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This example shows how to compute and visualize state redistributions, which show the evolution of the deterministic state distributions over time from an initial distribution.

Consider this theoretical, right-stochastic transition matrix of a stochastic process.

$$P=\left[\begin{array}{ccccccc}0& 0& 1/2& 1/4& 1/4& 0& 0\\ 0& 0& 1/3& 0& 2/3& 0& 0\\ 0& 0& 0& 0& 0& 1/3& 2/3\\ 0& 0& 0& 0& 0& 1/2& 1/2\\ 0& 0& 0& 0& 0& 3/4& 1/4\\ 1/2& 1/2& 0& 0& 0& 0& 0\\ 1/4& 3/4& 0& 0& 0& 0& 0\end{array}\right].$$

Create the Markov chain that is characterized by the transition matrix *P*.

P = [ 0 0 1/2 1/4 1/4 0 0 ; 0 0 1/3 0 2/3 0 0 ; 0 0 0 0 0 1/3 2/3; 0 0 0 0 0 1/2 1/2; 0 0 0 0 0 3/4 1/4; 1/2 1/2 0 0 0 0 0 ; 1/4 3/4 0 0 0 0 0 ]; mc = dtmc(P);

Plot a directed graph of the Markov chain and identify classes using node colors and markers.

```
figure;
graphplot(mc,'ColorNodes',true);
```

`mc`

represents a single recurrent class with a period of 3.

Suppose that the initial state distribution is uniform. Compute the evolution of the distribution for 20 time steps.

numSteps = 20; X = redistribute(mc,numSteps);

`X`

is a 21-by-7 matrix. Row *t* contains the evolved state distribution at time step *t*.

Visualize the redistributions in a heatmap.

figure; distplot(mc,X);

The periodicity of the chain is apparent.

Remove the periodicity from the Markov chain by transforming it to a lazy chain. Plot the transition matrix of the lazy chain as a heatmap.

lc = lazy(mc); figure; imagesc(lc.P); colormap('jet'); axis square; colorbar; title('Theoretical Lazy Chain Transition Matrix')

`lc`

is a `dtmc`

object. `lazy`

creates the lazy chain by adding weight to the probability of persistence, that is, `lazy`

enforces self-loops.

Compute the evolution of the distribution in the lazy chain for 20 time steps. Plot the redistributions in a heatmap.

X1 = redistribute(lc,numSteps); figure; distplot(lc,X1);

View the evolution of the state distribution as an animated histogram. Specify a frame rate of 1 second.

figure; distplot(lc,X1,'Type','histogram','FrameRate',1)

Compute the stationary distribution of the lazy chain. Compare it to the final redistribution in the animated histogram.

xFix = asymptotics(lc)

`xFix = `*1×7*
0.1300 0.2034 0.1328 0.0325 0.1681 0.1866 0.1468

The stationary distribution and the final redistribution are nearly identical.