Additive Markov Chain
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An Additive Markov Chain is a Markov Chain with an additive conditional probability function.
References
2015
- (Wikipedia, 2015) ⇒ https://www.wikiwand.com/en/Additive_Markov_chain Retrieved 2016-07-10
- In probability theory, an additive Markov chain is a Markov chain with an additive conditional probability function. Here the process is a discrete-time Markov chain of order m and the transition probability to a state at the next time is a sum of functions, each depending on the next state and one of the m previous states.
- Definition
- An additive Markov chain of order m is a sequence of random variables X1, X2, X3, ..., possessing the following property: the probability that a random variable Xn has a certain value xn under the condition that the values of all previous variables are fixed depends on the values of m previous variables only (Markov chain of order m), and the influence of previous variables on a generated one is additive,
- [math]\displaystyle{ \Pr(X_n=x_n|X_{n-1}=x_{n-1}, X_{n-2}=x_{n-2}, \dots, X_{n-m}=x_{n-m}) = \sum_{r=1}^{m} f(x_n,x_{n-r},r) }[/math].
- Binary case
- A binary additive Markov chain is where the state space of the chain consists on two values only, Xn ∈ { x1, x2 }. For example, Xn ∈ { 0, 1 }. The conditional probability function of a binary additive Markov chain can be represented as
- [math]\displaystyle{ \Pr(X_n=1|X_{n-1}=x_{n-1}, X_{n-2}=x_{n-2}, \dots) = \bar{X} + \sum_{r=1}^{m} F(r) (x_{n-r}-\bar{X}), }[/math]
- [math]\displaystyle{ \Pr(X_n=0|X_{n-1}=x_{n-1}, X_{n-2}=x_{n-2}, \dots) = 1 - \Pr(X_n=1|X_{n-1} = x_{n-1}, X_{n-2} = x_{n-2}, \dots). }[/math]
- Here [math]\displaystyle{ \bar{X} }[/math] is the probability to find Xn = 1 in the sequence and
- F(r) is referred to as the memory function. The value of [math]\displaystyle{ \bar{X} }[/math] and the function F(r) contain all the information about correlation properties of the Markov chain.
- Relation between the memory function and the correlation function
- In the binary case, the correlation function between the variables [math]\displaystyle{ X_n }[/math] and [math]\displaystyle{ X_k }[/math] of the chain depends on the distance [math]\displaystyle{ n - k }[/math] only. It is defined as follows:
- [math]\displaystyle{ K(r) = \langle (X_n-\bar{X})(X_{n+r}-\bar{X}) \rangle = \langle X_n X_{n+r} \rangle -{\bar{X}}^2, }[/math]
- where the symbol [math]\displaystyle{ \langle \cdots \rangle }[/math] denotes averaging over all n. By definition,
- [math]\displaystyle{ K(-r)=K(r), K(0)=\bar{X}(1-\bar{X}). }[/math]
- There is a relation between the memory function and the correlation function of the binary additive Markov chain:
- [math]\displaystyle{ K(r)=\sum_{s=1}^m K(r-s)F(s), \, \, \, \, r=1, 2, \dots\,. }[/math]
2006
- (Izrailev ey al. 2006) ⇒ Izrailev, F. M., Krokhin, A. A., Makarov, N. M., Melnyk, S. S., Usatenko, O. V., & Yampol'skii, V. A. (2006). “Memory function versus binary correlator in additive Markov chains". Physica A: Statistical Mechanics and its Applications, 372(2), 279-297. http://arxiv.org/pdf/cond-mat/0610387.pdf
2006
- (Melnyk, 2006) ⇒ Melnyk, S. S., Usatenko, O. V., & Yampol'skii, V. A. (2006). “Memory functions of the additive Markov chains: applications to complex dynamic systems". Physica A: Statistical Mechanics and its Applications, 361(2), 405-415. http://arxiv.org/pdf/physics/0412169.pdf
2003
- (Usatenko et al. ,2003) ⇒ Usatenko, O. V., Yampolskii, V. A., Kechedzhy, K. E., & Melnyk, S. S. (2003). Symbolic stochastic dynamical systems viewed as binary N-step Markov chains. Physical Review E, 68(6), 061107. https://www.researchgate.net/profile/Oleg_Usatenko/publication/8894034_Symbolic_stochastic_dynamical_systems_viewed_as_binary_N_-step_Markov_chains/links/56165ee608ae0f2140070ee3.pdf