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==Example==
Consider for example a sequence of [[random variablesvariable]]s each of which takes value from the ternary state space[[alphabet]] {a,b,c}.
To construct the Markov chain of order 1 for the next state in this sequence, one needs to 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)}. ▼Consider for example the string aaabcaaabcaaabcaaabc…aaabc constructed from infinite concatenations of the sub-string aaabc.
To construct the Markov chain of order 2 for the next state in this sequence, one needs to estimate the following 27 conditional probability components {Pr(a|aa), Pr(a|ab), …, Pr(c|cc)}. ▼
To construct the Markov chain of order three for the next state in this sequence, one needs to estimate the following 81 conditional probability components {Pr(a|aaa), Pr(a|aab), …, Pr(c|ccc)}.
InTo practicalconstruct settingsthe there[[Markov ischain]] seldomof sufficientorder data1 tofor accuratelythe estimatenext thecharacter polynomialin growingthis numbersequence, ofone needs to estimate the following 9 conditional probability components as{Pr(a|a), thePr(a|b), orderPr(a|c), ofPr(b|a), thePr(b|a), MarkovPr(b|a), chainPr(c|a), Pr(c|a), increasesPr(c|a)}.
As indicated above, in 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. ▼▲To construct the [[Markov chain ]] of order 12 for the next statecharacter in this sequence, one needs to estimate the following 927 [[conditional probability ]] components {Pr(a| aaa), Pr(a| bab), Pr(a|c), Pr(b|a), Pr(b|a), Pr(b|a)…, Pr(c| a), Pr(c|a), Pr(c|acc)}.
Consider for example the string aaabcaaabcaaabcaaabc…aaabc constructed from infinite concatenations of the sub-string aaabc. The VOM model of maximal order 2 can approximate the string using only the following four conditional probability components {Pr(a|aa)=0.5, Pr(b|aa)=0.5, Pr(c|b)=1.0, Pr(a|c)= 1.0}. In this example, Pr(c|ab)=Pr(c|b)=1.0, therefore, the shorter context b is sufficient to determine the future state. Similarly, the VOM model of maximal order 3 can approximate the string using only four conditional probability components. ▼
▲To construct the [[Markov chain ]] of order 2three for the next statecharacter in this sequence, one needs to estimate the following 2781 [[conditional probability ]] components {Pr(a| aaaaa), Pr(a| abaab), …, Pr(c| ccccc)}.
In practical settings there is seldom sufficient data to accurately estimate the [[exponent|exponential]] growing number of [[conditional probability]] components as the order of the [[Markov chain]] increases.
▲As indicated above, in theThe 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.
▲Consider for example 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 four [[conditional probability ]] components {Pr(a|aa)=0.5, Pr(b|aa)=0.5, Pr(c|b)=1.0, Pr(a|c)= 1.0}. In this example, Pr(c|ab)=Pr(c|b)=1.0, therefore, the shorter context b is sufficient to determine the future state. Similarly, the VOM model of maximal order 3 can approximate the string using only four [[conditional probability ]] components.
==Application Areas==
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