Variable-order Markov model

{{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|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, 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>

==Example==

Consider for example a sequence of random variables, 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 exponentially increasing number of conditional probability components as the order of the Markov chain increases.

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."<ref name="Rissanen"/>

==Definition== Let {{mvar|A}} be a state space (finite 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 variables, 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>.

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. Specifically, the learner generates a 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 distributions 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. In contrast, conventional Markov models attempt to estimate these conditional distributions by assuming a fixed contexts' length <math>|s| = 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.<ref name="Shmilovici"/><ref name="Begleiter"/><ref name="Ben-Gal"/>

==Application areas== 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 coding and data compression,<ref name="Rissanen"/> document compression,<ref name="Begleiter"/> classification and identification of DNA and 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>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.

==References== {{reflist}}

Category:Markov models