Variable-order Markov model: Difference between revisions

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{{Short description|Markov-based processes with variable "memory"}}

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In the mathematical theory of [[stochastic processes]], '''Variable variable-order Markov (VOM) models''' Models are an important class of models that extend the well known [[Markov Chainchain]] Modelsmodels. In contrast to the Markov Chainchain Modelsmodels, 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'' and thus the VOM models are also called ''Context Trees'' [1]. The flexibility in the number of conditioning random variables turns out to be of real advantage for many applications, such as statistical analysis, classification and prediction.

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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|doi = 10.1109/TIT.1983.1056741}}</ref> VOM models are nicely rendered by colorized probabilistic suffix trees (PST).<ref name=":0">{{Cite journal|last1=Gabadinho|first1=Alexis|last2=Ritschard|first2=Gilbert|date=2016|title=Analyzing State Sequences with Probabilistic Suffix Trees: The PST R Package|url=http://www.jstatsoft.org/v72/i03/|journal=Journal of Statistical Software|language=en|volume=72|issue=3|doi=10.18637/jss.v072.i03|s2cid=63681202 |issn=1548-7660|doi-access=free}}</ref> The flexibility in the number of conditioning random variables turns out to be of real advantage for many applications, such as [[statistical analysis]], [[Statistical classification|classification]] and [[prediction]].<ref name="Shmilovici">{{cite journal|last = Shmilovici|first = A.|author2=Ben-Gal, I. |title = Using a VOM Model for Reconstructing Potential Coding Regions in EST Sequences|journal = Computational Statistics|volume = 22|issue = 1|year = 2007|pages = 49–69|doi = 10.1007/s00180-007-0021-8| s2cid=2737235 }}</ref><ref name="Begleiter">{{cite journal|last = Begleiter|first = R.|author2 = El-Yaniv, R.|author3 = Yona, G.|title = On Prediction Using Variable Order Markov models|journal = Journal of Artificial Intelligence Research|volume = 22|year = 2004|pages = 385–421|doi = 10.1613/jair.1491|doi-access = free|arxiv = 1107.0051}}</ref><ref name="Ben-Gal">{{cite journal|last=Ben-Gal|first=I.|author2=Morag, G.|author3=Shmilovici, A.|year=2003|title=Context-Based Statistical Process Control: A Monitoring Procedure for State-Dependent Processes|url=http://www.eng.tau.ac.il/~bengal/Technometrics_final.pdf|journal=Technometrics|volume=45|issue=4|pages=293–311|doi=10.1198/004017003000000122|s2cid=5227793 |issn=0040-1706}}</ref>

'''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'' and thus the VOM models are also called ''Context Trees'' [1]. The flexibility in the number of conditioning random variables turns out to be of real advantage for many applications, such as statistical analysis, classification and prediction.

==Example==

Consider for example a sequence of [[random variable]]s, each of which takes a value from the ternary [[alphabet]] {{math|{{mset|''a'', ''b'', ''c''}}}}. Specifically, consider the string constructed from infinite concatenations of the sub-string {{math|''aaabc''}}: {{math|''aaabcaaabcaaabcaaabc…aaabc''}}.

The VOM model of maximal order 2 can approximate the above string using ''only'' the following five [[conditional probability]] components: {{math|Pr(''a'' {{!}} ''aa'') {{=}} 0.5}}, {{math|Pr(''b'' {{!}} ''aa'') {{=}} 0.5}}, {{math|Pr(''c'' {{!}} ''b'') {{=}} 1.0}}, {{math|Pr(''a'' {{!}} ''c''){{=}} 1.0}}, {{math|Pr(''a'' {{!}} ''ca'') {{=}} 1.0}}.

In this example, {{math|Pr(''c'' {{!}} ''ab'') {{=}} Pr(''c'' {{!}} ''b'') {{=}} 1.0}}; therefore, the shorter context {{math|''b''}} is sufficient to determine the next character. Similarly, the VOM model of maximal order 3 can generate the string exactly using only five conditional probability components, which are all equal to 1.0.

