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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 is given by:
which may be thought of as a generalization of a harmonic number. In the formula, k is the rank of the data, and q and s are parameters of the distribution. In the limit as approaches infinity, this becomes the Hurwitz zeta function . For finite and the Zipf–Mandelbrot law becomes Zipf's law. For infinite and it becomes a Zeta distribution.
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.
Within music, many metrics of measuring "pleasing" music conform to Zipf–Mandlebrot distributions.
- ↑ 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 63 (2): 279–295.
- ↑ Manris, B, Vaughan, D, Wagner, CS, Romero, J, Davis, RB. Evolutionary Music and the Zipf-Mandelbrot Law: Developing Fitness Functions for Pleasant Music. Proceedings of 1st European Workshop on Evolutionary Music and Art (EvoMUSART2003) 611.
- Mandelbrot, Benoît (1965). "Information Theory and Psycholinguistics" B.B. Wolman and E. Nagel Scientific psychology, Basic Books. Reprinted as
- Mandelbrot, Benoît  (1968). "Information Theory and Psycholinguistics" 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.
- 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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