# Documentation

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# Markov Chain Models

Discrete state-space processes characterized by transition matrices

For an overview of the Markov chain analysis tools, see Markov Chain Modeling.

## Using Objects

 `dtmc` Create discrete-time Markov chain

## Functions

 `dtmc` Create discrete-time Markov chain `mcmix` Create random Markov chain with specified mixing structure
 `asymptotics` Determine Markov chain asymptotics `isergodic` Check Markov chain for ergodicity `isreducible` Check Markov chain for reducibility `classify` Classify Markov chain states `lazy` Adjust Markov chain state inertia `subchain` Extract Markov subchain
 `redistribute` Compute Markov chain redistributions `simulate` Simulate Markov chain state walks
 `distplot` Plot Markov chain redistributions `eigplot` Plot Markov chain eigenvalues `graphplot` Plot Markov chain directed graph `simplot` Plot Markov chain simulations

## Topics

Discrete-Time Markov Chains

Markov chains are discrete-state Markov processes described by a right-stochastic transition matrix and represented by a directed graph.

Markov Chain Modeling

The `dtmc` class provides basic tools for modeling and analysis of discrete-time Markov chains. The class supports chains with a finite number of states that evolve in discrete time with a time-homogeneous transition structure.

Create and Modify Markov Chain Model Objects

Create a Markov chain model object from a state transition matrix of probabilities or observed counts, or a random Markov chain with a specified structure.

Visualize Markov Chain Structure and Evolution

Visualize the structure and evolution of a Markov chain model using `dtmc` plotting functions.

Determine Asymptotic Behavior of Markov Chain

Compute the stationary distribution of a Markov chain, estimate its mixing time, and determine whether the chain is ergodic and reducible.

Compare Markov Chain Mixing Times

Compare the estimated mixing times of several Markov chains having differing structures.

Identify Classes in Markov Chain

Programmatically and visually identify classes in a Markov chain.

Simulate Random Walks Through Markov Chain

Generate and visualize random walks through a Markov chain.

Compute State Distribution of Markov Chain at Each Time Step

Compute and visualize state redistributions, which is the evolution of the deterministic state distributions over time from an initial distribution.