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Classify Markov chain states

`bins = classify(mc)`

`[bins,ClassStates,ClassRecurrence,ClassPeriod] = classify(mc) `

partitions states of the discrete-time Markov chain `bins`

= classify(`mc`

)`mc`

into
disjoint communicating
classes and returns the class labels `bins`

identifying
the communicating class to which each state belongs.

`[`

additionally
returns the states in each class `bins`

,`ClassStates`

,`ClassRecurrence`

,`ClassPeriod`

] = classify(`mc`

) `ClassStates`

, whether the classes
are recurrent `ClassRecurrence`

, and class periods
`ClassPeriod`

.

`classify`

determines recurrence and transience from the outdegree of the*supernode*associated with each communicating class in the condensed digraph [1]. An outdegree of 0 corresponds to recurrence; an outdegree that is greater than 0 corresponds to transience. See`graphplot`

.`classify`

determines periodicity using a breadth-first search of cycles in the associated digraph, as in [3]. Class period is the greatest common divisor of the lengths of all cycles originating at any state in the class.

[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.

[3]
Jarvis, J. P. and D.
R. Shier. "Graph-Theoretic Analysis of Finite Markov Chains." In *Applied Mathematical
Modeling: A Multidisciplinary Approach.* Boca Raton: CRC Press, 2000.

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