gives a summary of properties for the finite state Markov process mproc.
gives the specified "property" for the process mproc.
- MarkovProcessProperties can be used for finite state Markov processes such as DiscreteMarkovProcess and ContinuousMarkovProcess.
- MarkovProcessProperties[mproc,"Properties"] gives a list of available properties.
- MarkovProcessProperties[mproc,"property","Description"] gives a description of the property as a string.
- Basic properties include:
"InitialProbabilities" initial state probability vector "TransitionMatrix" conditional transition probabilities m "TransitionRateMatrix" conditional transition rates q "TransitionRateVector" state transition rates μ "HoldingTimeMean" mean holding time for a state "HoldingTimeVariance" variance of holding time for a state "SummaryTable" summary of properties
- For a continuous-time Markov process "TransitionMatrix" gives the transition matrix of the embedded discrete-time Markov process.
- The holding time is the time spent in each state before transitioning to a different state. This takes into account self-loops which may cause the process to transition to the same state several times.
- Structural properties include:
"CommunicatingClasses" sets of states accessible from each other "RecurrentClasses" communicating classes that cannot be left "TransientClasses" communicating classes that can be left "AbsorbingClasses" recurrent classes with a single element "PeriodicClasses" communicating classes with finite period greater than 1 "Periods" period for each of the periodic classes "Irreducible" whether the process has a single recurrent class "Aperiodic" whether all classes are aperiodic "Primitive" whether the process is irreducible and aperiodic
- The states of a finite Markov process can be grouped into communicating classes where from each state in a class there is a path to every other state in the class.
- A communicating class can be transient when there is a path from the class to another class or recurrent when there is not. A special type of recurrent class, called absorbing, consist of a single element.
- A state is periodic is if there is a non-zero probability that you return to the state after two or more steps. All the states in a class have the same period.
- Transient properties before the process enters a recurrent class:
"TransientVisitMean" mean number of visits to each transient state "TransientVisitVariance" variance of number of visits to each transient state "TransientTotalVisitMean" mean total number of transient states visited
- A Markov process will eventually enter a recurrent class. The transient properties characterize how many times each transient state is visited or how many different transient states are visited.
- Limiting properties include:
"ReachabilityProbability" probability of ever reaching a state "LimitTransitionMatrix" Cesaro limit of the transition matrix "Reversible" whether the process is reversible
- If a property is not available, this is indicated by Missing["reason"].
Examplesopen allclose all
Basic Examples (2)
Properties & Relations (1)
Wolfram Research (2012), MarkovProcessProperties, Wolfram Language function, https://reference.wolfram.com/language/ref/MarkovProcessProperties.html.
Wolfram Language. 2012. "MarkovProcessProperties." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MarkovProcessProperties.html.
Wolfram Language. (2012). MarkovProcessProperties. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MarkovProcessProperties.html