Zipf's law

Probability distribution
name =Zipf's law
type =mass
pdf_

Zipf PMF for "N" = 10 on a log-log scale. The horizontal axis is the index "k" . (Note that the function is only defined at integer values of "k". The connecting lines do not indicate continuity.)
cdf_

Zipf CMF for "N" = 10. The horizontal axis is the index "k" . (Note that the function is only defined at integer values of "k". The connecting lines do not indicate continuity.)
parameters =$s>0,$ (real)
$N\; in\; \{1,2,3ldots\}$ (integer)
support =$k\; in\; \{1,2,ldots,N\}$
pdf =$frac\{1/k^s\}\{H\_\{N,s$
cdf =$frac\{H\_\{k,s\{H\_\{N,s$
mean =$frac\{H\_\{N,s-1\{H\_\{N,s$
median =
mode =$1,$
variance =
skewness =
kurtosis =
entropy =$frac\{s\}\{H\_\{N,ssum\_\{k=1\}^Nfrac\{ln(k)\}\{k^s\}+ln(H\_\{N,s\})$
mgf =$frac\{1\}\{H\_\{N,ssum\_\{n=1\}^N\; frac\{e^\{nt\{n^s\}$
char =$frac\{1\}\{H\_\{N,ssum\_\{n=1\}^N\; frac\{e^\{int\{n^s\}$

Zipf's law, an empirical law formulated using mathematical statistics, refers to the fact that many types of data studied in the physical and social sciences can be approximated with a Zipfian distribution, one of a family of related discrete power law probability distributions. The law is named after the linguist George Kingsley Zipf (pronEng|ˈzɪf) who first proposed it (Zipf 1935, 1949), though J.B. Estoup appears to have noticed the regularity before Zipf. [ Christopher D. Manning, Hinrich Schütze "Foundations of Statistical Natural Language Processing", MIT Press (1999), ISBN 978-0262133609, p. 24] .

Motivation

Zipf's law states that given some corpus of natural language utterances, the frequency of any word is inversely proportional to its rank in the frequency table. Thus the most frequent word will occur approximately twice as often as the second most frequent word, which occurs twice as often as the fourth most frequent word, etc. For example, in the Brown Corpus "the" is the most frequently occurring word, and by itself accounts for nearly 7% of all word occurrences (69971 out of slightly over 1 million). True to Zipf's Law, the second-place word "of" accounts for slightly over 3.5% of words (36411 occurrences), followed by "and" (28852). Only 135 vocabulary items are needed to account for half the Brown Corpus.

Theoretical issues

Zipf's law is most easily observed by scatterplotting the data, with the axes being log(rank order) and log(frequency). For example, "the" as described above would appear at "x" = log(1), "y" = log(69971). The data conform to Zipf's law to the extent that the plotted points appear to fall along a single line segment.

Formally, let:
* "N" be the number of elements;
* "k" be their rank;
* "s" be the value of the exponent characterizing the distribution.Zipf's law then predicts that out of a population of "N" elements, the frequency of elements of rank "k", "f"("k";"s","N"), is:

:$f(k;s,N)=frac\{1/k^s\}\{sum\_\{n=1\}^N\; (1/n^s)\}.$

In the example of the frequency of words in the English language, "N" is the number of words in the English language and, if we use the classic version of Zipf's law, the exponent "s" is 1. "f"("k"; "s","N") will then be the fraction of the time the "k"th most common word occurs.

It is easily seen that the distribution is normalized, i.e., the predicted frequencies sum to 1:

:$sum\_\{k=1\}^N\; f(k;s,N)=1.$

The law may also be written:

:$f(k;s,N)=frac\{1\}\{k^s\; H\_\{N,s$

where "H_N,s" is the "N"th generalized harmonic number.

The simplest case of Zipf's law is a "1/"f" function". Given a set of Zipfian distributed frequencies, sorted from most common to least common, the second most common frequency will occur ½ as often as the first. The third most common frequency will occur 1/3 as often as the first. The "n"^th most common frequency will occur 1/"n" as often as the first. However, this cannot hold exactly, because items must occur an integer number of times: there cannot be 2.5 occurrences of a word. Nevertheless, over fairly wide ranges, and to a fairly good approximation, many natural phenomena obey Zipf's law.

Mathematically, it is impossible for the classic version of Zipf's law to hold exactly if there are infinitely many words in a language, since the sum of all relative frequencies in the denominator above is equal to the harmonic series and therefore:

:$sum\_\{n=1\}^infty\; frac\{1\}\{n\}=infty.!$

In English, the frequencies of the approximately 1000 most-frequently-used words are approximately proportional to 1/"n"^"s" where "s" is just slightly more than one. Fact|date=June 2007

As long as the exponent "s" exceeds 1, it is possible for such a law to hold with infinitely many words, since if "s" > 1 then:

Just why data conform to Zipfian distributions is a matter of some controversy. That Zipfian distributions arise in randomly-generated texts with no linguistic structure suggests that in linguistic contexts, the law may be a statistical artifact. [cite journal |author=Wentian Li |title=Random Texts Exhibit Zipf's-Law-Like Word Frequency Distribution |url=http://www.nslij-genetics.org/wli/pub/ieee92_pre.pdf |journal=IEEE Transactions on Information Theory |volume=38 |issue=6 |year=1992 |pages=1842–1845 |doi=10.1109/18.165464]

Related laws

"Zipf's law" now refers more generally to frequency distributions of "rank data," in which the relative frequency of the "n"th-ranked item is given by the Zeta distribution, 1/("n"^"s"ζ("s")), where the parameter "s" > 1 indexes the members of this family of probability distributions. Indeed, "Zipf's law" is sometimes synonymous with "zeta distribution," since probability distributions are sometimes called "laws". This distribution is sometimes called the Zipfian or Yule distribution.

