mcmix

Generate a six-state Markov chain from a random transition matrix.

rng(1); % For reproducibility
mc = mcmix(6);

mc is a dtmc object.

Display the transition matrix.

mc.P

ans = 6×6

    0.2732    0.1116    0.1145    0.1957    0.0407    0.2642
    0.3050    0.2885    0.0475    0.0195    0.1513    0.1882
    0.0078    0.0439    0.0082    0.2439    0.2950    0.4013
    0.2480    0.1481    0.2245    0.0485    0.1369    0.1939
    0.2708    0.2488    0.0580    0.1614    0.0137    0.2474
    0.2791    0.1095    0.0991    0.2611    0.1999    0.0513

Plot a digraph of the Markov chain. Specify coloring the edges according to the probability of transition.

figure;
graphplot(mc,ColorEdges=true);

Generate random transition matrices containing a specified number of zeros in random locations. A zero in location (i, j) indicates that state i does not transition to state j.

Generate two 10-state Markov chains from random transition matrices. Specify the random placement of 10 zeros within one chain and 30 zeros within the other chain.

rng(1); % For reproducibility
numStates = 10;
mc1 = mcmix(numStates,Zeros=10);
mc2 = mcmix(numStates,Zeros=30);

mc1 and mc2 are dtmc objects.

Estimate the mixing times for each Markov chain.

[~,tMix1] = asymptotics(mc1)

tMix1 = 
0.7567

[~,tMix2] = asymptotics(mc2)

tMix2 = 
0.8137

mc1, the Markov chain with higher connectivity, mixes more quickly than mc2.

Generate a Markov chain characterized by a partially random transition matrix. Also, decrease the number of feasible transitions.

Generate a 4-by-4 matrix of missing (NaN) values, which represents the transition matrix.

P = NaN(4);

Specify that state 1 transitions to state 2 with probability 0.5, and that state 2 transitions to state 1 with the same probability.

P(1,2) = 0.5;
P(2,1) = 0.5;

Create a Markov chain characterized by the partially known transition matrix. For the remaining unknown transition probabilities, specify that five transitions are infeasible for 5 random transitions. An infeasible transition is a transition whose probability of occurring is zero.

rng(1); % For reproducibility
mc = mcmix(4,Fix=P,Zeros=5);

mc is a dtmc object. With the exception of the fixed elements (1,2) and (2,1) of the transition matrix, mcmix places five zeros in random locations and generates random probabilities for the remaining nine locations. The probabilities in a particular row sum to 1.

Display the transition matrix and plot a digraph of the Markov chain. In the plot, indicate transition probabilities by specifying edge colors.

P = mc.P

P = 4×4

         0    0.5000    0.1713    0.3287
    0.5000         0    0.1829    0.3171
    0.1632         0    0.8368         0
         0    0.5672    0.1676    0.2652

figure;
graphplot(mc,'ColorEdges',true);