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A Markov chain is a type of Markov process that has either a discrete state space or a discrete index set (often representing time), but the precise definition of a Markov chain varies. [6] For example, it is common to define a Markov chain as a Markov process in either discrete or continuous time with a countable state space (thus regardless ...
A finite-state machine can be used as a representation of a Markov chain. Assuming a sequence of independent and identically distributed input signals (for example, symbols from a binary alphabet chosen by coin tosses), if the machine is in state y at time n , then the probability that it moves to state x at time n + 1 depends only on the ...
In probability theory, a transition-rate matrix (also known as a Q-matrix, [1] intensity matrix, [2] or infinitesimal generator matrix [3]) is an array of numbers describing the instantaneous rate at which a continuous-time Markov chain transitions between states.
The Markov-modulated Poisson process or MMPP where m Poisson processes are switched between by an underlying continuous-time Markov chain. [8] If each of the m Poisson processes has rate λ i and the modulating continuous-time Markov has m × m transition rate matrix R , then the MAP representation is
For example, a series of simple observations, such as a person's location in a room, can be interpreted to determine more complex information, such as in what task or activity the person is performing. Two kinds of Hierarchical Markov Models are the Hierarchical hidden Markov model [2] and the Abstract Hidden Markov Model. [3]
[1] [2] Such models are often described as M/G/1 type Markov chains because they can describe transitions in an M/G/1 queue. [3] [4] The method is a more complicated version of the matrix geometric method and is the classical solution method for M/G/1 chains. [5]
A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a stochastic matrix. An equivalent formulation describes the process as changing state according to ...
If () = is the unit function and =, the interaction between the particle vanishes and the particle model reduces to a sequence of independent copies of the Markov chain . When ϵ = 0 {\displaystyle \epsilon =0} the mean field particle model described above reduces to a simple mutation-selection genetic algorithm with fitness function G and ...