Compute Markov chain redistributions
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
Plot a digraph of the Markov chain.
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 = 11×4 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 ⋮
X is an 11-by-4 matrix. Rows correspond to time steps, and columns correspond to states.
Visualize the state redistribution.
After 10 transitions, the distribution appears to settle with a majority of the probability mass in state
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 by 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.
Remove 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); figure; distplot(lc,lrd1);
The redistribution appears to settle after several steps.
mc— Discrete-time Markov chain
Discrete-time Markov chain with
NumStates states and transition matrix
P, specified as a
P must be fully specified (no
numSteps— Number of discrete time steps
Number of discrete time steps to compute, specified as a positive integer.
comma-separated pairs of
the argument name and
Value is the corresponding value.
Name must appear inside quotes. You can specify several name and value
pair arguments in any order as
'X0',[0.5 0.25 0.25]specifies an initial state distribution of
[0.5 0.25 0.25].
'X0'— Initial distribution
Initial distribution, specified as the comma-separated pair consisting of
'X0' and a nonnegative numeric vector of
X0 so that it sums to
The default is a uniform distribution of states.
'X0',[0.5 0.25 0.25]
X— Evolution of state probabilities
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 the transition matrix
mc is ergodic, and
numSteps is sufficiently large,
x = asymptotics(mc). See
To visualize the data created by