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The Markov chain forecasting models utilize a variety of settings, from discretizing the time series, [107] to hidden Markov models combined with wavelets, [106] and the Markov chain mixture distribution model (MCM). [110]
A Markov chain with two states, A and E. In probability, a discrete-time Markov chain (DTMC) is a sequence of random variables, known as a stochastic process, in which the value of the next variable depends only on the value of the current variable, and not any variables in the past.
A Markov chain is a stochastic process defined by a set of states and, for each state, a probability distribution on the states. Starting from an initial state, it follows a sequence of states where each state in the sequence is chosen randomly from the distribution associated with the previous state.
Consider this figure depicting a section of a Markov chain with states i, j, k and l and the corresponding transition probabilities. Here Kolmogorov's criterion implies that the product of probabilities when traversing through any closed loop must be equal, so the product around the loop i to j to l to k returning to i must be equal to the loop the other way round,
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.
In this context, the Markov property indicates that the distribution for this variable depends only on the distribution of a previous state. An example use of a Markov chain is Markov chain Monte Carlo , which uses the Markov property to prove that a particular method for performing a random walk will sample from the joint distribution .
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
Continuous-time Markov chain (CTMC), an example of a jump process and a generalization of the Poisson process; Counting process, an example of a jump process and a generalization of the Poisson process in a different direction than that of CTMCs; Interacting particle system, an example of a jump process