of their abundance) is often found to conform to a Zipf-Mandelbrot law.[1]
Notes
- ^ Mouillot, D; Lepretre, A (2000). "Introduction of relative abundance distribution (RAD) indices, estimated from the rank-frequency diagrams (RFD), to assess changes in community diversity". Environmental Monitoring and Assessment (Springer) 63 (2): 279-295. http://cat.inist.fr/?aModele=afficheN&cpsidt=1411186. Retrieved 24 Dec 2008.
References
- Mandelbrot, Benoît (1965). "Information Theory and Psycholinguistics". in B.B. Wolman and E. Nagel. Scientific psychology. Basic Books. Reprinted as
- Mandelbrot, Benoît (1968) [1965]. "Information Theory and Psycholinguistics". in R.C. Oldfield and J.C. Marchall. Language. Penguin Books.
- Zipf, George Kingsley (1932). Selected Studies of the Principle of Relative Frequency in Language. Cambridge, MA: Harvard University Press.
External links
- Z. K. Silagadze: Citations and the Zipf-Mandelbrot's law
- NIST: Zipf's law
- W. Li's References on Zipf's law
- Gelbukh & Sidorov, 2001: Zipf and Heaps Laws’ Coefficients Depend on Language
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