Ranté Markov: Béda antarrépisi

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m Ngarapihkeun éjahan, replaced: oge → ogé (2), nyaeta → nyaéta (4), rea → réa, ngarupakeun → mangrupa, ea → éa (7), eo → éo, Didinya → di dinya
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Ranté Markov aya sanggeus [[Andrei Andreevich Markov|A.A. Markov]], nu mimiti ngahasilkeun ieu proses dina taun (1906). Dijadikeun bisa diitung keur tetapan dina ruang anu "tidak tebatas" ku [[Andrey Nikolaevich Kolmogorov|Kolmogorov]] (1936).
Ranté Markov patali jeung [[Brownian motion|gerak Brown]] sarta [[ergodic hypothesis|hipotesa ergodik]], dua topik penting dina widang fisik di awal [[twentieth century|abad kaduapuluh]],
tapi Markov digunakeun ogeogé di luar widang matematika, saperti [[law of large numbers|hukum wilangan gede]] dina kajadian anu pakait.
 
== Sifat ranté Markov ==
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nu disebut prosés ''transition probability''.
Kadangkala disebut ogeogé "one-step" transition probability.
''Transition probability'' dua tahap, tilu tahap atawa leuwih dijentrekeun tina ''transition probability'' satahap sarta sifat Markov:
 
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Rumus ieu nyaruakeun keur kayaan waktu nu teu ''teratur'' di mangsa datang ''n''+''k'' ku cara ngalikeun transition probabilities sarta nga-''integral''-keun waktu ''k''.
 
[[Marginal distribution]] ''P''(''X''<sub>''n''</sub>) nyaetanyaéta distribusi nu ditangtukeun dina waktu ''n''.
Distiribusi awal nyaetanyaéta ''P''(''X''<sub>0</sub>).
Evolusi proses ngaliwatan sakali tahap waktu nu dijentrekeun ku
 
: <math> P(X_{n+1}) = \int P(X_{n+1}|X_n)\,P(X_n)\,dX_n </math>
 
Ieu ngarupakeunmangrupa versi [[Frobenius-Perron equation]].
Didinyadi dinya aya hiji atawa leuwih ''tetapan'' distribusi π saperti
 
:<math> \pi(X) = \int P(X|Y)\,\pi(Y)\,dY</math>
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numana ''Y'' ngan sakadar ngaran variabel integrasi.
Saperti distribution π disebut ''stationary distribution'' atawa ''steady-state distribution''.
Stationary distribution nyaetanyaéta [[eigenfunction]] tina fungsi ''conditional distribution'', nu pakait jeung [[eigenvalue]] 1.
 
Whether or not there is a stationary distribution,
and whether or not it is unique if it does exist,
are determined by certain properties of the process.
''Irreducible'' meansméans that every state is accessible from every other state.
''Aperiodic'' meansméans that there exists at leastléast one state for which the transition from that state to itself is possible. ''Positive recurrent'' meansméans that the expected return time is finite for every state.
Sometimes the terms ''indecomposable'', ''acyclic'', and ''persistent'' are used as synonyms for "irreducible", "aperiodic", and "recurrent", respectively.
 
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In particular,
this holds for ''f'' equal to the identity function.
Mangka nilai average sampel dina waktu nyaetanyaéta sarua jeung [[nilai ekspektasi]] tina sebaran ''stationary''.
 
Furthermore,
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= \int_A \pi(X)\,dX = \mu_{\pi}(A) </math>
 
where μ<sub>π</sub> is the measureméasure induced by π.
This makes it possible to approximate the stationary distribution by a [[histogram]] or other density estimate of a sequence of samples.
 
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the transition probability distribution can be represented as a matrix,
called the ''transition matrix'',
with the (''i'', ''j'')'th element equal to
 
:<math>P(X_{n+1}=i\mid X_n=j)</math>
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associated with the [[eigenvalue]] 1.
If the transition matrix ''P'' is positive recurrent, irreducible, and aperiodic,
then ''P''<sup>''k''</sup> converges elementwise to a matrix in which eachéach column is the unique stationary distribution.
 
A transition matrix which is positive (that is, every element of the matrix is positive)
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== Scientific applications ==
 
Markov chains are used to model various processes in [[queueing theory]] and [[statistics]], and can also be used as a signal model in [[entropy coding]] techniques such as [[arithmetic coding]]. Markov chains also have many biological applications, particularly [[population process]]es, which are useful in modelling processes that are (at leastléast) analogous to biological populations. Markov chains have been used in [[bioinformatics]] as well. An example is the [[genemark algorithm]] for coding region/gene prediction.
 
Markov processes can also be used to generate superficially "real-looking" text given a sample document: they are used in various pieces of recreationalrecréational "parody generator" software (see [[Jeff Harrison]]).
 
== Tempo oge ==
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== Rujukan ==
 
* A.A. Markov. "Rasprostranenie zakona bol'shih chisel na velichiny, zavisyaschie drug ot druga". ''Izvestiya Fiziko-matematicheskogo obschestva pri Kazanskom universitete'', 2-ya seriya, tom 15, pp 135-156135–156, 1906.
* A.A. Markov. "Extension of the limit theorems of probability theory to a sum of variables connected in a chain". reprinted in Appendix B of: R. Howard. ''Dynamic Probabilistic Systems, volume 1: Markov Chains''. John Wiley and Sons, 1971.
 
* LeoLéo Breiman. ''Probability''. Original edition published by Addison-Wesley, 1968; reprinted by Society for Industrial and Applied Mathematics, 1992. ISBN 0-89871-296-3. ''(See Chapter 7.)''
* A.A. Markov. "Extension of the limit theorems of probability theory to a sum of variables connected in a chain". reprinted in Appendix B of: R. Howard. ''Dynamic Probabilistic Systems, volume 1: Markov Chains''. John Wiley and Sons, 1971.
 
* Leo Breiman. ''Probability''. Original edition published by Addison-Wesley, 1968; reprinted by Society for Industrial and Applied Mathematics, 1992. ISBN 0-89871-296-3. ''(See Chapter 7.)''
 
* J.L. Doob. ''Stochastic Processes''. New York: John Wiley and Sons, 1953. ISBN 0-471-52369-0.
 
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* [http://crypto.mat.sbg.ac.at/~ste/diss/node6.html Markov Chains]
 
* [http://www.cs.bell-labs.com/cm/cs/pearls/sec153.html Generating Text] ''(About generating random text using a Markov chain.)''
 
* [http://www.mathworks.com/company/newsletters/news_notes/clevescorner/oct02_cleve.html The World's Largest Matrix Computation] ''(Google's PageRank as the stationary distribution of a random walk through the Web.)''
 
* [http://www.gnu.org/software/emacs/manual/html_node/emacs_473.html Disassociated Press] in [[Emacs]] approximates a Markov process