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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.[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.[2][3][4]

Example

Consider for example a sequence of random variables, each of which takes a value from the ternary alphabet {a, b, c}. Specifically, consider the string aaabcaaabcaaabcaaabc...aaabc constructed from infinite concatenations of the sub-string aaabc.

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

In this example, Pr(c|ab) = Pr(c|b) = 1.0; therefore, the shorter context 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: {Pr(a | a), Pr(a | b), Pr(a | c), Pr(b | a), Pr(b | b), Pr(b | c), Pr(c | a), Pr(c | b), Pr(c | c)}. To construct the Markov chain of order 2 for the next character, one must estimate 27 conditional probability components: {Pr(a | aa), Pr(a | ab), ..., 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: {Pr(a | aaa), Pr(a | aab), ..., 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."[1]
Definition

Let A be a state space (finite alphabet) of size |A|.

Consider a sequence with the Markov property $$x_1^{n}=x_1x_2\dots x_n$$ of n realizations of random variables, where $$x_i\in A$$ is the state (symbol) at position $${\displaystyle \scriptstyle (1\leq i\leq n)}$$ , and the concatenation of states $$x_{i}$$ and $$x_{i+1}$$ is denoted by $$x_ix_{i+1}.$$

Given a training set of observed states, $$x_1^{n}$$ , the construction algorithm of the VOM models[2][3][4] learns a model 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 $${\displaystyle P(x_{i}\mid s)}$$ for a symbol $$x_i \in A$$ given a context $$s\in A^*$$, 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 $${\displaystyle P(x_{i}\mid s)}$$ where the context length $${\displaystyle |s|\leq D}$$ varies depending on the available statistics. In contrast, conventional Markov models attempt to estimate these conditional distributions by assuming a fixed contexts' length$${\displaystyle |s|=D}$$ 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]
Application areas

Various efficient algorithms have been devised for estimating the parameters of the VOM model.[3]

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,[1] document compression,[3] classification and identification of DNA and protein sequences,[5] [1][2] statistical process control,[4] spam filtering,[6] haplotyping[7] and others.

Examples of Markov chains
Variable order Bayesian network
Markov process
Markov chain Monte Carlo
Semi-Markov process
Artificial intelligence

References

Rissanen, J. (Sep 1983). "A Universal Data Compression System". IEEE Transactions on Information Theory. 29 (5): 656–664. doi:10.1109/TIT.1983.1056741.
Shmilovici, A.; Ben-Gal, I. (2007). "Using a VOM Model for Reconstructing Potential Coding Regions in EST Sequences". Computational Statistics. 22 (1): 49–69. doi:10.1007/s00180-007-0021-8.
Begleiter, R.; El-Yaniv, R.; Yona, G. (2004). "On Prediction Using Variable Order Markov models" (PDF). Journal of Artificial Intelligence Research. 22: 385–421. doi:10.1613/jair.1491. Archived from the original (PDF) on 2007-09-28. Retrieved 2007-04-22.
Ben-Gal, I.; Morag, G.; Shmilovici, A. (2003). "CSPC: A Monitoring Procedure for State Dependent Processes" (PDF). Technometrics. 45 (4): 293–311. doi:10.1198/004017003000000122.
Grau J.; Ben-Gal I.; Posch S.; Grosse I. (2006). "VOMBAT: Prediction of Transcription Factor Binding Sites using Variable Order Bayesian Trees" (PDF). Nucleic Acids Research, vol. 34, issue W529–W533.
Bratko, A.; Cormack, G. V.; Filipic, B.; Lynam, T.; Zupan, B. (2006). "Spam Filtering Using Statistical Data Compression Models" (PDF). Journal of Machine Learning Research. 7: 2673–2698.

Browning, Sharon R. "Multilocus association mapping using variable-length Markov chains." The American Journal of Human Genetics 78.6 (2006): 903–913.

[1]

Smith, A. R.; Denenberg, J. N.; Slack, T. B.; Tan, C. C.; Wohlford, R. E (August 1985). "Application of a Sequential Pattern Learning System to Connected Speech Recognition" (PDF). Proceedings of the IEEE 1985 International Conference on Acoustics, Speech, and Signal Processing: 1201–1204.