Zipf–Mandelbrot law

Zipf–Mandelbrot

Parameters	$N \in \{1,2,3\ldots\}$ (integer) $q \in [0;\infty)$ (real) $s>0\,$ (real)
Support	$k \in \{1,2,\ldots,N\}$
PMF	$\frac{1/(k+q)^s}{H_{N,q,s}}$
CDF	$\frac{H_{k,q,s}}{H_{N,q,s}}$
Mean	$\frac{H_{N,q,s-1}}{H_{N,q,s}}-q$
Mode	$1\,$

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 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 & 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.

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