Variable-order Markov model: Difference between revisions

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

In [[stochasticthe processes]],mathematical atheory topic inof [[mathematicsstochastic processes]], '''Variablevariable-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|urldoi = 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://ieeexplorewww.ieeejstatsoft.org/xplv72/freeabs_all.jsp?isnumberi03/|journal=22734&arnumberJournal of Statistical Software|language=1056741en|volume=72|issue=3|doi = 10.110918637/TITjss.1983v072.1056741i03|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|url=http://www.springerlink.com/content/a447865604519210/|doi = 10.1007/s00180-007-0021-8| s2cid=2737235 }}</ref><ref name="Begleiter">{{cite journal|last = Begleiter|first = R.|coauthorsauthor2 = El-Yaniv, R.|author3 and= Yona, G.|title = On Prediction Using Variable Order Markov models|journal = Journal of Artificial Intelligence Research|volume = 22|year = 2004|pages = 385–421|urldoi = http:10.1613//www.jair.org/media/1491/live|doi-1491-2335-jairaccess = free|arxiv = 1107.pdf0051}}</ref><ref name="Ben-Gal">{{cite journal|last = Ben-Gal|first = I.|coauthors author2= Morag, G. and |author3=Shmilovici, A.|year=2003|title =Context-Based CSPCStatistical Process Control: A Monitoring Procedure for State -Dependent Processes|journal = Technometrics|volume = 45|issue = 4|year = 2003|pages = 293–311|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>

==Example==

Consider for example a sequence of [[random variable]]s, each of which takes a value from the ternary [[alphabet]] {{math|{{mset|''a'',&nbsp; ''b'',&nbsp; ''c''}}}}. Specifically, consider the string ''aaabcaaabcaaabcaaabc...aaabc'' 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 [[exponential growth|exponentially increasing]] number of conditional probability components as the order of the Markov chain increases.

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==Definition==

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

Consider a sequence with the [[Markov property]] <math>x_1^{n}=x_1x_2\dots x_n</math> of <math>{{mvar|n</math>}} realizations of [[random variable]]s, where <math> x_i\in A</math> is the state (symbol) at position <math>{{mvar|i}} </math>\scriptstyle (1 \le 1≤<math>i</math>≤<math> \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>.

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 <math>{{mvar|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 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.

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

In contrast, conventional [[Markov chain|Markov models]] attempt to estimate these [[conditional distribution]]s 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 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"/>

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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 paperjournal |url= http://www.eng.tau.ac.il/~bengal/VOMBAT.pdf |title= VOMBAT: Prediction of Transcription Factor Binding Sites using Variable Order Bayesian Trees, |authorauthor1= 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. |coauthors author2= Cormack, G. V., |author3=Filipic, B., |author4=Lynam, T. and |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): 903903–913.</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|chapter-913url=https://ieeexplore.ieee.org/document/1168282|___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.

==See also==

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

* [[Examples of Markov chains]]

* [[Variable order Bayesian network]]