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Compute Markov chain redistributions


X = redistribute(mc,numSteps)
X = redistribute(mc,numSteps,'X0',x0)



X = redistribute(mc,numSteps) returns data X on the evolution of a uniform distribution of states in the discrete-time Markov chain mc after it advances numSteps time steps.


X = redistribute(mc,numSteps,'X0',x0) optionally specifies the initial state distribution x0.


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Create a four-state Markov chain from a randomly generated transition matrix containing eight infeasible transitions.

rng('default'); % For reproducibility
mc = mcmix(4,'Zeros',8);

mc is a dtmc object.

Plot a digraph of the Markov chain.


State 4 is an absorbing state.

Compute the state redistributions at each step for 10 discrete time steps. Assume an initial uniform distribution over the states.

X = redistribute(mc,10)
X =

    0.2500    0.2500    0.2500    0.2500
    0.0869    0.2577    0.3088    0.3467
    0.1073    0.2990    0.1536    0.4402
    0.0533    0.2133    0.1844    0.5489
    0.0641    0.2010    0.1092    0.6257
    0.0379    0.1473    0.1162    0.6985
    0.0404    0.1316    0.0765    0.7515
    0.0266    0.0997    0.0746    0.7991
    0.0259    0.0864    0.0526    0.8351
    0.0183    0.0670    0.0484    0.8663
    0.0168    0.0569    0.0358    0.8905

X is an 11-by-4 matrix. Rows correspond to time steps and columns correspond to states.

Visualize the state redistribution.


After ten transitions, the distribution appears to settle with a majoritty of the probability mass in state 4.

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

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. Indicate the probability of transition using edge colors.


Compute a 20-step redistribution of the Markov chain using random initial values.

rng(1); % For reproducibility
x0 = rand(mc.NumStates,1);
rd = redistribute(mc,20,'X0',x0);

Plot the redistribution.


The redistribution suggests that the chain is periodic with a period of three.

Remedy periodicity by creating a lazy version of the Markov chain.

lc = lazy(mc);

Compute a 20-step redistribution of the lazy chain using random initial values. Plot the redistribution.

x0 = rand(mc.NumStates,1);
lrd1 = redistribute(lc,20,'X0',x0);


The redistribution appears to settle after several steps.

Input Arguments

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Discrete-time Markov chain with NumStates states and transition matrix P, specified as a dtmc object.

Number of discrete time steps to compute, specified as a positive integer.

Data Types: double

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'X0',[0.5 0.25 0.25] specifies an initial state distribution of [0.5 0.25 0.25].

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Initial distribution, specified as the comma-separated pair consisting of 'X0' and a nonnegative numeric vector of length NumStates. redistribute normalizes X0 so that it sums to 1.

The default is a uniform distribution of states.

Example: 'X0',[0.5 0.25 0.25]

Data Types: double

Output Arguments

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Evolution of state probabilities, returned as a (1 + numSteps)-by-NumStates nonnegative numeric matrix. The first row is X0. Subsequent rows are the redistributions at each step, which redistribute determines by transition matrix mc.P.


If mc is ergodic, and numSteps is sufficiently large, X(end,:) will approximate x = asymptotics(mc). See asymptotics.


To visualize the data created by redistribute, use distplot.

Introduced in R2017b

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