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**Zipf–Mandelbrot**
Probability mass function
Cumulative distribution function
Parameters	$N \in \{1,2,3\ldots\}$ (integer) $q \in [0;\infty)$ (real) $s>0\,$ (real)
Support	$k \in \{1,2,\ldots,N\}$
Template:Probability distribution/link mass	$\frac{1/(k+q)^s}{H_{N,q,s}}$
cdf	$\frac{H_{k,q,s}}{H_{N,q,s}}$
Mean	$\frac{H_{N,q,s-1}}{H_{N,q,s}}-q$
Median
Mode	$1\,$
Variance
Skewness
Kurtosis
Entropy
mgf
Char. func.

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:

$f(k;N,q,s)=\frac{1/(k+q)^s}{H_{N,q,s}}$

where $H_{N,q,s}$ is given by:

$H_{N,q,s}=\sum_{i=1}^N \frac{1}{(i+q)^s}$

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 $N$ approaches infinity, this becomes the Hurwitz zeta function $\zeta(q,s)$ . 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.

ApplicationsEdit

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]

Within music, many metrics of measuring "pleasing" music conform to Zipf–Mandlebrot distributions.^[2]

NotesEdit

↑ 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.

ReferencesEdit

Mandelbrot, Benoît (1965). "Information Theory and Psycholinguistics" B.B. Wolman and E. Nagel Scientific psychology, Basic Books. Reprinted as
- Mandelbrot, Benoît [1965] (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.

External linksEdit

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[0] 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.

[1] 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.

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[2]

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Zipf–Mandelbrot law

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