Simulate Markov chain state walks
Simulate Random Walk Through Markov Chain
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.
Simulate a 20-step random walk that starts from a random state.
rng(1); % For reproducibility numSteps = 20; X = simulate(mc,numSteps)
X = 21×1 3 7 1 3 6 1 3 7 2 5 ⋮
X is a 21-by-1 matrix. Rows correspond to steps in the random walk. Because
3, the random walk begins at state 3.
Visualize the random walk.
Specify Starting States for Multiple Simulations
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.
Run three 10-step simulations for each state.
x0 = 3*ones(1,mc.NumStates); numSteps = 10; X = simulate(mc,numSteps,'X0',x0);
X is an 11-by-12 matrix. Rows corresponds to steps in the random walk. Columns 1–3 are the simulations that start at state 1; column 4–6 are the simulations that start at state 2; columns 7–9 are the simulations that start at state 3; and columns 10–12 are the simulations that start at state 4.
For each time, plot the proportions states that are visited over all simulations.
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 in each simulation, specified as a positive integer.
Specify optional pairs of arguments as
the argument name and
Value is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name in quotes.
'X0',[1 0 2] specifies simulating three times, the first simulation starts in state 1 and the final two start in state 3.
X0 — Initial states of simulations
vector of nonnegative integers
Initial states of simulations, specified as the comma-separated pair consisting of
'X0' and a vector of nonnegative integers of
X0 provides counts for the number of simulations to begin in each state. The total number of simulations (
The default is a single simulation beginning from a random initial state.
'X0',[10 10 0 5]
X — Indices of states
numeric matrix of positive integers
Indices of states visited during the simulations, returned as a
(1 + numSteps)-by-
numSims numeric matrix of positive integers. The first row contains the initial states. Columns, in order, are all simulations beginning in the first state, then all simulations beginning in the second state, and so on.
nsimulations from state
X0 = zeros(1,NumStates); X0(k) = n;
To visualize the data created by