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] VOM models are nicely rendered by colorized probabilistic suffix trees (PST).^[2] 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.^[3][4][5]

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 constructed from infinite concatenations of the sub-string aaabc: aaabcaaabcaaabcaaabc…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 ${\displaystyle |A|}$.

Consider a sequence with the Markov property ${\displaystyle x_{1}^{n}=x_{1}x_{2}\dots x_{n}}$ of n realizations of random variables, where ${\displaystyle x_{i}\in A}$ is the state (symbol) at position i ${\displaystyle \scriptstyle (1\leq i\leq n)}$, and the concatenation of states ${\displaystyle x_{i}}$ and ${\displaystyle x_{i+1}}$ is denoted by ${\displaystyle x_{i}x_{i+1}}$.

Given a training set of observed states, ${\displaystyle x_{1}^{n}}$, the construction algorithm of the VOM models^[3][4][5] 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 ${\displaystyle x_{i}\in A}$ given a context ${\displaystyle s\in A^{*}}$, 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.^[3][4][5]

Application areas

Various efficient algorithms have been devised for estimating the parameters of the VOM model.^[4]

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,^[4] classification and identification of DNA and protein sequences,^{[6] [3]} statistical process control,^[5] spam filtering,^[7] haplotyping,^[8] speech recognition,^[9] sequence analysis in social sciences,^[2] and others.

See also

• Stochastic chains with memory of variable length
• Examples of Markov chains
• Variable order Bayesian network
• Markov process
• Markov chain Monte Carlo
• Semi-Markov process
• Artificial intelligence

References