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 Variable s turns out to be of real advantage for many applications, such as statistical analysis, classification and prediction.234
Consider for example a sequence of Random Variable s, 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''.
|
| |
Pr(''c''''b'') = 10 therefore, the shorter context ''b'' is sufficient to determine the next character Similarly, the VOM model of maximal order 3 can also approximate the string using only four conditional probability components
|
| |
"http://wwwinformationdelightinfo/information/entry/Markov_chain" class="copylinks">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''''a''), Pr(''b''''a''), Pr(''c''''a''), Pr(''c''''a''), Pr(''c''''a'')} 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''), , <br />Pr(''c''''ccc'')}
|
| |
"http://wwwinformationdelightinfo/information/entry/alphabet" class="copylinks">Alphabet ) of size <nowiki>A</nowiki>
|
| |
"http://wwwinformationdelightinfo/information/entry/conditional_distribution" class="copylinks">Conditional Probability Distribution <math>P(x_is)</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
|
| |
"http://wwwinformationdelightinfo/information/entry/conditional_distribution" class="copylinks">Conditional Distribution s of the form <math>P(x_is)</math> where the context length <math>s</math>≤<math>D</math> varies depending on the available statistics
|
| |
"http://wwwinformationdelightinfo/information/entry/Markov_chain" class="copylinks">Markov Models attempt to estimate these Conditional Distribution s by assuming a fixed contexts' length <math>s</math>=<math>D</math> and, hence, can be considered as special cases of the VOM models
|
