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Encyclopedia > Zipf's law
Parameters Probability mass function 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.) Cumulative distribution function 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.) $s>0,$ (real) $N in {1,2,3ldots}$ (integer) $k in {1,2,ldots,N}$ $frac{1/k^s}{H_{N,s}}$ $frac{H_{k,s}}{H_{N,s}}$ $frac{H_{N,s-1}}{H_{N,s}}$ $1,$ $frac{s}{H_{N,s}}sum_{k=1}^Nfrac{ln(k)}{k^s} +ln(H_{N,s})$ $frac{1}{H_{N,s}}sum_{n=1}^N frac{e^{nt}}{n^s}$ $frac{1}{H_{N,s}}sum_{n=1}^N frac{e^{int}}{n^s}$

## Contents

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 all 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. In linguistics, a corpus (plural corpora) or text corpus is a large and structured set of texts (now usually electronically stored and processed). ... In the philosophy of language, a natural language (or ordinary language) is a language that is spoken, written, or signed by humans for general-purpose communication, as distinguished from formal languages (such as computer-programming languages or the languages used in the study of formal logic, especially mathematical logic) and... This article is about proportionality, the mathematical relation. ... The Brown Corpus of Standard American English (or just Brown Corpus) was compiled by Henry Kucera and W. Nelson Francis at Brown University, Providence, RI as a general corpus (text collection) in the field of corpus linguistics. ...

## 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 confom to Zipf's law to the extent that the plotted points appear to fall along a single line segment. Waiting time between eruptions and the duration of the eruption for the Old Faithful geyser in Yellowstone National Park, Wyoming, USA. A scatterplot or scatter graph is a graph used in statistics to visually display and compare two or more sets of related quantitative, or numerical, data by displaying only... Look up logarithm in Wiktionary, the free dictionary. ...

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(ks,N) will then be the fraction of the time the kth 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 HN,s is the Nth generalized harmonic number. The harmonic number with (red line) with its asymptotic limit (blue line). ...

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 nth 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: See harmonic series (music) for the (related) musical concept. ...

In English, the frequencies of the approximately 1000 most-frequently-used words are approximately proportional to 1/ns where s is just slightly more than one.[citation needed]

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

$zeta (s) = sum_{n=1}^infty frac{1}{n^s}

where ζ is Riemann's zeta function. In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ...

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.[2]

## Related laws

A plot of word frequency in Wikipedia (November 27, 2006). The plot is in log-log coordinates. x  is rank of a word in the frequency table; y  is the total number of the word’s occurences. Most popular words are “the”, “of” and “and”, as expected. Zipf's law corresponds to the upper linear portion of the curve, roughly following the green (1/x)  line.

Zipf's law now refers more generally to frequency distributions of "rank data," in which the relative frequency of the nth-ranked item is given by the Zeta distribution, 1/(nsζ(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. In probability theory and statistics, the zeta distribution is a discrete probability distribution. ... A probability distribution describes the values and probabilities that a random event can take place. ...

A generalization of Zipf's law is the Zipf-Mandelbrot law, proposed by Benoît Mandelbrot, whose frequencies are: In probability theory and statistics, the Zipf-Mandelbrot law is a discrete probability distribution. ... BenoÃ®t B. Mandelbrot, PhD, (born November 20, 1924) is a Franco-American mathematician, best known as the father of fractal geometry. BenoÃ®t Mandelbrot was born in Poland, but his family moved to France when he was a child; he is a dual French and American citizen and was...

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

The "constant" is the reciprocal of the Hurwitz zeta function evaluated at s. In mathematics, the Hurwitz zeta function is one of the many zeta functions. ...

Zipfian distributions can be obtained from Pareto distributions by an exchange of variables.[3] The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations. ...

The tail frequencies of the Yule-Simon distribution are approximately In probability and statistics, the Yule-Simon distribution is a discrete probability distribution. ...

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

for any choice of ρ > 0.

If the natural log of some data are normally distributed, the data follow the log-normal distribution. This distribtuion is useful when random influences have an effect that is multiplicative rather than additive. The natural logarithm is the logarithm to the base e, where e is approximately equal to 2. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... In probability and statistics, the log-normal distribution is the single-tailed probability distribution of any random variable whose logarithm is normally distributed. ...

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.[3] Like fractal dimension, it is possible to calculate Zipf dimension, which is a useful parameter in the analysis of texts.[4] In the parabolic fractal distribution, the logarithm of the frequency or size of entities in a population is a quadratic polynomial of the logarithm of the rank. ...

It has been argued that Benford's law is a special case of Zipf's law.[3] A logarithmic scale bar. ...

A logarithmic scale bar. ... Bradfords law is a pattern first described by Samuel C. Bradford in 1934 that estimates the exponentially diminishing returns of extending a library search. ... In linguistics, Heaps law is an empirical law which describes the portion of a vocabulary which is represented by an instance document (or set of instance documents) consisting of words chosen from the vocabulary. ... The Lorenz curve is a graphical representation of the cumulative distribution function of a probability distribution; it is a graph showing the proportion of the distribution assumed by the bottom y% of the values. ... Lotkas law describes the frequency of publication by authors in a given field. ... The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations. ... For other uses of the word pareto, see Pareto. ... In probability theory and statistics, the Zipf-Mandelbrot law is a discrete probability distribution. ... Rank-size distribution or the rank-size rule or law describes the remarkable regularity in many phenomena including the distribution of city sizes around the world, size of businesses, particle sizes (such as sand), lengths of rivers, frequency of word usage, wealth among individuals, etc. ...

## References

1. ^ Christopher D. Manning, Hinrich Schütze Foundations of Statistical Natural Language Processing, MIT Press (1999), ISBN 978-0262133609, p. 24
2. ^ Wentian Li (1992). "Random Texts Exhibit Zipf's-Law-Like Word Frequency Distribution". IEEE Transactions on Information Theory 38 (6): 1842–1845.
3. ^ a b c Johan Gerard van der Galien (2003-11-08). Factorial randomness: the Laws of Benford and Zipf with respect to the first digit distribution of the factor sequence from the natural numbers.
4. ^ Ali Eftekhari (2006) Fractal geometry of texts. Journal of Quantitative Linguistic 13(2-3): 177 – 193.

The IEEE Transactions on Information Theory is a scientific journal published by the Institute of Electrical and Electronic Engineers (IEEE) It is dedicated to the study of information theory, the mathematics on communications. ... Year 2003 (MMIII) was a common year starting on Wednesday of the Gregorian calendar. ... is the 312th day of the year (313th in leap years) in the Gregorian calendar. ...

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: George Kingsley Zipf (IPA ), (1902-1950), was an American linguist and philologist who studied statistical occurrences in different languages. ...

• Gelbukh, Alexandr, and Sidorov, Grigori (2001) "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) "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)
• Gabaix, Xavier (August 1999). Zipf's Law for Cities: An Explanation. Quarterly Journal of Economics 114 (3): 739-67. ISSN 0033-5533.

is the 49th day of the year in the Gregorian calendar. ... Year 2001 (MMI) was a common year starting on Monday (link displays the 2001 Gregorian calendar). ... ISSN, or International Standard Serial Number, is the unique eight-digit number applied to a periodical publication including electronic serials. ...

Results from FactBites:

 Zipf's law - Wikipedia, the free encyclopedia (980 words) The "law" was publicized by Harvard linguist George Kingsley Zipf (IPA [zɪf]). Zipf's law is thus an experimental law, not a theoretical one. Zipf's law is most easily observed by scatterplotting the data, with the axes being log(rank order) and log(frequency).
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