Abstract. JAVIER, Rodríguez et al. Mathematical diagnosis of fetal monitoring using the Zipf-Mandelbrot law and dynamic systems’ theory applied to cardiac. RODRIGUEZ VELASQUEZ, Javier et al. Zipf/Mandelbrot Law and probability theory applied to the characterization of adverse reactions to medications among . Zipf’s Law. In the English language, the probability of encountering the r th most common word is given roughly by P(r)=/r for r up to or so. The law.

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Zipf distribution is related to the zeta distributionbut is not identical. For example, 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, three times as often as the third most frequent word, etc.: True to Zipf’s Law, the second-place word of accounts for slightly over 3.

Only vocabulary items are needed to account for half the Brown Corpus. The law is named after the American linguist George Kingsley Zipf —who popularized it and sought to explain it Zipf, though he did not claim to have originated it.

The same relationship occurs in many other rankings unrelated to language, such as the population ranks of cities in various countries, corporation sizes, income rankings, ranks of number of people watching the same TV channel, [5] and so on.

The appearance of the distribution in rankings of cities by population was first noticed by Felix Auerbach in Zipf’s law is most easily observed by plotting the data on a log-log graph, with the axes being log rank order and log frequency.

It is also possible to plot reciprocal rank against frequency or reciprocal frequency or interword interval against rank. Zipf’s law then predicts that out of a population of N elements, the normalized frequency of elements of rank kf k ; sNis:. It has been claimed that this representation of Zipf’s law is more suitable for statistical testing, and in this way it has been analyzed in more than 30, English texts. 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.

However, this cannot hold exactly, because items must occur an integer number of times; there cannot be 2.

## Zipf’s law

Nevertheless, over fairly wide ranges, and to a fairly good approximation, many natural phenomena obey Zipf’s law. In human languages, word frequencies have a very heavy-tailed distribution, and can therefore be modeled reasonably well by a Zipf distribution with an s close to 1.

Wentian Li has shown that in a document in which each character has been chosen randomly from a uniform distribution of all letters plus a space characterthe “words” follow the general trend of Zipf’s law appearing approximately linear on log-log plot.

He took a large class of well-behaved statistical distributions not only the normal distribution and expressed them in terms of rank. He then expanded each expression into a Taylor series.

## Zipf’s Law

In every case Belevitch obtained the remarkable result that a first-order truncation of the series resulted in Zipf’s law. Further, a second-order truncation of the Taylor series resulted in Mandelbrot’s law. The principle of least effort is another possible explanation: Zipf himself proposed that neither speakers nor hearers using a given language want to work any harder than necessary to reach understanding, and the process that results in approximately equal distribution of effort leads to the observed Zipf distribution.

Similarly, preferential attachment intuitively, “the rich get richer” or “success breeds success” that results in the Yule—Simon distribution has been shown to fit word frequency versus rank in language [16] and population versus city rank [17] better than Zipf’s law. It was originally derived to explain population versus rank in species by Yule, and applied to cities by Simon. Indeed, Zipf’s law is sometimes synonymous with “zeta distribution,” since probability distributions are sometimes called “laws”.

This distribution is sometimes called the Zipfian distribution. The “constant” is the reciprocal of the Hurwitz zeta function evaluated at s.

In practice, as easily observable in distribution plots for large corpora, the observed distribution can be modelled more accurately as a sum of separate distributions for different subsets or subtypes of words that follow different parameterizations of the Zipf—Mandelbrot distribution, in particular the closed class of functional words exhibit s lower than 1, while open-ended vocabulary growth with document size and corpus size require s greater than 1 for convergence of the Generalized Harmonic Series.

Zipfian distributions can be obtained from Pareto distributions by an exchange of variables. The Zipf distribution is sometimes called the discrete Pareto distribution [18] because it is analogous to the continuous Pareto distribution in the same way that the discrete uniform distribution is analogous to the continuous uniform distribution.

The tail frequencies of the Yule—Simon distribution are approximately. In the parabolic fractal distributionthe 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. It has been argued that Benford’s law is a special bounded case of Zipf’s law, [19] with the connection between these two laws being explained by their both originating from scale invariant functional relations from statistical physics and critical phenomena.

Hence, Zipf law for natural numbers: Zipf’s law also has been used for extraction of parallel fragments of texts out of comparable corpora. From Wikipedia, the free encyclopedia. Zipf’s law Probability mass function.

Association for Computational Linguistics: Power-Law Distributions in Empirical Data.

SIAM Review, 51 4— Artificial Intelligence and Applications. Archived PDF from the original on 5 March Archived from the original on Human Behavior and the Principle of Least Effort. Archived PDF from the original on Univariate Discrete Distributions second ed.

### Zipf’s law – Simple English Wikipedia, the free encyclopedia

Retrieved 8 July Journal of Quantitative Linguistic 13 Vespignani Explaining the uneven distribution of numbers in nature: The laws of Benford and Zipf. Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher soliton discrete uniform Zipf Zipf—Mandelbrot.

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The horizontal axis is the index k. Note that the function is only defined sipf integer values of k. The connecting lines do not indicate continuity.

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