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Mark V. Shaney is a third-order Markov chain program, and a Markov text generator. It ingests the sample text (the Tao Te Ching, or the posts of a Usenet group) and creates a massive list of every sequence of three successive words (triplet) which occurs in the text. It then chooses two words at random, and looks for a word which follows those ...
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 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.
A stochastic matrix describes a Markov chain X t over a finite state space S with cardinality α.. If the probability of moving from i to j in one time step is Pr(j|i) = P i,j, the stochastic matrix P is given by using P i,j as the i-th row and j-th column element, e.g.,
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
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.
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]
A game of snakes and ladders or any other game whose moves are determined entirely by dice is a Markov chain, indeed, an absorbing Markov chain. This is in contrast to card games such as blackjack, where the cards represent a 'memory' of the past moves. To see the difference, consider the probability for a certain event in the game.
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