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A '''bigram''' or '''digram''' is every sequence of two adjacent elements in a string of [[Token (parser)|tokens]], which are typically letters, syllables, or words; they are [[n-gram|''n''-grams]] for ''n''=2. The frequency distribution of bigrams in a string are commonly used for simple statistical analysis of text in many applications, including in computational linguistics, cryptography, speech recognition, and so on.
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''Gappy bigrams'' or ''skipping bigrams'' are word pairs which allow gaps (perhaps avoiding connecting words, or allowing some simulation of dependencies, as in a [[dependency grammar]]).  
 
''Head word bigrams'' are gappy bigrams with an explicit dependency relationship.  
 
Bigrams help provide the conditional probability of a token given the preceding token, when the relation of the [[conditional probability]] is applied:
 
<math> P(W_n|W_{n-1}) = { P(W_{n-1},W_n) \over P(W_{n-1}) } </math>
 
That is, the probability <math> P() </math> of a token <math>W_n</math> given the preceding token <math>W_{n-1}</math> is equal to the probability of their bigram, or the co-occurrence of the two tokens <math>P(W_{n-1},W_n)</math>, divided by the probability of the preceding token.
 
==Applications==
 
Bigrams are used in one of the most successful [[language model]]s for [[speech recognition]].<ref>Michael Collins. ''A new statistical parser based on bigram lexical dependencies''. In Proceedings of the 34th Annual Meeting of the Association of Computational Linguistics, Santa Cruz, CA. 1996. pp.184-191.</ref> They are a special case of [[N-gram]].
 
Bigram frequency attacks can be used in [[cryptography]] to solve [[cryptograms]]. See [[Frequency analysis (cryptanalysis)|frequency analysis]].
 
Bigram frequency is one approach to [[Language_detection#Statistical_approaches|statistical language identification]].
 
==Bigram Frequency in the English language==
The frequency of the most common letter bigrams in a small English corpus is:<ref>[http://www.math.cornell.edu/~mec/2003-2004/cryptography/subs/digraphs.html Cornell Math Explorer's Project &ndash; Substitution Ciphers]</ref>
 
th 1.52      en 0.55      ng 0.18
he 1.28      ed 0.53      of 0.16
in 0.94      to 0.52      al 0.09
er 0.94      it 0.50      de 0.09
an 0.82      ou 0.50      se 0.08
re 0.68      ea 0.47      le 0.08
nd 0.63      hi 0.46      sa 0.06
at 0.59      is 0.46      si 0.05
on 0.57      or 0.43      ar 0.04
nt 0.56      ti 0.34      ve 0.04
ha 0.56      as 0.33      ra 0.04
es 0.56      te 0.27      ld 0.02
st 0.55      et 0.19      ur 0.02
Complete bigram frequencies for a larger corpus are available.<ref>{{Cite journal | issn = 0743-3808 | volume = 36 | issue = 3 | pages = 388–396| last = Jones| first = Michael N| coauthors = D J K Mewhort| title = Case-sensitive letter and bigram frequency counts from large-scale English corpora| journal = Behavior Research Methods, Instruments, & Computers: A Journal of the Psychonomic Society, Inc| <!--accessdate = 2012-04-07|--> date = 2004-08
| pmid=15641428}}</ref>
 
==See also==
* [[Digraph (orthography)]]
* [[N-gram]]
* [[Letter frequency]]
 
==References==
<references/>
 
[[Category:Formal languages]]
[[Category:Classical cryptography]]
[[Category:Natural language processing]]

Latest revision as of 03:53, 15 December 2014

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