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'' {{!}} ''a'')}}, {{math|Pr(''a'' {{!}} ''b'')}}, {{math|Pr(''a'' {{!}} ''c'')}}, {{math|Pr(''b'' {{!}} ''a'')}}, {{math|Pr(''b'' {{!}} ''b'')}}, {{math|Pr(''b'' {{!}} ''c'')}}, {{math|Pr(''c'' {{!}} ''a'')}}, {{math|Pr(''c'' {{!}} ''b'')}}, {{math|Pr(''c'' {{!}} ''c'')}}. To construct the Markov chain of order 2 for the next character, one must estimate 27 conditional probability components: {{math|Pr(''a'' {{!}} ''aa'')}}, {{math|Pr(''a'' {{!}} ''ab'')}}, {{math|…}}, {{math|Pr(''c'' {{!}} ''cc'')}}. And to construct the Markov chain of order three for the next character one must estimate the following 81 conditional probability components: {{math|Pr(''a'' {{!}} ''aaa'')}}, {{math|Pr(''a'' {{!}} ''aab'')}}, {{math|…}}, {{math|Pr(''c'' {{!}} ''ccc'')}}.

In practical settings there is seldom sufficient data to accurately estimate the polynomial[[exponential growinggrowth|exponentially increasing]] number of conditional probability components as the order of the Markov chain increases.

AsThe indicated above, in the Variable Ordervariable-order Markov model assumes that in realistic settings, there are certain realizations of states (represented by contexts) in which some past states are independent from the future states,; accordingly, "a great reduction in the number of model parameters can be achieved."<ref name="Rissanen"/>

==Definition==

Let {{mvar|A}} be a state space (finite [[Alphabet (formal languages)|alphabet]]) of size <math>|A|</math>.

Let <math>A</math> be a state space (finite alphabet) of size <nowiki>|A|</nowiki>. Consider a sequence with the Markov property <math>x_1^{n}=x_1x_2...x_n</math> of <math>n</math> realizations of random variables, where <math> x_i\in A</math> is the state (symbol) at position <math>i</math> 1≤<math>i</math>≤<math>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>. Given a training set of observed states, <math>x_1^{n}</math>, the construction algorithm of the VOM models learns a model <math>P</math> that provides a probability assignment for each state in the sequence given its past (previously observed symbols) or future states. Specifically, the learner generates a conditional probability distribution <math>P(x|s)</math> for a symbol <math>x_i \in A</math> given a context <math>s\in A^*</math>, where the * sign represents a sequence of states of any length, including the empty context. VOM models attempt to estimate conditional distributions of the form <math>P(x|s)</math> where the context length |<math>s</math>|≤<math>D</math> varies depending on the available statistics. In contrast, conventional Markov models attempt to estimate these conditional distributions by assuming a fixed contexts' length |<math>s</math>|=<math>D</math> and, hence, can be considered as special cases of the VOM models. Effectively, for a given training sequence, the VOM models are found to obtain better model parameterization than the fixed-order Markov Models that leads to a better variance-bias tradeoff of the learned models [2,3,4].

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>.

==Example==

Consider for example a sequence of random variables each of which takes value from the ternary state space {a,b,c}.

Given a training set of observed states, <math>x_1^{n}</math>, the construction algorithm of the VOM models<ref name="Shmilovici"/><ref name="Begleiter"/><ref name="Ben-Gal"/> learns a model {{mvar|P}} that provides a [[probability]] assignment for each state in the sequence given its past (previously observed symbols) or future states.

To construct the Markov chain of order 1 for the next state in this sequence, one needs to estimate the following 9 conditional probability components {Pr(a|a), Pr(a|b), Pr(a|c), Pr(b|a), Pr(b|a), Pr(b|a), Pr(c|a), Pr(c|a), Pr(c|a)}.

To construct the Markov chain of order 2 for the next state in this sequence, one needs to estimate the following 27 conditional probability components {Pr(a|aa), Pr(a|ab), …, Pr(c|cc)}.

Specifically, the learner generates a [[conditional distribution|conditional probability distribution]] <math>P(x_i\mid s)</math> for a symbol <math>x_i \in A</math> given a context <math>s\in A^*</math>, where the * sign represents a sequence of states of any length, including the empty context.

To construct the Markov chain of order three for the next state in this sequence, one needs to estimate the following 81 conditional probability components {Pr(a|aaa), Pr(a|aab), …, Pr(c|ccc)}.

In practical settings there is seldom sufficient data to accurately estimate the polynomial growing number of conditional probability components as the order of the Markov chain increases.

VOM models attempt to estimate [[conditional distribution]]s of the form <math>P(x_i\mid s)</math> where the context length <math>|s| \le D</math> varies depending on the available statistics.

As indicated above, in the Variable Order Markov model assumes that in realistic settings, there are certain realizations of states (represented by contexts) in which some past states are independent from the future states, accordingly, a great reduction in the number of model parameters can be achieved.

In contrast, conventional [[Markov chain|Markov models]] attempt to estimate these [[conditional distribution]]s by assuming a fixed contexts' length <math>|s| = D</math> and, hence, can be considered as special cases of the VOM models.

