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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 ...
Notice that the general state space continuous-time Markov chain is general to such a degree that it has no designated term. While the time parameter is usually discrete, the state space of a Markov chain does not have any generally agreed-on restrictions: the term may refer to a process on an arbitrary state space. [15]
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 ...
A basic property about an absorbing Markov chain is the expected number of visits to a transient state j starting from a transient state i (before being absorbed). This can be established to be given by the (i, j) entry of so-called fundamental matrix N, obtained by summing Q k for all k (from 0 to ∞).
The original chain is scaled by the fastest transition rate γ, so that transitions occur at the same rate in every state, hence the name. The method is simple to program and efficiently calculates an approximation to the transient distribution at a single point in time (near zero). [1] The method was first introduced by Winfried Grassmann in 1977.
The distribution can be represented by a random variable describing the time until absorption of an absorbing Markov chain with one absorbing state. Each of the states of the Markov chain represents one of the phases. It has continuous time equivalent in the phase-type distribution.
Figure 1. Probabilistic parameters of a hidden Markov model (example) X — states y — possible observations a — state transition probabilities b — output probabilities. In its discrete form, a hidden Markov process can be visualized as a generalization of the urn problem with replacement (where each item from the urn is returned to the original urn before the next step). [7]
For a continuous time Markov chain (CTMC) with transition rate matrix, if can be found such that for every pair of states and = holds, then by summing over , the global balance equations are satisfied and is the stationary distribution of the process. [5]