Extract Markov subchain

State

is`j`

*reachable*from stateif there is a nonzero probability of moving from`i`

to`i`

in a finite number of steps.`j`

`subchain`

determines reachability by forming the transitive closure of the associated digraph, then enumerating one-step transitions.Subchains are closed under reachability to ensure that the transition matrix of

`sc`

remains stochastic (that is, rows sum to`1`

), with transition probabilities identical to the transition probabilities in`mc.P`

.If you specify a state in a recurrent communicating class, then

`subchain`

extracts the entire communicating class. If you specify a state in a transient communicating class, then`subchain`

extracts the transient class and all classes reachable from the transient class. To extract a unichain, specify a state in each component transient class. See`classify`

.

[1]
Gallager, R.G. *Stochastic Processes: Theory for Applications.* Cambridge, UK: Cambridge University Press, 2013.

[2]
Horn, R., and C. R.
Johnson. *Matrix Analysis.* Cambridge, UK: Cambridge University Press,
1985.