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EspañolZIPF'S LAW
(Redirected from Zipf\'s Law)
{{Probability distribution|
name =Zipf|
type =mass|
pdf_image =
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_image =
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 = (real)
(integer)|
support =|
pdf =|
cdf =|
mean =|
median =|
mode =|
variance =|
skewness =|
kurtosis =|
entropy =|
mgf =|
char =|
}}
Originally, 'Zipf's law' stated that, in a corpus of natural language utterances, the frequency of any word is roughly inversely proportional to its rank in the frequency table. So, 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. The term has come to be used to refer to any of a family of related power law probability distributions.
The "law" was publicized by Harvard linguist George Kingsley Zipf (IPA ).
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.
Zipf's law is thus an empirical law, not a theoretical one. Zipfian distributions are commonly observed, in many kinds of phenomena. The causes of Zipfian distributions in real life are a matter of some controversy, however. The fact that Zipfian distributions arise in randomly-generated texts with no linguistic structure suggests that the law as applied to languages may in part be a statistical artifact.[1]
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). If the points are close to a single straight line, the distribution follows Zipf's law.
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 precisely true, 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.
Zipf's law may be stated mathematically as:
:
where ''N'' is the number of elements, ''k'' is their rank, and ''s'' is the exponent characterizing the distribution. 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:
:
The law may also be written:
:
where ''HN,s'' is the ''N''th generalized harmonic number.
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:
:
Empirical research has found that 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.
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
:
{{Probability distribution|
name =Zipf|
type =mass|
pdf_image =
Plot of the Zipf PMF for N=10
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_image =
Plot of the Zipf CMF for N=10
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 = (real)
(integer)|
support =|
pdf =|
cdf =|
mean =|
median =|
mode =|
variance =|
skewness =|
kurtosis =|
entropy =|
mgf =|
char =|
}}
Originally, 'Zipf's law' stated that, in a corpus of natural language utterances, the frequency of any word is roughly inversely proportional to its rank in the frequency table. So, 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. The term has come to be used to refer to any of a family of related power law probability distributions.
The "law" was publicized by Harvard linguist George Kingsley Zipf (IPA ).
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.
Zipf's law is thus an empirical law, not a theoretical one. Zipfian distributions are commonly observed, in many kinds of phenomena. The causes of Zipfian distributions in real life are a matter of some controversy, however. The fact that Zipfian distributions arise in randomly-generated texts with no linguistic structure suggests that the law as applied to languages may in part be a statistical artifact.[1]
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). If the points are close to a single straight line, the distribution follows Zipf's law.
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 precisely true, 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.
| Contents |
| Theoretical issues |
| Related laws |
| See also |
| Further reading |
| External links |
| References |
Theoretical issues
Zipf's law may be stated mathematically as:
:
where ''N'' is the number of elements, ''k'' is their rank, and ''s'' is the exponent characterizing the distribution. 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:
:
The law may also be written:
:
where ''HN,s'' is the ''N''th generalized harmonic number.
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:
:
Empirical research has found that 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.
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
:
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psst.. try this: add to faves