A generalization of Zipf's law is the Zipf–Mandelbrot law, proposed by Benoît Mandelbrot, whose frequencies are:

:$f(k;N,q,s)=\; [mbox\{constant\}]\; /(k+q)^s.,$

The "constant" is the reciprocal of the Hurwitz zeta function evaluated at "s".

Zipfian distributions can be obtained from Pareto distributions by an exchange of variables.

The tail frequencies of the Yule–Simon distribution are approximately

:$f(k;\; ho)\; approx\; [mbox\{constant\}]\; /k^\{\; ho+1\}$

for any choice of "ρ" > 0.

If the natural log of some data are normally distributed, the data follow the log-normal distribution. This distribution is useful when random influences have an effect that is multiplicative rather than additive.

In the parabolic fractal distribution, the logarithm of the frequency is a quadratic polynomial of the logarithm of the rank. This can markedly improve the fit over a simple power-law relationship.cite web |url=http://home.zonnet.nl/galien8/factor/factor.html |title=Factorial randomness: the Laws of Benford and Zipf with respect to the first digit distribution of the factor sequence from the natural numbers |author=Johan Gerard van der Galien |date=2003-11-08] Like fractal dimension, it is possible to calculate Zipf dimension, which is a useful parameter in the analysis of texts. [Ali Eftekhari (2006) Fractal geometry of texts. "Journal of Quantitative Linguistic" 13(2-3): 177 – 193.]

It has been argued that Benford's law is a special case of Zipf's law.

See also

* Benford's law
* Bradford's law
* Finger fluting
* Heaps' law
* Lorenz curve
* Lotka's law
* Pareto distribution
* Pareto principle
* Zipf–Mandelbrot law
* Rank-size distribution

References

Further reading

Primary:
* George K. Zipf (1949) "Human Behavior and the Principle of Least-Effort". Addison-Wesley.
* -------- (1935) "The Psychobiology of Language". Houghton-Mifflin. (see citations at http://citeseer.ist.psu.edu/context/64879/0 )Secondary:
* Gelbukh, Alexander, and Sidorov, Grigori (2001) [http://www.gelbukh.com/CV/Publications/2001/CICLing-2001-Zipf.htm "Zipf and Heaps Laws’ Coefficients Depend on Language"] . Proc. CICLing-2001, "Conference on Intelligent Text Processing and Computational Linguistics", February 18–24, 2001, Mexico City. Lecture Notes in Computer Science N 2004, ISSN 0302-9743, ISBN 3-540-41687-0, Springer-Verlag: 332–335.
* Damián H. Zanette (2006) " [http://xxx.arxiv.org/abs/cs.CL/0406015 Zipf's law and the creation of musical context,] " "Musicae Scientiae 10": 3-18.
* Kali R. (2003) "The city as a giant component: a random graph approach to Zipf's law," "Applied Economics Letters 10": 717-720(4)
*cite journal |quotes= no |last= Gabaix|first= Xavier|authorlink= Xavier Gabaix|year= 1999|month= August|title= Zipf's Law for Cities: An Explanation |journal= Quarterly Journal of Economics|volume= 114|issue= 3|pages= 739–67|id= ISSN|0033-5533|url= http://pages.stern.nyu.edu/~xgabaix/papers/zipf.pdf|doi= 10.1162/003355399556133

External links

* [http://www.nslij-genetics.org/wli/zipf/ Comprehensive bibliography of Zipf's law]
* [http://www.hpl.hp.com/research/idl/papers/ranking/ranking.html Zipf, Power-laws, and Pareto - a ranking tutorial]
* [http://www.theatlantic.com/issues/2002/04/rauch.htm Seeing Around Corners (Artificial societies turn up Zipf's law)]
* [http://planetmath.org/encyclopedia/ZipfsLaw.html PlanetMath article on Zipf's law]
* [http://www.hubbertpeak.com/laherrere/fractal.htm Distributions de type “fractal parabolique” dans la Nature (French, with English summary)]
* [http://www.newscientist.com/article.ns?id=mg18524904.300 An analysis of income distribution]
* [http://www.lexique.org/listes/liste_mots.txt Zipf List of French words]
* [http://uk.arxiv.org/abs/physics/9901035 Citations and the Zipf-Mandelbrot's law]
* [http://demonstrations.wolfram.com/ZipfsLawForUSCities/ Zipf's Law for U.S. Cities] by Fiona Maclachlan, The Wolfram Demonstrations Project.
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