Consider for example the string aaabcaaabcaaabcaaabc…aaabc constructed from infinite concatenations of the sub-string aaabc. The VOM model of maximal order 2 can approximate the string using only the following four conditional probability components {Pr(a|aa)=0.5, Pr(b|aa)=0.5, Pr(c|b)=1.0, Pr(a|c)= 1.0}. In this example, Pr(c|ab)=Pr(c|b)=1.0, therefore, the shorter context b is sufficient to determine the future state. Similarly, the VOM model of maximal order 3 can approximate the string using only four conditional probability components.

Effectively, for a given training sequence, the VOM models are found to obtain better model parameterization than the fixed-order [[Markov chain|Markov models]] that leads to a better [[variance]]-bias tradeoff of the learned models.<ref name="Shmilovici"/><ref name="Begleiter"/><ref name="Ben-Gal"/>

==Application Areasareas==

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 |author1= Grau J. |author2= Ben-Gal I. |author3= Posch S. |author4= Grosse I. |journal= Nucleic Acids Research |publisher= Nucleic Acids Research, vol. 34, issue W529–W533. |year= 2006 |volume= 34 |issue= Web Server issue |pages= W529-33 |doi= 10.1093/nar/gkl212 |pmid= 16845064 |pmc= 1538886 |archive-date= 2018-09-30 |access-date= 2014-01-10 |archive-url= https://web.archive.org/web/20180930084306/http://www.eng.tau.ac.il/~bengal/VOMBAT.pdf |url-status= dead }}</ref> [http://www.eng.tau.ac.il/~bengal/VOMBAT.pdf]<ref name="Shmilovici"/> [[statistical process control]],<ref name="Ben-Gal"/> [[spam filtering]],<ref name="Bratko">{{cite journal|last = Bratko|first = A. |author2=Cormack, G. V. |author3=Filipic, B. |author4=Lynam, T. |author5=Zupan, B.|title = Spam Filtering Using Statistical Data Compression Models|journal = Journal of Machine Learning Research|volume = 7|year = 2006|pages = 2673–2698|url = http://www.jmlr.org/papers/volume7/bratko06a/bratko06a.pdf}}</ref> [[haplotyping]],<ref>[[Sharon R. Browning|Browning, Sharon R.]] "Multilocus association mapping using variable-length Markov chains." The American Journal of Human Genetics 78.6 (2006): 903–913.</ref> speech recognition,<ref>{{Cite book|last1=Smith|first1=A.|last2=Denenberg|first2=J.|last3=Slack|first3=T.|last4=Tan|first4=C.|last5=Wohlford|first5=R.|title=ICASSP '85. IEEE International Conference on Acoustics, Speech, and Signal Processing |chapter=Application of a sequential pattern learning system to connected speech recognition |date=1985|___location=Tampa, FL, USA|publisher=Institute of Electrical and Electronics Engineers|volume=10|pages=1201–1204|doi=10.1109/ICASSP.1985.1168282|s2cid=60991068 }}</ref> [[sequence analysis in social sciences]],<ref name=":0" /> and others.

==Application Areas==

Various efficient algorithms were devised for estimating the parameters of the VOM model [3]. The VOM models were successfully applied to areas such as Machine learning, Information theory and Bioinformatics including specific applications such as coding and data compression [1] document compression [3], classification and identification of DNA and protein sequences [2] Statistical Process Control [4] and more.

==See also==

* [[Stochastic chains with memory of variable length]]

* [[Markov chain]]

* [[Examples of Markov chains]]

* [[Variable order Bayesian network]]

* [[Markov process]]

* [[Markov chain Monte Carlo]]

* [[Semi-Markov process]]

* [[Artificial intelligence]]

==References==

{{reflist}}

[[Category:Markov models]]

[1] Rissanen J. (1983). “A Universal Data Compression System”. IEEE Transactions on Information Theory. 29 (5):656- 664.<br />

[2] Shmilovici A., Ben-Gal I. (2007), “Using a VOM Model for Reconstructing Potential Coding Regions in EST Sequences”, Computational Statistics vol. 22, no. 1, 49-69.<br />

[3] Begleiter R., El-Yaniv R., Yona G., (2004), "On Prediction Using Variable Order Markov Models", Journal of Artificial Intelligence, 22:385-421.<br />

[4] Ben-Gal I., Morag G., Shmilovici A., (2003) "CSPC: A Monitoring Procedure for State Dependent Processes", Technometrics, vol. 45, no. 4, 293-311.