Variable-order Markov model

In 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 {abc}. 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 $i$ 1≤$i$$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 $P(x_i|s)$ 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 $P(x_i|s)$ where the context length |$s$|≤$D$ varies depending on the available statistics. In contrast, conventional Markov models attempt to estimate these conditional distributions by assuming a fixed contexts' length |$s$|=$D$ 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.