Stochastic chains with memory of variable length are a family of stochastic chains of finite order in a finite alphabet, such as, for every time pass, only one finite suffix of the past, called context, is necessary to predict the next symbol.
These models were introduced in the information theory literature by Jorma Rissanen in 1983,[1] as a universal tool to data compression, but recently have been used to model data in different areas such as biology,[2] linguistics[3] and music.
[4] A stochastic chain with memory of variable length is a stochastic chain
, taking values in a finite alphabet
, and characterized by a probabilistic context tree
, so that The class of stochastic chains with memory of variable length was introduced by Jorma Rissanen in the article A universal data compression system.
[1] Such class of stochastic chains was popularized in the statistical and probabilistic community by P. Bühlmann and A. J. Wyner in 1999, in the article Variable Length Markov Chains.
Named by Bühlmann and Wyner as “variable length Markov chains” (VLMC), these chains are also known as “variable-order Markov models" (VOM), “probabilistic suffix trees”[2] and “context tree models”.
[5] The name “stochastic chains with memory of variable length” seems to have been introduced by Galves and Löcherbach, in 2008, in the article of the same name.
[6] Consider a system by a lamp, an observer and a door between both of them.
When the lamp is on, the observer may see the light through the door, depending on which state the door is at the time: open, 1, or closed, 0. such states are independent of the original state of the lamp.
a Markov chain that represents the state of the lamp, with values in
be a sequence of independent random variables that represents the door's states, also taking values in
Using a context tree it's possible to represent the past states of the sequence, showing which are relevant to identify the next state.
is, then, a chain with memory of variable length, taking values in
and compatible with the probabilistic context tree
, one can find the appropriated context tree using the following algorithms.
In the article A Universal Data Compression System,[1] Rissanen introduced a consistent algorithm to estimate the probabilistic context tree that generates the data.
This algorithm's function can be summarized in two steps: Be
a sample of a finite probabilistic tree
the number of occurrences of the sequence in the sample, i.e., Rissanen first built a context maximum candidate, given by
is an arbitrary positive constant.
comes from the impossibility of estimating the probabilities of sequence with lengths greater than
From there, Rissanen shortens the maximum candidate through successive cutting the branches according to a sequence of tests based in statistical likelihood ratio.
is the ratio of the log-likelihood to test the consistency of the sample with the probabilistic context tree
differ only by a set of sibling knots.
The length of the current estimated context is defined by where
of a finite probabilistic context tree
The estimator of the context tree by BIC with a penalty constant
is defined as The smallest maximizer criterion[3] is calculated by selecting the smallest tree τ of a set of champion trees C such that