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In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability . [ 1 ] [ 2 ] : 10 It is also called a probability matrix , transition matrix , substitution matrix , or Markov matrix .
If the Markov chain is time-homogeneous, then the transition matrix P is the same after each step, so the k-step transition probability can be computed as the k-th power of the transition matrix, P k. If the Markov chain is irreducible and aperiodic, then there is a unique stationary distribution π. [41]
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 Markov arrival process is defined by two matrices, D 0 and D 1 where elements of D 0 represent hidden transitions and elements of D 1 observable transitions. The block matrix Q below is a transition rate matrix for a continuous-time Markov chain. [5]
Change-of-basis matrix, associated with a change of basis for a vector space. Stochastic matrix , a square matrix used to describe the transitions of a Markov chain . State-transition matrix , a matrix whose product with the state vector x {\displaystyle x} at an initial time t 0 {\displaystyle t_{0}} gives x {\displaystyle x} at a later time t ...
The transition matrix is characterized entirely by its upper-left block . Definition. A distribution on { 0 , 1 , 2 , . . . } {\displaystyle \{0,1,2,...\}} is a discrete phase-type distribution if it is the distribution of the first passage time to the absorbing state of a terminating Markov chain with finitely many states.
In conclusion, in the long term about 83.3% of days are sunny. Not all Markov processes have a steady state vector. In particular, the transition matrix must be regular. Otherwise, the state vectors will oscillate over time without converging.
For a continuous time Markov chain (CTMC) with transition rate matrix, if can be found such that for every pair of states and π i q i j = π j q j i {\displaystyle \pi _{i}q_{ij}=\pi _{j}q_{ji}} holds, then by summing over j {\displaystyle j} , the global balance equations are satisfied and π {\displaystyle \pi } is the stationary ...