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{{Short description|Markov-based processes with variable "memory"}}
In the mathematical theory of [[stochastic processes]], '''variable-order Markov (VOM) models''' are an important class of models that extend the well known [[Markov chain]] models. In contrast to the Markov chain models, where each [[random variable]] in a sequence with a [[Markov property]] depends on a fixed number of random variables, in VOM models this number of conditioning random variables may vary based on the specific observed realization.
This realization sequence is often called the ''context''; therefore the VOM models are also called ''context trees''.<ref name="Rissanen">{{cite journal|last = Rissanen|first = J.|title = A Universal Data Compression System|journal = IEEE Transactions on Information Theory|volume = 29|issue = 5|date = Sep 1983|pages = 656–664|
==Example==
Consider for example a sequence of [[random variable]]s, each of which takes a value from the ternary [[alphabet]] {{math|{{mset|''a'',
The VOM model of maximal order 2 can approximate the above string using ''only'' the following five [[conditional probability]] components: {{math|Pr(''a''
In this example, {{math|Pr(''c''
To construct the [[Markov chain]] of order 1 for the next character in that string, one must estimate the following 9 conditional probability components: {{math|Pr(''a''
In practical settings there is seldom sufficient data to accurately estimate the [[exponential growth|exponentially increasing]] number of conditional probability components as the order of the Markov chain increases.
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==Definition==
Let {{mvar|A}} be a state space (finite [[Alphabet (formal languages)|alphabet]]) of size <math>|A|</math>.
Consider a sequence with the [[Markov property]] <math>x_1^{n}=x_1x_2\dots x_n</math> of {{mvar|n}} realizations of [[random variable]]s, where <math> x_i\in A</math> is the state (symbol) at position {{mvar|i}} <math>\scriptstyle (1 \le i \le n)</math>, and the concatenation of states <math>x_i</math> and <math>x_{i+1}</math> is denoted by <math>x_ix_{i+1}</math>.
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Various efficient algorithms have been devised for estimating the parameters of the VOM model.<ref name="Begleiter"/>
VOM models have been successfully applied to areas such as [[machine learning]], [[information theory]] and [[bioinformatics]], including specific applications such as [[code|coding]] and [[data compression]],<ref name="Rissanen"/> document compression,<ref name="Begleiter"/> classification and identification of [[DNA]] and [[protein|protein sequences]],<ref>{{cite journal |url= http://www.eng.tau.ac.il/~bengal/VOMBAT.pdf |title= VOMBAT: Prediction of Transcription Factor Binding Sites using Variable Order Bayesian Trees
==See also==
* [[Stochastic chains with memory of variable length]]
* [[Examples of Markov chains]]
* [[Variable order Bayesian network]]
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